Mathematical Physics, Analysis and Geometry2: 1–24, 1999. © 1999Kluwer Academic Publishers. Printed in the Netherlands.
1
Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation
LI-YENG SUNG Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
(Received: 20 February 1998; in final form: 27 November 1998)
Abstract. We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev spaceHk(R2) for k > 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.
Mathematics Subject Classification (1991):35Q53.
1. Introduction
(qt − 6qqx + qxxx)x = 3qyy, (1.1a)
q(x, y,0) = q0(x, y), (1.1b)
for the Kadomtsev–Petviashvili-I (KPI) equation (cf. [8]) is solved in [6] by the inverse spectral method, under the assumption thatq0 is a small Schwartz function. It is shown in [6] that the solutionq(x, y, t) obtained by the inverse spectral method is aC∞ classical solution of (1.1) for(x, y) ∈ R2, t 6= 0, andt → q(·, t) ∈ C1((−∞,0) ∪ (0,∞), C0(R2)).
We will show in this paper thatt → q(·, t) ∈ C(R,H∞(R2)) andq(x, y, t) is the unique generalized solution for the forward (respectively backward) prob- lem of (1.1) in the classC([0,∞),H 3(R2)) ∩ C1((0,∞), L∞(R2)) (respectively C((−∞,0],H 3(R2)) ∩ C1((−∞,0), L∞(R2))).
The inverse spectral method for KPI is studied formally in [10] and [5]. Rigor- ous aspects of this method have been investigated in [14] and [20]. The version of the inverse spectral method used in this paper is essentially that of [20]. However, the results in [14] and [20] are obtained under the additional assumption that∫
R dx q0(x, y) = 0, (1.2)
2 LI-YENG SUNG
which is also assumed in many of the papers that study (1.1) by PDE techniques (cf. [4, 12, 13, 16, 17]). As a consequence of the nonphysical constraint (1.2), the scattering data have decay in all directions, which greatly simplifies the analysis. When (1.2) is not assumed, the analysis is much more subtle due to the lack of decay of the scattering data in certain directions.
The implications of the constraint (1.2) are also studied in [1, 2] and [3] using the inverse spectral method. But the fact that the solution obtained by the inverse spectral method (without assuming (1.2)) is the unique solution of (1.1) in a general class of solutions has not been rigorously established until now.
Since there is an isomorphism (cf. [9]) between solutions of the KP equation and the Johnson equation (cf. [7]) in the case of rapidly decaying initial data, the results of this paper can also be applied to the Johnson equation.
The rest of the paper is organized as follows. The inverse spectral method for (1.1) is described in Section 2, where we give both the solution from the left and the solution from the right. We also recall some relevant results from [6]. Section 3 contains the proof that the inverse spectral solution from the left and from the right are identical. The integrability of the time-dependent Jost function is studied in Section 4. We prove in Section 5 thatt → q(·, t) ∈ C(R,H∞(R2)), and establish the uniqueness of solutions in Section 6.
For the convenience of the reader we collect here some notation frequently used in this paper.
(a) S(Rn) is the space of Schwartz functions inn real variables andS′(Rn) is the space of tempered distributions equipped with the weak∗-topology.
(b) Hk(R2), k = 0,1,2, . . ., are theL2 based Sobolev spaces in two real variables, andH∞(R2) = ∞
k=0H k(R2) is equipped with the natural Fréchet space
topology.Wm p (R) is theLp based Sobolev space in one real variable.
(c) C(X, Y ) is the space of continuous maps from the topological spaceX into the topological spaceY andCb(X, Y ) is the space of bounded continuous maps from the Banach spaceX into the Banach spaceY . Cb(Rn) is the space of bounded continuous functions onRn equipped with the sup-norm andC0(Rn) is the subspace ofCb(Rn) whose members vanish at infinity.
(d) Let I be an open interval andY be a topological vector space.C1(I, Y ) is the space of continuously differentiableY -valued functions onI .
(e) LetX be a measure space andY be a Banach space.L1(X, Y ) is the space of Y -valued Bochner integrable functions onX.
(f) The Fourier transformsf andf are defined by
f (ξ, η) = ∫ R2
dx dy e−i(xξ+yη)f (x, y) and f (ξ, y) = ∫ R
dx e−ixξf (x, y).
(g) The operatorsP± are defined by
(P±f )(k) = ±1
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 3
They always act on functions in thek variable in this paper. (h) A . B meansA 6 constant× B, where the constant is independent of the
variables inA andB.
2. The Inverse Spectral Method
We describe in this section the solution of(1.1) obtained by the inverse spectral method. We consider the following time-dependent Schrödinger equation defined by the initial datumq0:
iφy − φxx + q0φ = 0, (2.1)
and introduce the Jost functionµ(x, y, k) by the relation
φ(x, y, k) = µ(x, y, k)e−i(xk−yk2). (2.2)
The equation forµ is
iµy − µxx + 2ikµx + q0µ = 0. (2.3)
Taking the Fourier transform in thex variable, we can write(2.3) as
µy(ξ, y, k) = i(ξ2− 2kξ)µ(ξ, y, k)+ i
2π (q0 ∗ µ)(ξ, y, k), (2.4)
where∗ denotes the convolution in theξ variable. The Jost functionµ+ (resp.µ−) is defined for Imk > 0 (resp. Imk 6 0) by
µ+(ξ, y, k) = 2πδ(ξ)+ i
2π
× ei(ξ 2−2kξ)(y−y ′)(q0 ∗ µ+)(ξ, y′, k), (2.5)
µ−(ξ, y, k) = 2πδ(ξ)+ i
2π
× ei(ξ 2−2kξ)(y−y ′)(q0 ∗ µ−)(ξ, y′, k), (2.6)
whereδ is the Dirac function, andE± are defined by
E±(ξ) = {
(2.7)
The left scattering dataL±(k, l) (k, l ∈ R) are defined by
L−(k, l) =
i
2π
∫ R2
dx dy e−i[x(k−l)+y(l 2−k2)]q0(x, y)µ
−(x, y, k), k 6 l,
0, otherwise, (2.8)
4 LI-YENG SUNG
L+(k, l) =
i
2π
∫ R2
dx dy e−i[x(k−l)+y(l 2−k2)]q0(x, y)µ
+(x, y, k), k > l,
0, otherwise. (2.9)
UsingL± we define the time-dependent Jost function from the leftλ(x, y, t, k), k ∈ R, by the following equation:
λ(x, y, t, k) = 1+ [ P− ∫ k
−∞ dlL+(k, l)+ P+
dlL−(k, l) ] ×
× ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]λ(x, y, t, l). (2.10)
We can now write down the solution of(1.1) from the left:
q(x, y, t) = 1
2−k2)] ×
× e4it (k3−l3)[i(k − l)λ(x, y, t, l) + λx(x, y, t, l)]. (2.11)
In summary, given a smallq0 ∈ S(R2), the inverse spectral solution from the left for (1.1) is obtained through the following steps: (I) Solve(2.5) and (2.6) for µ±(x, y, k). (II) Define L±(k, l) by (2.8) and (2.9) usingµ±(x, y, k). (III) Solve(2.10) for λ(x, y, t, k). (IV) The solutionq(x, y, t) is defined by(2.11) using L±(k, l) andλ(x, y, t, k).
Note that we can also define the solutionq(x, y, t) via theright scattering data:
R−(k, l) = −i 2π
∫ R2
dx dy e−i[x(k−l)+y(l 2−k2)]q0(x, y)µ
−(x, y, k), k > l,
0, otherwise, (2.12)
∫ R2
dx dy e−i[x(k−l)+y(l 2−k2)]q0(x, y)µ
+(x, y, k), k 6 l,
0, otherwise. (2.13)
Let the time-dependent Jost function from the rightρ(x, y, t, k), k ∈ R, be defined by the following equation:
ρ(x, y, t, k) = 1+ [ P− ∫ ∞ k
dlR+(k, l)+ P+ ∫ k
−∞ dlR−(k, l)
] ×
× ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]ρ(x, y, t, l). (2.14)
The solution of (1.1) from the right can then be expressed as
q(x, y, t) = 1
2−k2)] ×
× e4it (k3−l3)[i(k − l)ρ(x, y, t, l) + ρx(x, y, t, l)]. (2.15)
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 5
The relations amongL±(k, l),R±(k, l),λ(x, y, k) andρ(x, y, k), and the equiv- alence of(2.11) and(2.15) will be established in Section 3.
Finally we recall some relevant results from [6] which are obtained under the smallness assumption∫
R2 dy dξ(1+ ξ2)|q0(ξ, y)| 1. (2.16)
(i) The integral equation (2.5) (resp. (2.6)) is uniquely solvable in the Banach spaceCb(Ry, L1(Rξ )⊕ Cδ(ξ)) for Im k > 0 (resp. Imk 6 0).
[ ∂
. (1+ |k|)α+β for α, β,m, n > 0. (2.17)
(iii) In particular, we have
|F(k, l)| 6 Cq0(1+ |k − l|)−2, (2.18)
whereCq0 1 under the assumption (2.16), and the integral equations (2.10) and (2.14) are uniquely solvable inL2(Rk)⊕ C for (x, y, t) ∈ R3.
(iv) The Jost functionsµ± are related by
µ±(x, y, k) = µl(x, y, k)
∫ k
2−k2)]L±(k, l)µl(x, y, l), (2.19)
µ±(x, y, k) = µr(x, y, k)
∫ ±∞ k
dl ei[x(k−l)+y(l 2−k2)]R±(k, l)µr(x, y, l), (2.20)
where theleft andright Jost functionsµl andµr are the unique solutions of
µl(ξ, y, k) = 2πδ(ξ)+ + i
2π
∫ y
µr (ξ, y, k) = 2πδ(ξ)− − i
2π
∫ ∞ y
dy′ ei(ξ 2−2kξ)(y−y ′)(q0 ∗ µr )(ξ, y′, k). (2.22)
(v) Let # bel, r, +, or−. Forαj > 0, we have the following estimates:
6 LI-YENG SUNG
∂α1+α2+α3
∂xα1∂yα2∂kα3 (µ#(x, y, k) − 1) . (1+ |y|α3) for (x, y, k) ∈ R3. (2.23)
(vi) Let q(x, y, t) be defined by either (2.11) or (2.15). Then we have
q(x, y,0) = q0(x, y). (2.24)
(vii) For t 6= 0, Equations (2.11) and (2.15) can be written as
q(x, y, t) = ∂
] ×
× ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]λ(x, y, t, l)
) , (2.25)
dlR+(k, l) ] ×
× ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]ρ(x, y, t, l)
) , (2.26)
where the integrals exist as iterated integrals. (viii) The functionq(x, y, t) defined by (2.11) (resp. (2.15)) isC∞ for t 6= 0, and
∂m+nq ∂xm∂yn
3. The Equivalence of the Left and Right Solutions
We first investigate the relations between the scattering dataL±(k, l) andR±(k, l). Let the integral operatorsL± andR± be defined by
[L±f ](k) = ± ∫ k
[R±f ](k) = ∫ k
±∞ dlR±(k, l)f (l). (3.1)
LEMMA 3.1. The operatorsI − L± and I − R± are bounded and invertible on Lp(R) (16 p <∞) andCb(R).
Proof. Let F(k, l) = L±(k, l) or R±(k, l). From (2.17) we have|F(k, l)| . [1+ (k − l)2]−1 which implies the boundedness ofL± andR±.
We will only show thatL+ is invertible. Letf ∈ Lp(R) (1 6 p < ∞) or Cb(R). From (2.17) we derive∫
R dl|L+(k, l)f (l)| .
∫ R
. [ ∫
+ ∫ lk<0
(1+ |k + l||k|)2 ] . (3.2)
Using (3.2) we obtain for anyK > 0 that, for 16 p 6∞,
L+f Lp(k>K) . ∫ ∞
∫ 0
−∞ dl
Lp(k>K)
∫ 0
−∞ dl
Lp(Rk)
. 1
K f Lp(R). (3.4)
It follows from (3.3), (3.4), the smoothness ofL+(k, l) and the Fréchet–Kolmo- gorov theorem (respectively the Ascoli–Arzelà theorem) (cf. [19]) thatL+ is a compact operator onLp(R) for 16 p <∞ (respectivelyCb(R)).
We note thatI − L+ has the same kernel onLp(R) andCb(R). Let f ∈ Cb(R) belong to Ker(I − L+). We derive from (2.17) that for anyK > 0
L+f L∞(k<−K) . ∫ k
−∞ dl
. ∫ k
−∞ dl
1
K f L∞(k<−K). (3.5)
Sincef (k) = (L+f )(k), it follows from (3.5) thatf (k) = 0 for k sufficiently negative, and then Gronwall’s inequality (cf. [18]) impliesf (k) = 0 for all k ∈ R. The invertibility ofL+ follows from the Fredholm alternative (cf. [19]). 2
Let (L±)t and(R±)t be the transposes ofL± andR± respectively, i.e.,
[(L±)tf ](l) = ∫ l
[(R±)tf ](l) = ± ∫ l
We have the following analog of Lemma 3.1.
LEMMA 3.2. The operatorsI − (L±)t and I − (R±)t are bounded and invertible onLp(R) (16 p <∞) andCb(R).
8 LI-YENG SUNG
We define
φ#(x, y, k) = e−i(xk−yk 2)µ#(x, y, k), where #= +,−, l, or r. (3.7)
Then (2.19) and (2.20) can be rewritten as
φ± = (I − L±)φl and φ± = (I − R±)φr . (3.8)
From(3.8) and Lemma 3.1 we derive the following equations inCb(R):
φr = (I − R+)−1(I − L+)φl and φr = (I − R−)−1(I − L−)φl. (3.9)
For k ∈ R, let the functionsηl(x, y, k) be defined by the following analog of (2.21):
ηl(ξ, y, k) = 2πδ(ξ)+ i
2π
∫ y
2−2kξ)(y−y ′)(q0 ∗ ηl ) (ξ, y′, k), (3.10)
which is uniquely solvable inCb(Ry, L1(Rξ )⊕Cδ(ξ)) under the assumption (2.16). Of courseηl enjoys the same properties asµl.
Let γ l(x, y, k) be defined by
γ l(x, y, k) = e−i(xk−yk 2)ηl(x, y, k), (3.11)
then we have (cf. (2.1)–(2.4))
φly = −iφlxx + iq0φ l and γ ly = −iγ lxx + iq0γ
l. (3.12)
LEMMA 3.3. Let ζ(ξ, y, k) (resp. τ(ξ, y, k)) be the Fourier transform of (µl(x, y, k)−1) (resp. the complex conjugate of(η(x, y, k) − 1)) in thex variable. Then we have
lim y→−∞
y→−∞
∫ R
dξ sup s∈R |τ(ξ, y, k)| = 0. (3.13)
Proof. It suffices to discuss the case ofζ(ξ, y, k). From (2.21) we obtain
ζ(ξ, y, k) = i
+ i
2π
∫ y
2−2kξ)(y−y ′)(q0 ∗ ζ )(ξ, y′, k). (3.14)
It follows immediately from (3.14) thatζ(ξ, y, k) is continuous in all the vari- ables and hence supy,k∈R |ζ(ξ, y, k)| is a measurable function inξ . Moreover, we find from the Neumann series solution of (3.14) that
sup y,k∈R |ζ(ξ, y, k)| 6
∞∑ j=0
ζjL1(R) 6 (
. (3.16)
Combining (2.16), (3.15) and (3.16) we have∫ R
dξ sup y,k∈R |ζ(ξ, y, k)| <∞. (3.17)
The first limit in (3.13) follows from (3.14) and (3.17). 2 LEMMA 3.4. Letf (k) ∈ S(R), then
1
2π
∫ R
dsγ l(x, y, s)f (s) = f (k) ∀y ∈ R. (3.18)
Proof.It follows from (2.23) and (3.12) that the iterated integral on the left-hand side of(3.18) is independent ofy. Using the notation in Lemma 3.3 we can also rewrite the left-hand side of (3.18) by the Fourier inversion formula as
1
2π
∫ R
= 1
2π
∫ R
∫ R
+ 1
2π
∫ R
∫ R
+ 1
2π
∫ R
∫ R
ds ei(xs−ys 2)(ηl(x, y, s) − 1)f (s)+ f (k)
= eiyk 2
dξζ(k − s − ξ, y, k)τ(ξ, y, s)e−iys2 f (s)+
+ eiyk 2
2π
∫ R
ds[ζ(k − s, y, k) + τ(k − s, y, s)]e−iys2 f (s)+ f (k). (3.19)
The lemma follows from (3.13) and (3.19). 2 LEMMA 3.5. The following identity holds as operators onL2(R):
(I − R−)−1(I − L−) = (I − R+)−1(I − L+). (3.20)
Proof.From(3.9) we have(I −R−)−1(I −L−)φl = (I −R+)−1(I −L+)φl. Let g ∈ L1(R). Then we have∫
R dk φl(x, y, k)G−(k) =
∫ R
where
G± = [I − (L±)t ][I − (R±)t ]−1g. (3.22)
SinceG± ∈ L1(R) by Lemma 3.2 andφl(x, y, k) is bounded and continuous in all the variables, we find by (3.18), (3.21) and Fubini’s theorem that∫
R dkf (k)G−(k) =
dk[(I − R−)−1(I − L−)f ](k)g(k)
= ∫ R
dk[(I − R+)−1(I − L+)f ](k)g(k). (3.24)
Sinceg ∈ L1(R) is arbitrary andS(R) is dense inL2(R), the identity (3.20) follows from (3.24). 2
We now show that (2.11) and (2.15) define the same functionq(x, y, t). We can rewrite (2.10) and (2.14) as[
P−E(x,y,t)(I − L+)E−1 (x,y,t)+ P+E(x,y,t)(I − L−)E−1
(x,y,t)
] λ = 1, (3.25)[
P−E(x,y,t)(I − R+)E−1 (x,y,t)+ P+E(x,y,t)(I − R−)E−1
(x,y,t)
] ρ = 1, (3.26)
where[E(x,y,t)f ](k) = ei(xk−yk2+4tk3)f (k). Combining (3.20), (3.25) and (3.26) we obtain the following lemma.
LEMMA 3.6. The functionsλ(x, y, t, k) andρ(x, y, t, k) are related by
ρ = E(x,y,t)(I − R+)−1(I − L+)E−1 (x,y,t)λ
= E(x,y,t)(I − R−)−1(I − L−)E−1 (x,y,t)λ. (3.27)
PROPOSITION 3.7. The formulas(2.11) and (2.15) define the same function q(x, y, t).
Proof. The case wheret = 0 follows from (2.24). Fort 6= 0, in view of (2.25) and (2.26), we can rewrite (2.11) and (2.15) as
q(x, y, t)
(x,y,t)
(x,y,t)
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 11
4. The Integrability of λ(x, y, t, k) and ρ(x, y, t, k) in the (x, k) Variables
We study the integrability of functions related to the time-dependent Jost functions in this section. The results of this section are crucial for proving in the next section that t → q(·, t) ∈ C(R,H∞(R2)). We will concentrate onλ(x, y, t, k).
Let L (y,t) be the operator defined by
(L (y,t)f )(x, k) = [ P− ∫ k
−∞ dlL+(k, l)+
2−k2)+4t (k3−l3)]f (x, l). (4.1)
LEMMA 4.1. For each(y, t), L (y,t) is a contraction onL2(A× Rk) for anyA ⊆ Rx.
Proof.We have, by (2.18), ∫ k
±∞ dlL(k, l)ei[x(k−l)+y(l
2−k2)+4t (k3−l3)]f (x, l) L2(A×Rk)
6 Cq0
L2(Rk)
. (4.2)
The lemma follows from (4.2) and the smallness ofCq0. 2 Next we have a technical lemma for functions in the Sobolev spacesWm
p (R).
LEMMA 4.2. Let1< p <∞ andf (k) ∈ Wm p (R) for m > 0. ThenP±(eiskf (k))
Wm p (R) . (1+ s)−jf
W m+j+1 p (R) for s > 0 andj,m > 0. (4.3)
Proof.We will only prove the estimate forg = P+(e−iskf (k)). Also, it suffices to prove(4.3) for j = 0 since
sP+(e−iskf (k)) = −iP+(e−iskf ′(k))+ i[P+(e−iskf (k))]′. Form = 0 or 06 s < 1, this follows immediately from the properties of the
Hilbert transform (cf. [15]). So we assumem, s > 1. Sinces > 0, a simple contour integration gives
p.v. 1
2 . (4.4)
On the other hand we obtain from the Plemelj formula (cf. [11])
g(k) = p.v. 1
l − k + 1
12 LI-YENG SUNG
g(k) = 1
2πi (R)
l − k , (4.6)
where(R) ∫
dl denotes the improper Riemann integral. A simple calculation using integration by parts shows that
∂m
for m > 0, |k| < L. (4.7)
By the partitionR = {l : |l| 6 L} ∪ {l : |l| > L} and (4.7) we see that
g(m)(k) = 1
2πi (R)
(l − k)m+1 −
m!f (j)(k) j !(l − k)m+1−j
] = g1(k)+ g2(k)+ g3(k). (4.8)
We have triviallyg2Lp . f Lp sincem > 1. The integrals(R) ∫ |l−k|>1 dl
e−isl (l−k)m+1−j , 06 j 6 m, are uniformly bounded fors > 1 andk ∈ R, which implies thatg3Lp . f Wm
p .
We find by Taylor’s formula that∫ |l−k|<1
dl e−isl ∂m
Hence we also haveg1Lp . f Wm+1 p
. 2 In the following two lemmas the functionF(k, l) is piecewiseC∞ on the half
planes{(k, l) : k > l} and{(k, l) : k 6 l}, and it satisfies the estimates in (2.17).
LEMMA 4.3. LetH(x, y, t, k) be eitherP− ∫ k −∞ dlF (k, l)ei[x(k−l)+y(l2−k2)+4t (k3−l3)]
or P+ ∫∞ k
dlF (k, l)ei[x(k−l)+y(l2−k2)+4t (k3−l3)], andH (ξ, y, t, k) be the Fourier trans- form ofH(x, y, t, k) in thex variable. Then, for1< p <∞ andm > 0,
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 13∫ R
dξH (ξ, y, t, k)Wm p (Rk) .
( 1+ |t|m+2
1+ |y|1−(1/p) )
Moreover, ifB ⊂ R2 is bounded andε > 0, then
{H (ξ, y, t, k) : ε < |ξ | < ε−1, (y, t) ∈ B} is a precompact subset ofWm
p (Rk). (4.11)
Proof.We will treat only the case whereH is defined by the first formula. By a change of variables we can write
H(x, y, t, k) = P− ∫ ∞
0 dl eixle−ik(2yl)G(l, y, t, k), (4.12)
where
[ F(k, k − l)e12it (lk2−l2k)]. (4.13)
The estimates in (2.17) and the piecewise smoothness ofF imply that, for each (y, t), the mapl→ e−ik(2yl)G(l, y, t, k) belongs toC((0,∞), L2(Rk))∩L1((0,∞), L2(Rk)). Therefore the bounded operatorP− can be moved inside the integral in (4.12) and we obtain from the Fourier inversion formula that
H (ξ, y, t, k) = 2πE+(ξ)P− [ e−ik(2yξ)G(ξ, y, t, k)
] , (4.14)
whereE+(·) is defined in (2.7). From (2.17) we also obtain the following estimate: ∂`∂k`G(l, y, t, k)
. (1+ |t|` 1+ |kl|
for ` > 0. (4.15)
Combining (4.3) (withj = 0,1) and (4.15) we find, forξ > 0 andy 6 0,P− [ e−ik(2yξ)G(ξ, y, t, k)
] Wm p
1+ |yξ | , 1
1+ |ξ | ) . (4.16)
The estimate (4.10) follows immediately from (4.14) and (4.16). The estimate (4.15) and the Lebesgue dominated convergence theorem imply
that the map(ξ, y, t)→ e−ik(2yξ)G(ξ, y, t, k) is continuous from the set{(ξ, y, t) : ξ 6= 0} intoWm
p (R). The precompactness statement follows immediately from the boundedness ofP− onWm
p (R). 2 LEMMA 4.4. LetV (x, y, t, k) be either
P− ∫ k
−∞ dl(k − l)F (k, l)ei[x(k−l)+y(l2−k2)+4t (k3−l3)]
14 LI-YENG SUNG
P+ ∫ ∞ k
dl(k − l)F (k, l)ei[x(k−l)+y(l2−k2)+4t (k3−l3)].
Then the following estimate holds:
V (x, y, t, k)L2(R2 x,k) . (
1+ |t|3 1+ |y|
) for t ∈ R, y 6 0. (4.17)
Moreover, ifB is a bounded subset ofR2, we have
lim r→∞V (x, y, t, k)L2(r) = 0 uniformly for(y, t) ∈ B, (4.18)
where
r = {(x, k) ∈ R2 : |x| > r}. (4.19)
Proof. It suffices to discuss the case whereV is defined by the first formula. We have
V (x, y, t, k) = ∫ ∞
0 dl eixlP−
] , (4.20)
whereG is defined in (4.13). Applying (4.3) (with j = 0,2) we obtain the following analog of(4.16) for
l > 0 andy 6 0:P− [ e−ik(2yl)lG(l, y, t, k)
] L2(Rk)
1+ l2 ) . (4.21)
The estimate (4.17) follows from (4.20), (4.21) and the Plancherel theorem. From (4.15) (with` = 0) we see that the map(y, t) → e−ik(2yl)lG(l, y, t, k)
is continuous fromR2 y,t to L2(R2
l.k). It follows from (4.20), the boundedness of P− on L2(R) and the Plancherel theorem that{V−(x, y, t, k) : (y, t) ∈ B} is also a precompact subset ofL2(R2
x,k), which then implies (4.18) by the Fréchet– Kolmogorov theorem. 2 LEMMA 4.5. LetH(x, y, t, k) be as in Lemma4.3. Then
L (y,t)HL2(R2 x,k) . (
1+ t4 1+ |y|
) for t ∈ R, y 6 0. (4.22)
Moreover, ifB is a bounded subset ofR2 andr is defined by(4.19), we have
lim r→∞(L (y,t)H )(x, k)L2(r) = 0 uniformly for(y, t) ∈ B. (4.23)
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 15
Proof.We have
−∞ dlL+(k, l)+ P+
dlL−(k, l) ] ×
× ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]H(x, y, t, l)
= g−(x, y, t, k) + g+(x, y, t, k), (4.24)
where
= 1
2π
∫ R
dξ eixξ ∫ R
dl eixlE±(l)P[K±(l, y, t, k)H (ξ, y, t, k − l)], (4.25)
and
K±(l, y, t, k) = ei(yl 2+4t l3)[e−ik(2yl)L±(k, k − l)e12it (lk2−l2k)]. (4.26)
We note that (2.17) and (4.10) together imply(ξ, l)→ K±(l, y, t, k)H (ξ, y, t, k−l) ∈ L1(R2
ξ,l, L 2(Rk)) for fixed(y, t). Therefore by integrating against arbitrary
L2(Rk) functions we obtain (4.25) as an identity onC(Rx, L2(Rk)) for each fixed (y, t).
From (2.17) and (4.3) we obtain the following estimate fory 6 0:P[K±(l, y, t, k)H (ξ, y, t, k − l)] L2(Rk)
. [ (1+ t2)H (ξ, y, t, ·)L2(R) + (1+ |t|)H (ξ, y, t, ·)H1(R)+ + H (ξ, y, t, ·)H2(R)
] min
( 1
1+ |yl| , 1
1+ |l| ) . (4.27)
Combining (4.25), (4.27), (4.10) (withp = 2) and the Plancherel theorem we find
gL2(Rx,k) . ∫ R
dξP[K±(l, y, t, k)H (ξ, y, t, k − l)]L2(R2 k,l )
. 1+ t4 1+ |y| for y 6 0. (4.28)
The estimate (4.22) follows from (4.24) and (4.28). Let ε be an arbitrary positive number. From (4.25) we have
g(x, y, t, k)
= 1
2π
dξ
] eixξ×
× ∫ R
dl eixlE±(l)P[K±(l, y, t, k)H (ξ, y, t, k − l)]+
+ 1
2π
dξ eixξ ∫ R
dl eixlE±(l)P[K±(l, y, t, k)H (ξ, y, t, k − l)] = I1(x, y, t, k) + I2(x, y, t, k). (4.29)
16 LI-YENG SUNG
It follows from (4.14), (4.15) and (4.27) that for boundedt we can make I1(x, y, t, k)L2(Rx,k) arbitrarily small by choosingε small enough.
On the other hand, since|K±(l, y, t, k)| . (1 + |l|)−1 by (2.17), the map (y, t) → K±(l, y, t, k)f (k) ∈ C(R2, L2(R2
k,l)) for anyf ∈ L2(R). This together
with (4.11) show thatP±[K(l, y, t, k)H (ξ, y, t, k− l)] forms a precompact subset of L2(R2
l,k) for ε < |ξ | < ε−1 and(y, t) ∈ B. The Plancherel theorem then implies
∫ R dl eixlE±(l)P[K(l, y, t, k)H (ξ, y, t,
k − l)] forms a precompact subset ofL2(Rx,k) for ε < |ξ | < ε−1 and(y, t) ∈ B. Hence, we have ∫
R dl eixlE±(l)P[K(l, y, t, k)H (ξ, y, t, k − l)]
L2(r)
→ 0 (4.30)
uniformly for ε < |ξ | < ε−1 and(y, t) ∈ B asr →∞. The estimate (4.30) implies that theL2(r) norm ofI2(x, y, t, k) tends to zero
uniformly for (y, t) ∈ B asr →∞, and (4.23) follows. 2 LEMMA 4.6. The following estimate holds:
λ(x, y, t, k) − 1− (L (y,t)1)(x, k)L2(R2 x,k) . (
1+ t4 1+ |y|
) (4.31)
for t ∈ R andy 6 0. Moreover, ifB is a bounded subset ofR2 andr is defined by (4.19), we have
lim r→∞λ(x, y, t, k) − 1− (L (y,t)1)(x, k)L2(r ) = 0
uniformly for(y, t) ∈ B. (4.32)
Proof.Let χy,t (x, k) = (1+ |y|)[λ(x, y, t, k)− 1− (L (y,t)1)(x, k)]. Thenχ(y,t) satisfies
χ(y,t) = (1+ |y|)L2 (y,t)1+ L (y,t)χ(y,t). (4.33)
Note thatL (y,t)1 has the same properties asH in Lemma 4.3. The estimate (4.31) then follows from (4.22) and (4.33).
Letωy,t(x, k) = λ(x, y, t, k) − 1− (L (y,t)1)(x, k), then we have
ω(y,t) = L2 (y,t)1+ L (y,t)ω(y,t). (4.34)
The limit (4.32) follows from Lemma 4.1, Lemma 4.3, (4.23) and (4.34). 2 LEMMA 4.7. The following estimates hold: ∂j∂xj [λ(x, y, t, k)]
L2(R2
x,k)
6 Pj(|t|) 1+ |y| for t ∈ R, y 6 0, j > 1, (4.35)
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 17
wherePj is a polynomial. Moreover, ifB is a bounded subset ofR2 andr is defined by(4.19), we have
lim r→∞
= 0 uniformly for(y, t) ∈ B. (4.36)
Proof.Differentiating (2.10) we find
where
∞ dlei[x(k−l)+y(l
2−k2)+4t (k3−l3)] × × i(k − l)L±(k, l)λ(x, y, t, l). (4.38)
We can writeh−(x, y, t, k) as
h−(x, y, t, k) = h1(x, y, t, k) + h2(x, y, t, k) + h3(x, y, t, k), (4.39)
where
−∞ dl ei[x(k−l)+y(l
2−k2)+4t (k3−l3)]i(k − l)L+(k, l), (4.40)
h2(x, y, t, k) = P− ∫ k
−∞ dl ei[x(k−l)+y(l
2−k2)+4t (k3−l3)] × × i(k − l)L+(k, l)(L (y,t)1)(x, l), (4.41)
h3(x, y, t, k) = P− ∫ k
−∞ dl ei[x(k−l)+y(l
2−k2)+4t (k3−l3)]i(k − l)L+(k, l)× × [λ(x, y, t, l) − 1− (L (y,t)1)(x, l)]. (4.42)
We can estimateh1(x, y, t, k) by Lemma 4.4,h2(x, y, t, k) by Lemma 4.5 (with L±(k, l) replaced byi(k − l)L±(k, l) in (4.24)), andh3(x, y, t, k) by Lemma 4.6. The results are, forj = 1,2,3,
hj(x, y, t, k)L2(R2 x,k) . (
1+ t4 1+ |y|
) for y 6 0, t ∈ R, (4.43)
lim r→∞hj (x, y, t, k)L2(r) = 0 uniformly for (y, t) ∈ B. (4.44)
For the casej = 1, the lemma follows from Lemma 4.1, (4.37), (4.39), (4.43), (4.44) and the corresponding results forh+(x, y, t, k). The higher order cases are established by similar techniques and mathematical induction. 2
The following proposition is obtained by using Lemmas 4.1–4.5, mathematical induction and the techniques in the proofs of Lemmas 4.6 and 4.7.
18 LI-YENG SUNG
PROPOSITION 4.8.Lemma4.7 (resp. Lemma4.6) is valid if λ(x, y, t, k) is re- placed by ∂`
∂y` λ(x, y, t, k) (resp.λ(x, y, t, k) − 1− (L (y,t)1)(x, k) is replaced by
∂j+` ∂xj ∂y`
[λ(x, y, t, k) − 1− (L (y,t)1)(x, k)]) for ` > 0 (resp.j, ` > 0).
REMARK 4.9. We have so far obtained results concerningλ(x, y, t). But of course similar results (fory > 0) also hold forρ(x, y, k, t). This is the reason why we need both (2.11) and (2.15) to represent the solution.
5. The Square Integrability of q(x, y, t) in the (x, y) Variables
We first establish a useful estimate. LetF(k, l) satisfy the estimates in (2.17) and f ∈ L2(R). Then we have∫
R2 dk dl|F(k, l)f (l)| .
∫ R2
1+ |k2− l2| )
1+ |kl| )
. f L2(R). (5.1)
We can now prove one of the main results of this paper.
THEOREM 5.1. Let q0 be a Schwartz function which satisfies the smallness as- sumption(2.16) and q(x, y, t) be the solution of(1.1) obtained by the inverse spectral method. Then for eacht , q(·, t) ∈ Hj(R2) for j > 0, and the map t → q(·, t) is continuous fromR intoHj(R2).
Proof. Throughout this proofQj is a polynomial in one real variable. From (2.11) we can write
q(x, y, t) = q1,+(x, y, t) + q1,−(x, y, t) + q2,+(x, y, t) + q2,−(x, y, t),(5.2)
where
dl ei[x(k−l)+y(l 2−k2)+4t (k3−l3)] ×
× (k − l)L(k, l)λ(x, y, t, l), (5.3)
q2,(x, y, t)
dk ∫ ±∞ k
dl ei[x(k−l)+y(l 2−k2)+4t (k3−l3)]L(k, l)λx(x, y, t, l). (5.4)
It follows from (5.1), (4.35) and (5.4) that
q2,±(x, y, t)L2(Rx) 6 Q1(|t|)(1+ |y|)−1 for t ∈ R, y 6 0, (5.5)
and hence
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 19
q2,±(x, y, t)L2({(x,y):y60}) 6 Q2(|t|) for t ∈ R. (5.6)
In order to estimateq1,−(x, y, t), we break it into three terms:
q1,−(x, y, t) = 3∑ j=1
vj (x, y, t), (5.7)
where
dk ∫ ∞ k
dl ei[x(k−l)+y(l 2−k2)+4t (k3−l3)](k − l)L−(k, l), (5.8)
v2(x, y, t) = i
dl ei[x(k−l)+y(l 2−k2)+4t (k3−l3)] ×
× (k − l)L−(k, l)(L (y,t)1)(x, l), (5.9)
v3(x, y, t) = i
dk ∫ ∞ k
dl ei[x(k−l)+y(l 2−k2)+4t (k3−l3)](k − l)L−(k, l)×
× [λ(x, y, t, l) − 1− (L (y,t)1)(x, l)]. (5.10)
As in the case ofq2,±, by applying (5.1) and (4.31) to (5.10) we obtain
v3(x, y, t)L2(Rx) 6 Q3(|t|)(1+ |y|)−1 for t ∈ R, y 6 0, (5.11)
and hence
v3(x, y, t)L2({(x,y):y60}) 6 Q4(|t|) for t ∈ R. (5.12)
From (2.17), (5.8) and the change of variablesξ = k − l andη = l2 − k2 we find
v1(x, y, t) = ∫ R2
dξ dηG(ξ, η)ei(ξ 3+3η2/ξ)tei(xξ+yη), (5.13)
whereGL2(R2) <∞. It then follows from (5.13) and the Plancherel theorem that
v1(x, y, t)L2(R2 x,y ) . 1 for t ∈ R. (5.14)
Next we investigate the square integrability ofv2(x, y, t). LetH(k, l) = i(k − l)L−(k, l) andM(ξ, y, t, k) be the Fourier transform of(L (y,t)1)(x, k) in the x variable. Note that the results forH(ξ, y, t, k) in Lemma 4.3 are also valid for M(ξ, y, t, k).
We can rewritev2(x, y, t) as
v2(x, y, t) = 1
dl dk ei(xl+yl 2+4t l3)e−i2klye12it (lk2−l2k) ×
× E−(l)H(k, k − l)M(ξ, y, t, k − l), (5.15)
From (2.17) we have
20 LI-YENG SUNG∫ R
<∞. (5.16)
By the Plancherel theorem, Lemma 4.3, (5.15), (5.16) and Hölder’s inequality we have
v2(x, y, t)L2(Rx) . ∫ R
dξ
∫ R
. [ ∫
×
dξM(ξ, y, t, ·)L4(R)
6 Q5(t)(1+ |y|)−3/4 for t ∈ R, y 6 0. (5.17)
It follows from (5.17) that
v2(x, y, t)L2({(x,y):y60}) 6 Q6(|t|) for t ∈ R. (5.18)
Moreover, by splitting the integral in (5.15) over the sets{ξ : |ξ | < ε−1 or ε < |ξ |} and{ξ : ε < |ξ | < ε−1}, we obtain from Lemma 4.3 (cf. the arguments in the proof of Lemma 4.5) that, for any bounded subsetB of R2,
lim r→∞v2(x, y, t)L2(|x|>r) = 0 uniformly for (y, t) ∈ B. (5.19)
It follows from (5.7), (5.12), (5.14), (5.18) and their analogs forq1,+ that
q1,±(x, y, t)L2({(x,y):y60}) 6 Q7(|t|) for t ∈ R. (5.20)
The estimates (5.6) and (5.20) together with(5.2) show that
q(x, y, t)L2({(x,y):y60}) 6 Q8(|t|) for t ∈ R. (5.21)
Similarly, using Proposition 3.7 and Remark 4.9, we find
q(x, y, t)L2({(x,y):y>0}) 6 Q9(|t|) for t ∈ R. (5.22)
From (5.5), (5.11), (5.13) and (5.17) we see that theL2({(x, y) : y 6 0}) norm of q(x, y, t) becomes arbitrarily small for|y| large andt bounded. So to prove the continuous dependence ofq(x, y, t) in t with respect toL2({(x, y) : y 6 0}), it suffices to look at the case where(y, t) ∈ B andB is a bounded subset of {(y, t) : y 6 0}. Then (4.32), (4.36), (5.13) and (5.19) further show that we may also assumex be bounded. The continuity ofq(x, y, t) in t with respect to the L2({(x, y) : y 6 0}) norm therefore follows from (2.27). In view of Remark 4.9 and Proposition 3.7 this is also true in theL2(R2
x,y) norm. We have proved the case wherej = 0. The other cases are established by similar
techniques using Lemmas 4.3–4.5 and Proposition 4.8. 2
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 21
6. The Uniqueness of Solutions
The following lemma whose proof can be found in [6] enables us to write (1.1) in a different form.
LEMMA 6.1. Supposef, g ∈ L2(R2) andh ∈ L∞(R2) satisfyfy = (g + h)x in the sense of distributions and the weak derivativefy ∈ L2(R2), then there exists 8 ∈ S′(R2) such that8x = f and8y = g + h in the sense of distributions. Moreover, iff , g and h depend continuously on some parameters, then8 also depends continuously on the same parameters.
Let q(x, y, t) ∈ C([0,∞),H 3(R2)) ∩ C1((0,∞), L∞(R2)) be a generalized solution of (1.1) fort > 0. By applying Lemma 6.1 tof = 3qy , g = qxxx − 6qqx andh = qt , we obtain
qt − 6qqx + qxxx = 3ry, (6.1)
rx = qy, (6.2)
in the sense of distributions fort > 0, wheret → r(·, t) belongs toC((0,∞), S′(R2)).
REMARK 6.2. For eacht , the distributionr(·, t) is only unique up to a constant.
The uniqueness of the generalized solution for the forward problem of (1.1) in the classC([0,∞),H 3(R2)) ∩ C1((0,∞), L∞(R2)) therefore follows from the next proposition.
PROPOSITION 6.3.The Cauchy problem(1.1) has at most one generalized solu- tion q in the sense of(6.1)–(6.2) for t > 0 such that
t → q(·, t) ∈ C([0,∞),H 3(R2) ) ∩ C1((0,∞),S′(R2)
) , (6.3)
Proof.The following arguments generalize those in [17]. We denote theL2(R2)
inner product and norm by(·, ·) and · respectively. Let ε be an arbitrary positive number,βε(ξ, η) be aC∞ function such that (i)
0 6 βε(ξ, η) 6 1, (ii) βε(ξ, η) = 1 on the setEε = {(ξ, η) ∈ R2 : ε < |ξ | < ε−1
and|η| < ε−1}, and (iii)βε(ξ, η) = 0 onR2 \Eε/2. Letαε ∈ S(R2) be the function such thatαε = βε.
Assume thatq1(x, y, t) andq2(x, y, t) are two such solutions of (1.1). We define qj,ε, (q2
j )ε andrj,ε to be the convolutions ofqj , q2 j andrj with αε. Then forj = 1,2
we obtain from (6.1)–(6.4)
22 LI-YENG SUNG
which together with (6.3) imply that
t → rj,ε(·, t) ∈ C ( (0,∞),H∞(R2)
) . (6.8)
t → qj,ε(·, t) ∈ C1 ( (0,∞),H∞(R2)
) ∩ C([0,∞),H∞(R2) ) . (6.9)
Let 1ε(x, y, t) = q1,ε(x, y, t) − q2,ε(x, y, t), ωε(x, y, t) = (q2 1)ε(x, y, t) −
(q2 2)ε(x, y, t), andRε(x, y, t) = r1,ε(x, y, t)− r2,ε(x, y, t). From (6.5), (6.6), (6.8)
and (6.9) we have
((1ε)t ,1ε) = −3(ωε, (1ε)x)+ (1ε, (1ε)xxx)− 3(Rε, (Rε)x). (6.10)
Taking the real part of(6.10) we find d dt (1ε,1ε) = −3Re(ωε, (1ε)x), which
together with (6.9) imply that
1ε(·, t)2 = −3 ∫ t
0 dt ′ Re(ωε(·, t ′), (1ε)x(·, t ′)). (6.11)
Letting ε→ 0 we obtain from (6.11)
1(·, t)2 = −3 ∫ t
0 dt ′ Re(σ (·, t ′)1(·, t ′),1x(·, t ′)), (6.12)
where1(x, y, t) = q1(x, y, t)−q2(x, y, t) andσ(x, y, t) = q1(x, y, t)+q2(x, y, t). Let T > 0 be arbitrary. It follows from (6.3), (6.12) and the Sobolev inequality
(cf. [18]) that
0 dt ′ [1(·, t ′)2+ 1x(·, t ′)2
] for 06 t 6 T . (6.13)
A similar argument yields
∫ t
]1xx(·, t) for 06 t 6 T . (6.14)
From (6.5) and (6.6) we also find, forR = r1− r2, (1xx,1xx)+ 3(Rx,Rx) = 3Re(σ1,1xx) for t 6= 0. (6.15)
It follows from (6.3), (6.15) and the Sobolev inequality that
1xx . 1 for t 6= 0. (6.16)
SOLUTIONS OF THE KADOMTSEV–PETVIASHVILI-I EQUATION 23
We deduce from (6.13), (6.14), (6.16) and Gronwall’s inequality that
1 = 1x = 0 for 06 t 6 T . (6.17)2 It is shown in [6] that the functionq(x, y, t) obtained by the inverse spectral
method has the property thatt → q(·, t) ∈ C1((−∞,0) ∪ (0,∞), C0(R2)). From Theorem 5.1, Proposition 6.3 and its analog fort < 0 we have the following theorem.
THEOREM 6.4. Let q0 be a Schwartz function which satisfies the smallness as- sumption(2.16). The solutionq(x, y, t) for (1.1) obtained by the inverse spectral method is the unique generalized solution of the forward problem for(1.1) in the classC([0,∞),H 3(R2)) ∩ C1((0,∞), L∞(R2)). It is also the unique gen- eralized solution of the backward problem in the classC((−∞,0],H 3(R2)) ∩ C1((−∞,0), L∞(R2)).
Finally we prove a conservation law.
LEMMA 6.5. Letq(x, y, t) be the solution obtained by the inverse spectral method. Then we have∫
R2 dx dy q2(x, y, t) =
∫ R2
dx dy q2 0(x, y). (6.18)
Proof.We use the notation in the proof of Proposition 6.3. From Equations (6.5) and (6.6) (forqε andrε) we have
(d/dt) ∫ R2
∫ R2
= ∫ R2
− 3 ∫ t
dx dy (q2)ε(x, y, t ′)(qε)x(x, y, t ′). (6.20)
The conservation law (6.18) follows by lettingε→ 0 in (6.20). 2 REMARK 6.6. Whenq0 is real, the uniqueness of solution for (1.1) implies that q(x, y, t) is real for allt ∈ R and henceq(x, y, t)L2(R2
x,y ) is conserved.
24 LI-YENG SUNG
Acknowledgment
This work was supported in part by the National Science Foundation under Grant DMS-94-96154.
References
1. Ablowitz, M. J. and Villarroel, J.: On the Kadomtsev–Petviashvili equation and associated constraints,Stud. Appl. Math.85 (1991), 195–213.
2. Boiti, M., Pempinelli, F. and Pogrebkov, A.: Solutions of the KPI equation with smooth initial data,Inverse Problems10 (1994), 505–519.
3. Boiti, M., Pempinelli, F. and Pogrebkov, A.: Properties of solutions of the Kadomtsev– Petviashvili I equation,J. Math. Phys.35 (1994), 4683–4718.
4. Faminskii, A. V.: The Cauchy problem for the Kadomtsev–Petviashvili equation,Russian Math. Surveys5 (1990), 203–204.
5. Fokas, A. S. and Ablowitz, M. J.: On the inverse scattering of the time-dependent Schrödinger equation and the associated Kadomtsev–Petviashvili (I) equation,Stud. Appl. Math.69 (1983), 211–228.
6. Fokas, A. S. and Sung, L.-Y.: The Cauchy problem for the Kadomtsev–Petviashvili-I equation without the zero mass constraint,Math. Proc. Camb. Phil. Soc.125(1999), 113–138.
7. Johnson, R. S.: Water waves and Korteweg–de Vries equations,Fluid Mech.97 (1980), 701– 719.
8. Kadomtsev, B. B. and Petviashvili, V. I.: On the stability of solitary waves in weakly dispersive media,Soviet Phys. Dokl.15 (1970), 539–541.
9. Lipovskij, V. D., Matveev, V. B. and Smirnov, A. O.: On the isomorphism between KP-equation and Johnson equation,Zap. Semin. LOMI150(1986), 70–75.
10. Manakov, S. V.: The inverse scattering transform for the time-dependent Schrödinger equation and Kadomtsev–Petviashvili equation,Physica D3 (1981), 420–427.
11. Muskhelishvili, N. I.:Singular Integral Equations, Dover, New York, 1992. 12. Saut, J.-C.: Remarks on the generalized Kadomtsev–Petviashvili equations,Indiana Math. J.
42 (1993), 1011–1027. 13. Saut, J.-C.: Recent results on the generalized Kadomtsev–Petviashvili equations,Acta Appl.
Math.39 (1995), 477–487. 14. Segur, H.: Comments on IS for the Kadomtsev–Petviashvili equation, in: M. Tabor and Y. M.
Treve (eds.),Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, AIP Conference Proceedings, No. 88, 1982, pp. 211–228.
15. Stein, E.:Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
16. Tom, M. M.: On a generalized Kadomtsev–Petviashvili equation,Contemp. Math.200(1996), 193–210.
17. Ukai, S.: Local solutions of the Kadomtsev–Petviashvili equation,J. Fac. Sci. Univ. Tokyo, Sect. IA Math.36 (1989), 193–209.
18. Wloka, J. T.:Partial Differential Equations, Cambridge University Press, Cambridge, 1987. 19. Yosida, K.:Functional Analysis, Springer-Verlag, Berlin, 1995. 20. Zhou, X.: Inverse scattering transform for the time dependent Schrödinger equation with
applications to the KP-I equation,Comm. Math. Phys.128(1990), 551–564.
Mathematical Physics, Analysis and Geometry2: 25–51, 1999. © 1999Kluwer Academic Publishers. Printed in the Netherlands.
25
WERNER KIRSCH Institute of Mathematics of the Ruhr-University Bochum, D-44780 Bochum, Germany
VLADIMIR KOTLYAROV Mathematical Division, B. Verkin Institute for Low Temperature Physics, 310164 Kharkov, Ukraine
(Received: 26 November 1997; accepted in final form: 18 February 1999)
Abstract. An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.
Mathematics Subject Classifications (1991):35Q51, 35Q53, 81U40, 35B40.
Key words: solitons, sine-Gordon equation, Marchenko equation.
Introduction
It has been known for a long time [1 – 3] that the asymptotics of the solution of the initial value problem with decreasing initial data is a superposition of solitons. This superposition is the main term of the asymptotics. The next term tends to zero whent →∞. Sometimes, the phases of these solitons are the same as in the reflec- tionless case. They depend on the eigenvalues and the normalization coefficients only. This situation takes place for those nonlinear evolution equations integrated by the inverse scattering transform, for which the kernel of the Marchenko integral equation has no stationary point in the soliton domain. These are, for example, the Korteweg–de Vries equation and the modified one. For many other integrable equations, the kernel of the Marchenko integral equation has a stationary point. In this case, the phases of solitons depend, in addition, on the reflection coefficient [4 – 6]. It is important to remark that the additional phase shift, contributed by non-zero reflection coefficient, has a finite order fort →∞.
The first regorous results on this subject were obtained in the well-known papers by A. B. Shabat [7] for the Korteweg–de Vries equation with decreasing initial data, and by E. Ya. Khruslov for a step-like initial function [8]. Recently, Deift, Its
26 WERNER KIRSCH AND VLADIMIR KOTLYAROV
and Zhou [9, pp. 181–204] formulated the steepest descent method for the study of long-time asymptotics for integrable nonlinear wave equations, based on the oscillatory Riemann–Hilbert problem. This is a modern and very powerful method for the asymptotic investigation of decreasing solutions of nonlinear wave equa- tions as well as for the Painlevé equations, and for some models of quantum and statistical physics. The references to this approach can be found in the book [9]. The Riemann–Hilbert method was also applied by Fokas and Its to the study of initial boundary value problems on the semi-axis [10, 11]. Nevertheless, in the case of non-decreasing solutions it is more convenient to use the associated Marchenko integral equations.
The aim of this paper is twofold. The first is to present rigorous results on the soliton asymptotics of a solution of the sine-Gordon equation with decreas- ing and some non-decreasing initial data based on direct asymptotic analysis of the Marchenko integral equation. The second is to obtain explicitly the additional phase shift caused by the non-zero reflection coefficient.
Taking into account the equivalence [2] of the sine-Gordon equation given in laboratory coordinates and in light-cone coordinates, we restrict our consideration to the equation
uxt + 4 sinu = 0. (1)
A large class of solutions of Equation (1) can be constructed as follows. Let the system of integral equations
K1(x, y, t) + ∫ ∞ x
K2(x, y, t) + ∫ ∞ x
where
eiλy+2it/λ dρ(λ), ⊂ C,
have a unique solution, which is sufficiently smooth. Then the functionu(x, t)
defined by the equation
ux(x, t) = 4iK2(x, x, t)
is a solution of Equation (1). In what follows, we consider two cases. The first is the problem for Equation
(1) which we rewrite in the form
vt + 4 sin ( ∂−1 x v
) = 0, ∂−1 x v =
with decreasing initial data
u(x,0) = w(x), x ∈ R, w(x)→ 0, |x| → ∞. (4)
In this case, the kernelH(y, t) coincides with (1.2) (see below). The second one is the case for which
w(x)→ 0, asx →∞, w(x)→ wn(x), asx →−∞, (5)
wherewn(x) is an almost periodic function of a finite-gap type. The problem (3), (5) was studied in [12] for the reflectionless case.
Let us formulate the main results of the present paper.
THEOREM 1. Letw(x) be a real Schwarz type function(w(x) ∈ S(R)). Then, uniformly with respect tox ∈ R for t → ∞, the solution of the problem(3), (4) has the asymptotic form
v(x, t) = 4 ∂
∂x arctan
Im det[I +D(x, t)] Re det[I +D(x, t)] + o(1), t →∞.
The entries of the matrixD have the form
Djl(x, t) = imj exp[2it/λj ] λj + λl exp
[ i(λjx + θj (X)+ λlx + θl(X))
] ,
s − λj , α(X) = 1√ 2X , X = x
2t .
COROLLARY 1. The main term of the asymptotics of any decreasing solution u(x, t) can be found by the same scheme as in the reflectionless case. To this end we have to replaceeiλj x with eiλj x+iθj (X).
COROLLARY 2. The above mentioned solution has the form
u(x, t) = n∑ j=1
vj (x, t) + p∑ j=1
wj(x, t) + o(1), t →∞,
( x − ν−2
σj = sign Im(mj)
xj = x0 j −
j
and
{ Im λn+j Reλn+j
sin[2 Reλn+j (x + |λn+j |−2t + βj)] cosh[2 Imλn+j (x − |λn+j |−2t − λj)]
} is the soliton-breather with the phases
βj = β0 j +
,
ln[1+ |r(s)|2]ds s − λj , j = 1,2, . . . , p.
The soliton phasesx0 j , β
0 j and γ 0
j have to be taken in the same form as in the reflectionless case(r(λ) ≡ 0).
In the non-decreasing case, the kernel of the Marchenko integral equation is as follows
H(y) = 1
mj(t)eiλj y. (6)
This form of the kernel corresponds to the following structure of the spectrum of the Dirac operator with a “step-like” potentialw(x): this operator has a continuous spectrum of multiplicity two on the real lineR of the λ-plane, and a continuous spectrum of multiplicity one on a set of analytic arcsγj (j = 1,2, . . . ,2n) with end-points that are zeros of the polynomial
P(λ2) = k∏ i=1
( λ2− E2
) ,
ReEi = 0, i 6 k, ImEj > 0, k + 2m = n, which corresponds to the Riemann surface of an almost periodic potentialwn(x)
[12] and, possibly, a finite number of eigenvaluesλj which lie on the imaginary axis of theλ-plane and symmetrically with respect to this axis.
Let σ be the complete spectrum of the Dirac operator
L = iσ3 d
dx + i w(x)
C = max λ∈σ\R
|λ|−2 > 0 (7)
holds and the maximal value is attained in a finite number of pointsEj ∈ σ \ R, j = 1,2, . . . ,2p.
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 29
Let
2 ImE ln tN+1, t > t(N)
} , (8)
where ImE = max16j6n Im Ej .
THEOREM 2. Let condition(7) be fulfilled and for all eigenvaluesλj , |λj | > 1√ C
. Then for any natural numberN , the solution of problem(3), (5) in the domain(8) has the asymptotic form
v(x, t) = 4 ∂
∂x arctan
Im det[I +(ξ, t)] Re det[I +(ξ, t)] + o(1), ξ = x − Ct, t →∞, (9)
where(ξ, t) = Blr(ξ, t)pl,r=1 is a block matrix with the entries
Blrkj (x, t) = N−k−1∑ m=0
w (k+m) l (t)
k!m!tk+m+3/2
m∑ p=0
j∑ q=0
J lrp = ∫ ∞ ξ
The functionsω(n)l (t) andGlrpq
mj (t) are determined by the scattering data of the initial functionw(x) and they are uniformly bounded with respect tot .
THEOREM 3. Under the conditions of Theorem2, let us suppose that the max- imal value(7) is attained in two pointsE1 = ia and E2 = −ia, a > 0. Then formula(9) takes the form
v(x, t) = ∂
)]}+ o(1), t →∞, (10)
2π
. (11)
The numbersα0 n are determined(Equation (5.11)) by the scattering data of the
initial functionw(x), andr(µ) is the reflection coefficient of the Dirac operatorL.
THEOREM 4. Under the conditions of Theorem2, let us suppose that the maxi- mal value in(7) is attained in four pointsE1,2 = ±E, E3,4 = ±E. Then formula (9) takes the form
v(x, t) = ∂
{ ImE
ReE
} +
βn(x, t) = β0 n +
ln[1+ |r(µ)|2]dµ µ− E
) , (13)
1
− 1
) . (14)
data of the initial functionw(x).
REMARK. If the Dirac operator has a discrete spectrum which lies inside the circle of radius 1/
√ C, then the superposition of usual solitons, i.e. kinks(Reλk = 0) and
breathers(Reλk 6= 0), should be added to Equation (10) and (12) of asymptotic solitons. Each asymptotic soliton gets an additional phase shift caused by solitons generated by the discrete spectrum.
1. Representation of the Solution
Due to the inverse scattering transform, the solution of the problem (3), (4) is represented as follows [1, 2]
v(x, t) = 4iK2(x, x, t),
∫ ∞ x
]2 ds = 8K1(x, x, t), (1.1)
whereK1(x, y, t) = K1(x, y, t), K2(x, y, t) = −K2(x, y, t), and these functions solve the integral Marchenko equation (2). For this case, the kernel of Equation (2) has the form
H(y, t) = N∑ j=1
mj(t)eiλj y + 1
mj(t) = mj e2it/λj , r(λ, t) = r(λ)e2it/λ.
Here the numbersλj (j = 1,2, . . . , N) are the eigenvalues of the Dirac operator L. Sincew(x) is a real function, the eigenvalues are distributed in two subsets [2]:
λj = iνj , νj > 0, j = 1,2, . . . , n; λn+p+j = −λn+j , Reλn+j > 0, Im λn+j > 0, j = 1,2, . . . , p.
ThereforeN = n + 2p. The coefficientsmj (j = 1,2, . . . , N) corresponding to the eigenvaluesλj have the following properties:
mj = −mj, j = 1,2, . . . , n;
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 31
mn+p+j = −mn+j , mn+j ∈ C, j = 1,2, . . . , p.
The reflection coefficientr(λ) satisfies the equation
r(−λ) = −r(λ), λ ∈ R.
So the kernel (1.2) possesses the property
H(y, t) = −H(y, t). (1.3)
Let H be an integral operator which acts in the spaceL (i)[x,∞) (i = 1,2,∞) of vector-functionsf (y) = (f1(y), f2(y)) according to the formula
Hf (y) = ∫ ∞ x
)( f1(y)
f2(y)
) dz.
Then the Marchenko integral equation (2) can be written in the form( I + H )
K = G, K = (K1,K2), G = (0,−H). (1.4)
It is known [2] that for anyt ∈ R Equation (1.4) has a unique solution in each spaceL (i)[x,∞) (i = 1,2,∞) and if r(λ) ∈ S(R), thenK(x, y, t) is a Schwarz type vector-function for eacht, too.
2. Degenerate Integral Equation
Let us write the kernel (1.2) in the form
H(y, t) = HN(y, t)+ R(y, t) and letL = (L1(x, y, t), L2(x, y, t)) be the solution of the integral equation(
I + R )L = GR, GR = (0,−R). (2.1)
Since the operatorR is compact in any spaceL (i)[x,∞) (i = 1,2,∞) andR is skew-symmetric (1.3) in the spaceL2[x,∞), Equation (2.1) has a unique solution in each spaceL (i)[x,∞) (i = 1,2,∞) for everyt ∈ R. Therefore(I + R )−1
2 6 1
and it is easy to obtain an estimate, uniform with respect tox, t , for the vector- functionLL(x, ., t)2
2 6 GR22 = R22 = σ(2x, t) = ∫ ∞
2x
whereL(x, ., t)2 2 =
∫ ∞ x
2) dy.
Moreover, there is a uniform estimate in the spaceL∞[x,∞) for the components Lj(x, y, t)L1(x, ., t)
∞ 6 R2L22 6 σ(2x, t) 6 C2 2, (2.3)
L2(x, ., t) ∞ 6 R∞ + R2L12 6 sup
y>2x |R(y, t)| + σ(2x, t) 6 C1+ C2
2, (2.4)
∫ ∞ −∞ |r(λ)|dλ.
Let us find the solution of Equation (1.4) in the form
K = L+ (I + Q )M, (2.5)
where an integral operatorQ satisfies the equation( I + Q )(I + R ) = (I + R )(I + Q ) = I. (2.6)
LEMMA 1. The operatorQ is a bounded integral operator, and the correspond- ing kernel has the form
Q(y, z) = ( Q1(y, z) Q2(y, z)
Q2(y, z) Q1(y, z)
Q1(y, z) = { L1(z, y)+
∫ z x [L1(s, y)L1(s, z)+ L2(s, y)L2(s, z)]ds, z < y,
L1(y, z)+ ∫ y x [L1(s, y)L1(s, z)+ L2(s, y)L2(s, z)]ds, z > y,
Q2(y, z) = { L2(z, y)+
∫ z x [L2(s, y)L1(s, z)+ L1(s, y)L2(s, z)]ds, z < y,
L2(y, z)+ ∫ y x [L2(s, y)L1(s, z)+ L1(s, y)L2(s, z)]ds, z > y
and
Q1(y, z) = Q1(z, y) = Q1(z, y), Q2(y, z) = Q2(z, y) = −Q2(z, y).
Proof.The operator identities (2.6) are equivalent to pair equations of Marchenko type on functionsQ1(y, z) andQ2(y, z). Since the functionsL1(y, z) andL2(y, z)
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 33
are the solution of Equation (2.1), it is easy to check that the representations for Q1(y, z) andQ2(y, z) are valid. 2
Let us introduce some notations:
h (1) j (x, t) = eiλj x +
∫ ∞ x
h2 j (x, t) =
g (1) j (x, y, t) = h
(1) j (y, t) +
+h(2)j (s, t)L2(s, y, t) ]
ds,
(2) j (y, t) +
+h(2)j (s, t)L1(s, y, t) ]
ds.
The next lemma presents a degenerate integral equation for the problem (3), (4).
LEMMA 2. The vector-functionM(x, y, t) is the solution of the equation
M + FNM = −GN. (2.7)
The kernel of this equation is degenerate and has the form
FN(x, y, z, t) = ( F (1) N (x, y, z, t) F
(2) N (x, y, z, t)
F (2) N (x, y, z, t) F
(1) N (x, y, z, t)
) ,
N∑ j=1
(2.8)
N∑ j=1
The vector-functionGN(x, y, t) is given by the relations
G (1) N (x, y, t) = F
(1) N (x, y, x, t) =
N∑ j=1
(2) N (x, y, x, t) =
N∑ j=1
34 WERNER KIRSCH AND VLADIMIR KOTLYAROV
Proof. It follows from Equations (1.4), (2.1), (2.5) and (2.6) that
M + HN ( I + Q )M = −HNL+GHN .
Then the operatorFN = HN(I+Q) is degenerate and the corresponding kernel has the form (2.8). The vector-functionGN = −HNL + GHN and the corresponding components have the form (2.9). 2 LEMMA 3. The solution of Equation(2) with kernel(1.2)has the representation
K1(x, y, t) = L1(x, y, t) + N∑ j=1
[ Xj(x, t)g
(2.10)
[ Xj(x, t)g
(2) j (x, y, t) +
+Yj(x, t)g(1)j (x, y, t) ] ,
whereL = (L1, L2) is the solution of Equation(2.1) and theN-dimensional vec- tors X = (X1,X2, . . . , XN) and Y = (Y1, Y2, . . . , YN) are the solution of the system of linear algebraic equations
Xj + N∑ l=1
(AjlXl + BjlYl) = −mj(t)h(2)j , (2.11)
Yj + N∑ l=1
Ajl(x, t) = mj(t)
(2) l (s, t)+ h(2)j (s, t)h(1)l (s, t)
] ds,
(2.12)
(1) l (s, t)+ h(2)j (s, t)h(2)l (s, t)
] ds.
Xj(x, t)eiλj y, M2(x, y, t) = N∑ j=1
Yj(x, t)eiλj y.
Then Equation (2.7) is equivalent to the system (2.11) of linear algebraic equations, and Equation (2.5) leads to the representation (2.10). 2
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 35
COROLLARY 3. The solution of the problem(3), (4)has the form
v(x, t) = 4iL2(x, x, t) + 4i N∑ j=1
[ Xj(x, t)h
] .
So, the solution of the initial value problem (3), (4) is completely defined via the solutionL of Equation (2.1) and Jost type functionsh(1)j (x, t) andh(2)j (x, t). Therefore we have to study the asymptotic behaviour of these functions for a large time.
3. Asymptotic Analysis of the Marchenko Equation
In what follows it is convenient to introduce new (“slow”) variablesX = x/2t , Y = y/2t , Z = z/2t , etc. The kernelR(y, t) has a stationary point. Therefore one can obtain both the representation
R(y, t) = R0(y, t)+ R1(y, t),
where
and the estimateR1(y, t) 6 Ct−3/2ρ(Y ), (3.1)
whereρ(Y ) is the absolute value of a function of the Schwarz type. Let us look for the solution of Equation (2.1) in the form
L = P + S. (3.2)
If a vector-functionP is the solution of the equation( I + R0
) P = GR0 (3.3)
then the vector-functionS has to satisfy the equation( I + R )S = GR1 − R1P. (3.4)
LEMMA 4. The estimates
t (3.5)
are valid for the vector-functionS. Proof. It follows from Equation (3.4) that
S2 6 (I + R )−1
2
2P2.
Estimate (3.1) implies that
t
We find from Equation (3.3) and estimates (3.6), (3.7) that
P2 6 GR02 = R02 6 r(Y )2 =
(∫ ∞ X
6 C2. (3.8)
So, the first estimate (3.5) is valid. For the second one we can write
S∞ 6 GR1∞ + R12P2 + R2S2.
By using Equations (3.1), (3.6), (3.8), (2.2) and taking into account the first esti- mate (3.5) we finally obtain
S∞ 6 C
t ) for t →∞.
Let us find the solutionP = (P1, P2) in the form
P1(x, y, t) = N1(X, Y, t), (3.9)
P2(x, y, t) = 1√ t
[ N2(X, Y, t)e4it
√ X+Y
[ N2(X,Z, t)r(Z + Y )e4it (
] dZ +
] dZ (3.10)
for N1(X, Y, t) and the integral equation forN2(X, Y, t)
N2(X, Y, t) = ∫ ∞ X
N2(X, S, t)e4it ( √ X+S−√X+Y)0(X, Y, S, t)dS −
− ∫ ∞ X
N2(X, S, t)e−4it ( √ X+S+√X+Y)0(X, Y, S, t)dS
= −r(X + Y ). (3.11)
The kernel of this equation is
0(X, Y, S, t) = 4t ∫ ∞ X
r(Y + Z) ¯r(Z + S)e4it ( √ Y+Z−√Z+S) dZ −
−4t ∫ ∞ X
= 01+ 02.
02 t f (Y ) =
(3.12)
e−4it ( √ X+S+√X+Y)02(X, Y, S, t)f (S)dS,
04 t f (Y ) =
e−4it ( √ X+S+√X+Y)01(X, Y, S, t)f (S)dS.
Then Equation (3.11) and the complex conjugated one may be written as follows
N2+ ( 01 t + 02
)∗ = 01 t ,
)∗ = 03
t .
Therefore the first summand in (3.14) is a self-adjoint operator inL2[X,∞). The second summand is not self-adjoint in the same space.
LEMMA 5. For anyX > X0 > −∞ and for any fixedt , the operator0t is a compact operator in the spaceL2[X,∞), and Re(0tf, f ) > 0 for any vector- functionf (Z) ∈ L2[X,∞) and for sufficiently larget .
Proof.The compactness of the operator0t follows from the fact that it is an op- erator of the Hilbert–Schmidt type in the spaceL2[X,∞). This statement follows from the inequality∫ ∞
0 Y r(Y + 2X)
Sincer(λ) ∈ S(R), (3.15) holds. Moreover, we find that( 0tf, f
) = 4t ∫ ∞ X
−4t ∫ ∞ X
F2(X,Z, t) = ∫ ∞ X
G1(X,Z, t) = ∫ ∞ X
G2(X,Z, t) = ∫ ∞ X
¯r(Z + Y )f2(Y )e−4it ( √ Y+Z+√X+Y) dY.
By virtue of condition (3.15) these functions belong to the spaceL2[X,∞) in the variableZ, andFj (X,Z, t) = O(1) andGj(X,Z, t) = o(1), for t → ∞. Therefore
Re ( 0tf, f
) > 4tF1 + F2
(F1 + F2 − G1+G2 ) > 0.
The lemma is proved. 2 COROLLARY 5. For sufficiently larget , Equation(4.14)has a unique solution in the spaceL2[X,∞), and(I + 0t)−1
2 6 1.
LEMMA 6. For any function(Y ) ∈ D[X,∞) = L2[X,∞) ∩ C1[X,∞) and operators(3.12), the relations
lim t→∞ 0
lim t→∞ 0
i t =
× r0(X + Y )r0(X + S) 4πi(S − Y + i0) (S)dS, i = 1,
− ∫ ∞ X
(3.16)
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 39
holds withr0(Y ) = r(1/ √ Y ) andr(λ) the reflection coefficient; the equalities are
valid in the metric of the spaceL2[X,∞). This lemma can be proved as in [13]. Let
0∞ = ( 01∞ −03∞ −03
∞
) be an operator in the space of vector-functionsL2[X,∞), which is determined by the r.h.s. of Equation (3.16). It is easy to check that this operator is self-adjoint and nonnegative. Therefore the operator(I + 0∞)−1 exists, and(I + 0∞)−1 6 1.
LEMMA 7. For any vector-functionf (Y ) ∈ L2[X,∞),(I + 0t)−1 f − (I + 0∞)−1
f = o(1), ast →∞.
Proof.Let (Y ) ∈ D[X,∞), h = (I + 0∞). Then the value(I + 0∞)−1 h− (I + 0t)−1
h 6 (I + 0t)−10t − 0∞ 6 0t − 0∞
tends to zero ast →∞ by virtue of Lemma 6. The set of functionsh = (I+ 0∞) is the dense set in the spaceL2[X,∞), the operator(I + 0∞)−1 exists and is bounded. Hence, due to the resolvent convergence [14],(
I + 0t )−1 f → (
I + 0∞ )−1 f, ∀f ∈ L2[X,∞),
Lemma 7 is proved. 2 It follows from the given proof that for the solutionN2(X, Y, t) of Equation
(3.11) the following estimate is validN2(X, Y, t)−N(X, Y )
2 = o(1), t →∞, (3.17)
N(X, Y )− ∫ ∞ X
∫ ∞ X
× r0(X + Y )r0(X + S) 4π(S − Y ) dS = −r(X + Y ). (3.18)
40 WERNER KIRSCH AND VLADIMIR KOTLYAROV
LEMMA 8. Equation(3.18)has a unique solution in the spaceL2[X,∞) which can be represented in the explicit form
N(X, Y ) = −r(X + Y )× × exp
{ i
2π
∫ ∞ X
)1/2 ln[1+ |r0(X + S)|2] S − Y + i0 dS
} . (3.19)
The functionN(X, Y ) is bounded forY ∈ [X,∞). Proof.We look for the solutionN(X, Y ) in the form
N(X, Y ) = −r(X + Y )A(X, Y ), where
r(X + Y ) = eiπ/4√ 2π (X + Y )−3/4r0(X + Y ).
Then
√ X + Y X + S
) |r0(X + S)|2A(X, S)
4πi(S − Y ) dS = 1.
Let us takeλ = (X + Y )−1/2, µ = (X + S)−1/2, B(X, λ) = A(X,1/(λ2 − X)), then we obtain
B(X, λ) = 1− ∫ α
t x . In the last summand we takeµ = −τ and
B0(X, λ) = { B(X, λ), λ > 0, B(X,−λ), λ < 0.
Then
Due to the Riemann–Hilbert problem we find that
B0(X, λ) = exp
µ− λ− i0 ] .
If we come back to the initial variable, we obtain
A(X, Y ) = exp
} .
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 41
SinceA(X, Y ) ∈ L∞[X,∞) is a continuous function with respect toY 6= X, N(X, Y ) is bounded forY ∈ [X,∞), belongs to the spaceL2[X,∞), and is a continuous function with respect toY 6= X. This lemma is proved. 2
Let us introduce new functions
L0 1(x, y, t) = −2
∫ ∞ X
] dZ +
] dZ,
√ X+Y
] , (3.20)
whereN(X, Y ) is determined by formula (3.19). These functions belong to the spaceL2[X,∞) ∩ L∞[X,∞), andL0
1(x, y, t) is continuous with respect toY ∈ [X,∞), andL0
2(x, y, t) is continuous with respect toY 6= X, Y ∈ [X,∞). LEMMA 9. LetL1(x, y, t), L2(x, y, t) be the solution of Equation(2.1),and let L (0) 1 (x, y, t), L
(0) 2 (x, y, t) be defined by Equation(3.20). Then the estimatesL1(x, y, t) − L0
1(x, y, t) ∞ = o(1), t →∞, (3.21)L2(x, y, t) − L0
2(x, y, t)
2 = o(1), t →∞, (3.22)
are valid. Proof. It follows from Equations (3.2), (3.9), (3.10) and estimates (3.5), (3.17)
that L2(x, . , t) − L(0)2 (x, ., t)
L2[x,∞) 6 2
N2(x, ., t) −N(x, ., t)
L2[X,∞) + S2(x, ., t)
L2[x,∞)
= o(1), t →∞. Estimate (3.22) can be easily deduced from Equation (3.11). 2
4. Soliton Asymptotics of the Solution
The further analysis will be connected with the study of the asymptotic behaviour of the functionsh1
j (x, t), h 2 j (x, t), Ajl (x, t) andBjl(x, t).
LEMMA 10. LetX > X0 > −∞. Then fort →∞ h1 j (x, t) = exp
( iλjx + iθj (X)
2π
∫ α
µ− λj ;
h2 j (x, t) =
1√ t ηj (X, t)exp(iλjx). (4.2)
The functionsεj (X, t) = o(1) ast →∞, and the functionsηj (X, t) are uniformly bounded with respect toX and t .
Proof.Forh1 j (x, t), by virtue of (3.20) and (3.21), we have
h1 j (x, t)e−iλj x − 1 =
∫ ∞ x
− ∫ ∞ x
] eiλj (y−x) dy
= ∫ ∞ x
L (0) 1 (x, y, t)eiλj (y−x) dy + o(1), t →∞.
For the main integral, we write the relation∫ ∞ x
L (0) 1 (x, y, t)eiλj (y−x) dy
= 4t ∫ ∞ x
2it (1/ √ X + Y − λj)
+ c.c.(−λj)+ +4t
(−λj)+O ( t−1)
= J 1(λj )+ J 1(− λj)+ J 2(λj )+ J 2(−λj)+O ( t−1 ) .
The functionA(X,Z) does not have any limit asZ→ X, namely
A(X,Z) = exp
ν(X) ln(Z −X) ] A0(X,Z),
whereν(X) = 1 2π ln[1+ |r0(2X)|2], and the functionA0(X,Z) is continuous and
bounded whenZ→ X. By using the last equality, it is easy to obtain the estimate
J 1(λj )+ J 1(−λj) = O ( t−1 ) , t →∞.
The integralsJ 2(λj ) andJ 2 (−λj ) give us
J 2(λj )+ J 2(−λj) = − ∫ α
0
h1 j (x, t) = exp(iλjx)B0(X, λj )
[ 1+ εj (X, t)
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 43
Forh2 j (x, t) we write
√ t e−iλj xh2
√ X+Y
S2(x, y, t)eiλj (y−x) dy.
SinceN2(X, Y, t)∞ is uniformly bounded (Corollary 4) with respect toX,Y, t , and since estimate (3.5) holds, we come to Equation (4.2) with some functions ηj (X, t) for which the estimateηj (X, t) < C
Im λj
is valid. The lemma is proved. Bearing in mind the previous lemmas, it is easy to prove the following
LEMMA 11. For the entries(2.13)of the matricesA andB, the relations
Ajl(x, t) = imj (t)√ t
jl(X, t)
Bjl(x, t) = imj (t)[1+ εjl(X, t)] λj + λl exp
[ i(λjx + θj (X)+ λlx + θl(X))
] are true, and the functionsjl(X, t) are uniformly bounded with respect toX and t .
From the previous results, we deduce the following statement.
THEOREM 1′. For t →∞ the solution of the problems(3), (4)has the asymptotic form
v(x, t) = 4i N∑ j=1
Y (0) j (x, t)eiλj x+iθj (X) + o(1), x > X0t (X0 > −∞), (4.3)
where the functionsY (0)j (x, t) (together withX(0) j ) solve the system of linear alge-
braic equations
N∑ l=1
(4.4) N∑ l=1
Djl(x, t)X (0) l (x, t) + Y (0)j (x, t) = −mj(t)eiλj x+iθj (X)
44 WERNER KIRSCH AND VLADIMIR KOTLYAROV
with the entries
] ,
s − λj .
Proof.Due to Lemma 11, it is natural to look for the solution of Equation (2.11) in the form
X = X(0) + 1√ t X(1), Y = Y (0) + 1√
t Y (1).
By using Equation (2.10) we find that Equation (4.3) together with (4.4) hold. As in the reflectionless case one can obtain the determinant formula (4).
We have obtained the estimates uniformly with respect toX > X0 > −∞. Therefore, it is necessary to consider the integral Marchenko equation in which the variabley ∈ (−∞, x], and then we obtain the estimates uniformly with respect to X 6 X0 <∞.
Theorems 1 and 1′ are proved. 2
5. Non-Decreasing Initial Problem
The reflectionless case of this problem was studied in [12]. Letx = Ct + ξ, y = Ct + η, where the constantC is defined by Equation (7). In what follows, we restrict our consideration to the domainDN (Equation (8)). It was shown [12] that for t →∞, the kernel (6) in the domainDN has the asymptotic form
H(x + y, t) = HN(x + y, t) + RN(x + y, t), (5.1)
where
exp[iEk(ξ + η)] N−1∑ n=0
(ξ + η)nω(n)k (t) tn+3/2
= p∑ l=1
ηk N−k−1∑ m=0
ω (l) km(t)ξ
k!m!tk+m+3/2 ,
and
RN(x + y, t) = R0(x + y, t) + R1 N(x + y, t). (5.3)
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 45
The functionsω(m)k (t) were defined in [12]. It is important to remark here that they are uniformly bounded with respect tot . The functionR0(y, t) is the same as that in Equation (3.1), i.e.,
R0(y, t) = 1√ t
N(y, t) 6 C1
tN+3/2 (5.4)
holds. Therefore, ifL = (L1, L2) is the solution of Equation (2.1) with kernel (5.3), then forL = P + S we obtain now the estimates
P1∞ = O(1), P2∞ = O
) (5.5)
ast →∞. Due to Equations (5.1)–(5.3) and estimates (5.4), (5.5), we are in the same
situation as in the previous case. But, since the degenerate kernel (5.2) has a form different from (1.2), we need to introduce new functions of Jost type. Namely,
hl1n (x, t) = (x − Ct)n eiEl (x−Ct) + ∫ ∞ x
L1(x, y, t)(y − Ct)n eiEl (y−Ct) dy, (5.6)
hl2n (x, t) = ∫ ∞ x
L2(x, y, t)(y − Ct)n eiEl (y−Ct) dy,
whereL = (L1, L2) is the solution of Equation (2.1) with the kernel (5.3). Instead of Lemma 3 we obtain the following statement:
LEMMA 12. The solution of the initial value problem(3), (5)has the representa- tion
v(x, t) = 4iL2(x, x, t) + 4i p∑ l=1
N−1∑ n=0
+Y ln(x, t)hl1n (x, t) ] , (5.7)
where the functionsXl n(x, t), Y
l n(x, t) are the solution of the system of linear alge-
braic equations
Xl k +
p∑ r=1
p∑ r=1
46 WERNER KIRSCH AND VLADIMIR KOTLYAROV
The coefficients of this system are as follows.
Alrks(x, t) = N−k−1∑ m=0
ω (l) km(t)
] dy,
ω (l) km(t)
] dy,
ω (l) km(t)h
ω (l) km(t)h
l1 m(x, t).
Since estimates (5.4) and (5.5) hold, Lemmas 5–9 are true under the restriction that x, t ∈ DN . For the Jost type function (5.6), instead of Lemma 10 we obtain the following one.
LEMMA 13. Letx, t ∈ DN . Then fort →∞
hl1k (x, t) = exp(iElξ) k∑
m=0
hl2k (x, t) = t−1/2 exp(iElξ) k∑
m=0
m!(k −m)! ∂k−m
] , X = x/2t.
The functionεlkm(X, t) = o(1), as t → ∞, and the functionshl2km(X, t) are uniformly bounded with respect toX and t .
Sketch of the proof. First of all, we notice that
h lj
∂Ekl h lj
0 (x, t).
As in Lemma 10, we obtain
hl10 (x, t) = eiElξB0(X,El) [ 1+ ε1
l (X, t) ] , hl20 (x, t) =
1√ t
eiEl ξhl200(X, t),
whereε1 l (X, t) = o(1) for t →∞, hl200(X, t) is uniformly bounded. These relations
may be differentiated with respect toEl, and the estimates will be still valid.
COROLLARY 6. Since
t
( ln t
Alrks(x, t) = t−1/2 N−k−1∑ m=0
ω (l) km(t)
m∑ p=0
s∑ q=0
Hlrpq ms (t)J
lr p+q(ξ),
ω (l) km(t)
m∑ p=0
s∑ q=0
Glrpq ms (t)J
lr p+q(ξ),
H lrpq ms (t) = hl1mp(t)hr2sq(t)+ hl2mp(t)hr1sq(t),
Glrpq ms (t) = hl1mp(t)hr1sq(t)+ hl2mp(t)hr2sq(t)
are valid with the functionsω(l)km(t), H lrpq ms (t), andGlrpq
ms (t), that are uniformly bounded with respect tot .
The system of Equations (5.8) reduces to the following system
Xl k +
48 WERNER KIRSCH AND VLADIMIR KOTLYAROV
p∑ r=1
BlrksX r s + Y lk = blk, k = 0,1, . . . , N − 1.
Let us introduce the block matrix(ξ, t) = Blr(ξ, t)plr=1. Then the system of these equations takes the form
X +Y = 0,
v(x, t) = 2i ∂
∂ξ ln
det[I +(ξ, t)] det[I +(ξ, t)] + o(1), ξ = x − Ct, t →∞,
for the solution of the problem (3), (5). Hence Equation (7) is valid and Theorem 2 is proved.
Now let us shortly consider Theorem 3. In this case, the entries of the matrix (x, t) have the form
kj(ξ, t) = N−k−1∑ m=0
ωkm(t)wmj(ξ, t),
j∑ q=0
where
ωkm(t) = −ih00(k +m+ 3/2) [ k!m!(4a)2(k+m+1)tk+m+3/2]−1[1+ ψkm(t)]
with some constanth0, the gamma-function0(k +m+ 3/2) and an estimate for
ψkm(t) = O
( k2+m2
) .
The determinant1(ξ, t) = det[I +(ξ, t)] can be written in a standard way
1(ξ, t) = N∑ k=1
Dk(ξ, t), Dk(ξ, t) = ∑
SOLITON ASYMPTOTICS OF SOLUTIONS OF THE SINE-GORDON EQUATION 49
If n 6 [N+1 2 ], then the determinant of the matrixwmj (x, t) of n+1 order can be
represented as follows
det wmj(x, t) =
... ...
)2n+2 det Im+j (ξ).
By using the technique from [12] we find
1(ξ, t) = 1+ N∑ n=1
inPn(X, t)t −n(n+1/2) e−2naξ
[ 1+ δn(t)
, n 6 [ N + 1
2
] with someδn(t) = O(n2/t). This expansion leads to Equations (10), (11) with
α0 n =
1
] , (5.11)
where0(n) and0(n)1 are the determinants with the gamma-function entries0(k + j + 3/2) and0(k + j + 1). Theorem 3 is proved.
Below we give a sketch of the proof of Theorem 4. In this case
(ξ, t) = ( A(ξ, t) B(ξ, t)
−B(ξ, t) −A(ξ, t) ) ,
ωkm(t)w 1 mj(ξ, t),
ωkm(t)w 2 mj(ξ, t),
w1 mj(ξ, t) =
m∑ p=0
j∑ q=0
ωkm(t) = inh0 eid0t cn+1 1 0(n+ 3/2)
(k!m!)tn+3/2(1− id1)n+3/2 ×
Using the same scheme as in [12], we notice that
1(k,l)(ξ, t) = A B
−B −A = ω 0
0 −ω = h− 0
,
,
0 0 · · · h1 k−1k−1
. This allows us to obtain an expansion analogous to (5.10), which leads to Equa-
tions (12)–(14) with
1/2−2n ]} , (5.12)
γ 0 n =
[(n− 1)!2]|1− id1|2n−1/20 (n−1) 1
[ |ReE| 2|E| ImE
Acknowledgements
V.K. thanks the Fakultät und Institut für Mathematik, Ruhr Universität Bochum for kind hospitality and the DFG for financial support.
References
1. Novikov, S. P., Manakov, S. V., Pitaevsky, L. P. and Zakharov, V. E.:Theory of Solitons. The Inverse Scattering Method, Nauka, Moscow, 1980, 319 pp. English transl. Plenum, New York, 1984.
2. Faddeev, L. D. and Takhtadjan, L. A.:Hamiltonian Methods in the Theory of Solitons, Nauka, Moscow, 1986. English transl. Springer-Verlag, 1987.
3. Schuur, P. C.: Asymptotic analysis of soliton problems,Lect. Notes in Math.1232, Springer- Verlag, 1986.
4. Its, A. R.: Method of the isomonodromic deformation in the theory of completely integrable nonlinear evolution systems, Doctoral dissertation, Leningrad University, 1986.
5. Novokshenov, V. Yu.: Asymptotics ast → ∞ of the solution of the Cauchy problem for the nonlinear Schrödinger equation,Sov. Math. Dokl.21(2) (1980), v. 251, 529–533.
6. Kotlyarov, V. P.: Asymptotic analysis of the Marchenko integral equation and soliton asymptotic of a solution of the nonlinear Schrödinger equation,Physica D87(1–4) (1995), 176–185.
7. Shabat, A. B.: On the Korteweg–de Vries equation,Sov. Math. Dokl.14 (1973), 1266–1269. 8. Khruslov, E. Ya.: Asymptotics of the solution of the Cauchy problem for the Korteweg–de
Vries equation with the step-like initial data,Mat. Sbornik99(2) (1976), 261–281. 9. Fokas, A. S. and Zakharov, V. E. (eds.):Important Developments in Soliton Theory, Springer-
Verlag, 1993. 10. Fokas, A. S. and Its, A. R.: An initial-boundary value problem for the sine-Gordon equation in
laboratory coordinates,Teoreticheskaya i Matematicheskaya Fizika92(3) (1992), 387–403. 11. Fokas, A. S. and Its, A. R.: The linearization of the initial-boundary value problem of the
nonlinear Schrödinger equation,SIAM J. Math. Anal.27(3) (1996), 738–764. 12. Kotlyarov, V. P.: Asymptotic solitons of the sine-Gordon equation,Theor. and Math. Physics
80(1) (1989), 15–28. 13. Kotlyarov, V. P.: Influence of double continuous spectrum of the Dirac operator on the nonlinear
Schrödinger equation asymptotic solitons,Matem. Zametki49(2) (1991), 84–94. (Math. Notes (1991) (1–2), 172–180.)
14. Reed, M. and Simon, B.:Methods of Modern Mathematical Physics. 1. Functional Analysis, Academic Press, NY, 1972.
Mathematical Physics, Analysis and Geometry2: 53–81, 1999. © 1999Kluwer Academic Publishers. Printed in the Netherlands.
53
NAKAO HAYASHI Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjukuku, Tokyo 162, Japan; e-mail: nhayashi@rs.kagu.sut.ac.jp
PAVEL I. NAUMKIN Instituto de Física y Matemáticas, Universidad Michoacana AP 2-82, CP 58040, Morelia, Michoacan, Mexico; e-mail: naumkin@ifm1.ifm.umich.mx
(Received: 30 September 1998, accepted: 16 March 1999)
Abstract. We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödin- ger equations
iut +1u = N (v), (t, x, y) ∈ R3, u(0, x, y) = u0(x, y), (x, y) ∈ R2, (A)
where the Laplacian1 = ∂2 x + ∂2
y , the solutionu is a complex valued function, the nonlinear term N = N1+N2 consists of the local nonlinear partN1(v)which is cubic with respect to the vectorv = (u, ux, uy, u, ux, uy) in the neighborhood of the origin, and the nonlocal nonlinear partN2(v) = (v, ∂−1
x Kx(v)) + (v, ∂−1 y Ky(v)), where(·, ·) denotes the inner product,∂−1
x = ∫ x−∞ dx′, ∂−1 y =∫ y
−∞ dx′ and the vectorsKx ∈ (C4(C6;C))6 andKy ∈ (C4(C6;C))6 are quadratic with respect
to the vectorv in the neighborhood of the origin. We assume that the componentsK (2) x = K
(4) x ≡
0, K(3) y =K(6)
y ≡ 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with
N1 = |u|2u, K(1) x = ∂y(|u|2), K(1)
y = ∂x(|u|2), and all the rest components of the vectorsKx
andKy are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when
N1 = (1+|u|2)−1u(∇u)2, andK(3) x = −K(2)
y = (1+|u|2)−2(uxuy −uxuy). Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely ifN (eiθ v) = eiθN (v) for all θ ∈ R.
Mathematics Subject Classification (1991):35Q55.
Key words: Davey–Stewartson system, Ishimori system, elliptic–hyperbolic case, nonlocal nonlinear Schrödinger equation.
1. Introduction
We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations{
iut +1u = N (v), (x, y) ∈ R2, t ∈ R, u(0, x, y) = u0(x, y), (x, y) ∈ R2,
(1.1)
54 N. HAYASHI AND P. I. NAUMKIN
where the solutionu is a complex valued function, the Laplacian is1 = ∂2 x + ∂2
y , and the nonlinear termN (v) = N1(v)+N2(v) consists of the local nonlinear part N1(v), which is cubic with respect to the vectorv = (u, ux, uy, u, ux, uy) in the neighborhood of the origin, and the nonlocal nonlinear partN2(v) = (v, ∂−1
x Kx(v))
+ (v, ∂−1 y Ky(v)), where(·, ·) denotes the inner product,∂−1
x = ∫ x −∞ dx′, ∂−1
y =∫ y −∞ dy′ and the vectorsKx andKy are quadratic with respect to the vectorv in the
neighborhood of the origin. We assume thatN1(z) ∈ C4(C6;C), N1(z) = O(|z|3) asz → 0, Kx(z), Ky(z) ∈ (C4(C6;C))6, Kx(z) = O(|z|2), Ky(z) = O(|z|2) asz → 0. We also assume that the componentsK(2)
x ≡ K(4) x ≡ 0, andK(3)
y ≡ K(6) x ≡ 0. We show how to translate the Davey–Stewartson and Ishimori systems to the
nonlinear nonlocal Schrödinger Equation (1.1) and we include some historical comments and remarks about previous results for these systems. The Davey–Ste- wartson (D-S) system is written as iut + αuxx + uyy = a|u|2u+ buωx, (x, y) ∈ R2, t ∈ R,
ωxx + βωyy = ∂x |u|2, (x, y) ∈ R2, t ∈ R, u(0, x, y) = u0(x, y), (x, y) ∈ R2,
(1.2)
whereα, β ∈ R, a, b ∈ C. This system was derived by Davey and Stewartson [11], Benney and Roskes [6] and Djordjevic and Redekopp [12] for describing the evolution of weakly nonlinear water waves that travel predominantly in one direction, and where the wave amplitude is modulated slowly in the horizontal direction. Independently, Ablowitz and Haberman [1] and Cornille [10] obtained a particular form of system (1.2) as an example of a completely integrable system generalizing the one-dimensional Schrödinger eguation. Djordjevic and Redekopp in [12] have shown that the parameterβ can become negative when the capillarity effects are important.
In the literature on