Symmetry Methods for Differential Equations and Their Applications...
Transcript of Symmetry Methods for Differential Equations and Their Applications...
Symmetry Methods for Differential Equations and Their Applications in Mathematical Modeling
Alexey Shevyakov, University of Saskatchewan
Symmetry methods:
Applicability to virtually any DE model, linear/nonlinear Usefulness for DE analysis and solution Further problems of mathematical interest
Mathematicalmodel
Nonlinear PDE problem
Solution
Analysis
Exact
Approximate
Numerical
Symmetry transformation: maps an object into itself.
Symmetries of an equilateral triangle:
e g1 g2 g3 g4 g5
Rotations of a circle: a continuous group
1-parameter Lie group of point transformations
Composition:
(x, y)
(x1, y1)
x
y
a
x1 = f(x, y; a) = x cos a− y sin a,y1 = g(x, y; a) = x sin a+ y cos a.
f¡f(x, y; a), g(x, y; a); b
¢= f(x, y; a+ b)
g¡f(x, y; a), g(x, y; a); b
¢= g(x, y; a+ b)
Symmetry Transformations: Geometrical Picture
Global symmetry action
(x, y)(x1, y1)
Flow generated by a tangent vector field (TVF)
X = [ξ, η]
(x, y)
X = ξ∂
∂x+ η
∂
∂y.
M M
x1(a) = f(x, y; a) = x+ aξ(x, y) +O(a2) = eaXx
y1(a) = g(x, y; a) = y + aη(x, y) +O(a2) = eaXy
Group action:
Tangent vector field (TVF):
One-parameter groups and tangent vector fieldsOne-parameter Groups and Tangent Vector Fields
Group TVF :
ξ(x, y) =¡∂∂af(x, y; a)
¢ ¯a=0,
η(x, y) =¡∂∂ag(x, y; a)
¢ ¯a=0
Invariance condition:
Finding tangent vector fields:
Curve
Example: circle
XF (x, y)|F (x,y)=0 = 0.M : F (x, y) = 0.
x
yX = [−y, x]
XF (x, y) =
µ−y ∂
∂x+ x
∂
∂y
¶(x2 + y2 − 1) ≡ 0.
x2 + y2 = 1.
One-parameter groups and tangent vector fieldsOne-parameter Groups and Tangent Vector Fields
Flow generated by a tangent vector field (TVF)
X = [ξ, η]
(x, y)M
Global symmetry action
(x, y)(x1, y1)
M
Invariants:
is an invariant ifI(x, y) XI(x, y) = 0.
Invariance condition:
Finding tangent vector fields:
Curve
XF (x, y)|F (x,y)=0 = 0.M : F (x, y) = 0.
One-parameter groups and tangent vector fieldsOne-parameter Groups and Tangent Vector Fields
Flow generated by a tangent vector field (TVF)
X = [ξ, η]
(x, y)M
Global symmetry action
(x, y)(x1, y1)
M
Given an ODE
find groups of transformationspreserving (1):
½x1 = f(x, y; a)y1 = g(x, y; a)
Then solution is transformed into a solutiony(x) y1(x1).
y(n)(x) = G(x, y(x), y0(x), ..., y(n−1)(x)), (1)
Example 1: y0(x) = y(x).
x
y Cex
Translation:½x1 = x+ ay1 = y
Ce−aex
y = Cex ⇒ y1(x1) = y(x) = y(x1 − a) = Ce−aex1 .
TVF: X = [1, 0] =∂
∂x.
Point Transformations Admitted by ODEs
Point transformations admitted by ODEs
Example 1: y0(x) = y(x).
x
y Cex
Translation:½x1 = x+ ay1 = y
Ce−aex
TVF:
y = 0: invariant solution.
X = [1, 0] =∂
∂x.
Point Transformations Admitted by ODEs
Given an ODE
find groups of transformationspreserving (1):
½x1 = f(x, y; a)y1 = g(x, y; a)
Then solution is transformed into a solutiony(x) y1(x1).
y(n)(x) = G(x, y(x), y0(x), ..., y(n−1)(x)), (1)
Point transformations admitted by ODEsPoint Transformations Admitted by ODEs
Example 2: y0(x) = y(x).
x
y
Cex
Scaling:
TVF:
½x1 = xy1 = e
ay
Ceaex
y = 0: invariant solution.
X = [0, y] = y∂
∂y.
y = Cex ⇒ y1(x1) = eay(x1) = Ce
aex1 .
Given an ODE
find groups of transformationspreserving (1):
½x1 = f(x, y; a)y1 = g(x, y; a)
Then a solution is transformed into a solutiony(x) y1(x1).
y(n)(x) = G(x, y(x), y0(x), ..., y(n−1)(x)), (1)
Reduction of order:ODEs: each point symmetry reduction of order by 1.
Example: 2y000 + yy00 = 0 admits two point symmetries
X1 =∂
∂x, X2 = x
∂
∂x− y ∂
∂y,
and can be mapped into dV
dU=V
U
µ 12 + V + U
2U − V
¶.
Applications of Point Transformations to ODEs
All point symmetries of ODE (and PDE) systems can be algorithmically computed (theoretically).
Applications: • Reduction of order;• Invertible mappings to other ODEs;• Construction of exact invariant solutions.
Invertible mappings: solution set is preserved
E.g.: Any 2nd-order ODE with 8 point symmetries
Example: A nonlinear Liénard system
has 8 symmetries, is invertibly mapped into
y00 = 0.
x(t) +hb+ 3kx(t)
ix(t) + k2x3(t) + bkx2(t) + λx(t) = 0
solution is obtained [Bluman, Shev., Senthilvelan; J. Math. An. App. (2008)]
and a general
Application: astrophysics, expansion / collapse of a spherical gas cloud.
All point symmetries of ODE (and PDE) systems can be algorithmically computed (theoretically).
Applications: • Reduction of order;• Invertible mappings to other ODEs;• Construction of exact invariant solutions.
Applications of Point Transformations to ODEs
X00(T ) = 0,
Applications of Point Transformations to PDEs
Applications:• Reduction of # of variables; exact invariant solutions;• Generation of new solutions from known ones;• Infinite # of symmetries invertible mapping
into a linear system.
Symmetries of PDEs: e.g.
Tangent vector field:
x1 = f(x, t, u; a) = x+ aξ(x, t, u) +O(a2);
t1 = g(x, t, u; a) = t+ aτ(x, t, u) +O(a2);
u1 = h(x, t, u; a) = u+ aη(x, t, u) +O(a2);
X = ξ∂
∂x+ τ
∂
∂t+ η
∂
∂u.
ut = (uνux)x, u = u(x, t).
Example 1: solitons of KdV as invariant solutions.
KdV: ut + 6uux + uxxx = 0, u = u(x, t).
Admitted translations:
Linear combination:
Invariants:
Invariant solution: u = ϕ(x− ct); − cϕ0 + 6ϕϕ0 + ϕ000 = 0.
I1 = x− ct, I2 = u.
x→ x+ a : X1 =∂∂x ,
t→ t+ b : X2 =∂∂t .
X = c ∂∂x +∂∂t , c = const.
u(x, t) = c2 cosh
−2h√
c2 (x− ct)
i.
Traveling wave (soliton) solution:
Applications of Point Transformations to PDEs
Example 2: source solution of the heat equation (infinite rod).
0 x
Unit energy release at time 0
Temperature function: u(x, t), −∞ < x <∞, t > 0.
PDE Problem:
Two admitted symmetry transformations:
Solution invariant under both symmetries:
u(x, 0) = δ(x), limx→±∞ u(x, t) = 0.
1)
⎧⎨⎩x1 = αx,t1 = α2t,u1 =
1αu.
(the well-known 1D Green’s functionfor the heat equation.)
2)
⎧⎨⎩x1 = x− βt,t1 = t,
u1 = u eβx/2−β2t/4.
ut = uxx,
u(x, t) =C√te−x
2/4t.
(Without solving the PDE.)
Applications of Point Transformations to PDEs
Typical experimental setup [Pelce-Savornin et al (1988); Strehlow et al (1987)]
Observed properties:• Flame speed: 10 – 25 cm/s
• Flame tip: inside or on the wall
• Flame front:
Thin (~0.5 mm)
Flat or paraboloidal( + zero Neumann BCs )
10 cm
1-2
m
Gas mixture
Flame front
Combustion products
Flame tip
Premixed Flames: Experimental background
Flame front model
S = S(x, t), x ∈ ΩFlame front:
Nonlinear reaction-diffusion problem with a small parameter .[Rakib, Sivashinsky (1987)]
εModel:
Properties for rectangular domain:[Berestycki et al (2006)]
• When flame front is paraboloidal, the tip stays inside the tube exponentially long (proven);
• Flame tip moves towards the nearest wall (numerical)
S(x, t)
Ω
0 < ε¿ ε0,
Tube of general cross-section: what can we say?
An asymptotic estimate on the flame tip speed?
Flame front model
S = S(x, t), x ∈ ΩFlame front:
Change of variables:
Problem for(ut = ε2∆u+ u log u,
∂nu|∂Ω = 0.u(x, t):
S = 2ε2 log u(x, t) + f(t)S(x, t)
Ωx0
To estimate tip velocity1. Find a static solution and leading eigenpairs
far from the boundary.
2. Find first terms (in ) of solution and eigenpairs in a boundary layer; matching with far-field;
3. Search for a slowly moving solution
x0(t):
ε
u(x, t) = u(x;x0(t); ε) + E(x, t);
From the condition can be found.|E| ¿ |u|, x0(t)
u(x;x0)
ε2
∂Ω
Ω
x0
Flame front model
Point symmetries include:
How do we find an equilibrium solutionut = ε2∆u+ u logu
u(x;x0)
far from the boundary?of
X1 =∂∂t , X2 = −e
tx2ε2u
∂∂u + e
t ∂∂x , X3 = − e
ty2ε2u
∂∂u + e
t ∂∂y .
An exact solution invariant w.r.t. X1 ,X2 ,X3 :
u(x;x0) = exp
½1− |x− x0|
2
4ε2
¾.
Center: x0 ∈ Ω.
Width: ∼ ε.
Flame front model
Narrow spike (~ )ε
u(x;x0) = exp
½1− |x− x0|
2
4ε2
¾S(x;x0) ∼ − 1
2|x − x0|2
Parabolic flame front: S ∼ ln u,
Exponentially small error in BCs exponentially slow spike motion
x2
x1 Hom. Neumann BCs: ∂nu|∂Ω = 0
Flame front model
Tube cross-section
Principal result: equation of flame tip motion:
x0
x1
x2
d0
x00 ∼ − d0|d0|
q2π
d20ε√1−κ0d0 e
−d20/(2ε2)
• Flame tip moves asymptotically exponentially slowly in to the closest point on the wall. [Shev., Ward, Interfaces and Free Boundaries (2007)]
• Good agreement with numerical simulations (for rectangle).
ε
Euler equations of gas/fluid dynamics:
ρVt + ρ(V · grad)V = −grad P
(incompressibility)divV = 0
ρt + div ρV = 0
x∈Ω⊆R3
V:
P :
ρ:
gas velocity
pressure
density
x V
P, ρ
Symmetries in Plasma Models
A tokamak
Thermonuclear fusion:Plasma confinement(TOKAMAKs etc.)
• T ~ 107 – 109 K
• n ~ 1020 m-3
Symmetries in Plasma Models
−B× curl BBt = curl(V ×B)ρVt + ρ(V · grad)V = −grad P
(incompressible)divV = 0
ρt + div ρV = 0
Magnetohydrodynamics (MHD) equations:
divB = 0
x∈Ω⊆R3
x V
P, ρ
B
Symmetries in Plasma Models
Astrophysical jets:• L ~ 103 - 106 light years;
• Self - collimated (cone angle <20o)
−B× curl BBt = curl(V ×B)ρVt + ρ(V · grad)V = −grad P
(incompressible)divV = 0
ρt + div ρV = 0
Magnetohydrodynamics (MHD) equations:
divB = 0
x∈Ω⊆R3
x V
P, ρ
B
Symmetries in Plasma Models
Earth magnetosheath:• Deflects solar wind
−B× curl BBt = curl(V ×B)ρVt + ρ(V · grad)V = −grad P
(incompressible)divV = 0
ρt + div ρV = 0
Magnetohydrodynamics (MHD) equations:
divB = 0
x∈Ω⊆R3
x V
P, ρ
B
Symmetries in Plasma Models
B and V are tangent to 2D magnetic surfaces.
MHD equilibrium equations: No dependence on time.
System: 9 equations, 8 dep., 3 indep. variables.
Admitted point symmetries: • Translations• Rotations• Scalings• Two infinite families of symmetries (involving arbitrary functions)
In a bounded domain:
• nested tori[Alexandroff, Hopf (1935)]
divV = 0,
divB = 0,
div ρV = 0,
ρ(V · grad)V = −grad P −B× curl B,curl(V×B) = 0.
Arbitrary: a(x), b(x);
Applications: Any known solution family of solutions; Static (V = 0) Dynamic (V 0); Physically trivial Nontrivial.
B → B1 = b(x)B+ c(x)√ρV,
V → V1 =c(x)
a(x)√ρB+
b(x)
a(x)V,
ρ → ρ1 = a2(x)ρ,
P → P1 = CP + (CB2 −B21)/2,
Symmetries in Plasma Models
b2(x)− c2(x) = C
Infinite symmetries:[Bogoyavlenskij (2000)],also [Shev., Phys. Lett. A (2004)], [Shev. & Bogoyavlenskij, J. Phys. A (2004)]
Example 1: Earth Magnetosheath model
Start from another vacuum magnetic field:divB = 0, curlB = 0 ⇒ B = gradΦ, ∆Φ = 0.
Laplace’s equation is separable in many coordinate systems, e.g. ellipsoidal exact solution inΦ(x) R3.
Apply infinite symmetries a physical plasma equilibrium,B,V, P, ρ 6= 0.
• Model can be further extendedto anisotropic plasmas.[Shev., Phys. Rev. Lett. (2005)]
A common jet:
• Self-collimated MHD effects;• Helically-symmetric?
Helical symmetry:
Known: an exact static (V=0) MHD solution with helical symmetry[Bogoyavlenskij (2000)]
Helical magnetic surfaces
x
Example 2: Helical Astrophysical Jet model
After applying infinite symmetries:• An infinite family of physical exact MHD solutions with motion; • Helical symmetry;• Extended to anisotropic plasma case. [Shev. and Bogoyavlenskij,
J. Phys. A (2004)]
DE system:
Variables: x = (x1, ..., xn), u = u(x) = (u1, ..., um).
• Algebraic in and derivatives!x, u,
Example: ut + uux + xt2 = 0, u = u(x, t).
• Point transformation: X(1)
x, t
u
ux, ut
1st prolongation: X(1) = ξ∂∂x
+ τ∂∂t+ η
∂∂u
+ η(1)(x)∂∂ux
+ η(1)(t)∂∂ut
.
depend on
Gi(x, u, ∂u, ..., ∂(N)u) = 0, i = 1, ...,M.
Point symmetries of any DE system are found algorithmically.
ξ, τ, η.
Computation of Point Symmetries of DEs
x1 = x + aξ + O(a2),
t1 = t + aτ + O(a2),
u1 = u + aη + O(a2),
(ux)1 = ux + aη(1)(x) + O(a
2),
(ut)1 = ut + aη(1)(t) + O(a
2).
Finding point transformations for a general DE system:
x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Gi(x, u, ∂u, ..., ∂(N)u) = 0, i = 1, ...,M ;
Computation of Point Symmetries of DEs
Symbolic software: • CRACK (T.Wolf, for REDUCE),• GeM (for Maple)
[Shev., Comp. Phys. Comm. (2007)]
Point transformations for and
1. Write down extended components in terms of
2. Determining equations:
3. do not depend on derivatives split linear PDE system;
4. Solve for
η(q) j(...)
ξi, ηj
ξi, ηj .
ξi, ηj ;
Both packages: point symmetries and much more…
X(N)Gi|Gi=0 = 0, i = 1, ...,M ;
(xi)1 = xi + aξi + O(a2);
(uj)1 = u+ aηj + O(a2).
xi uj :
Computation of Point Symmetries of DEs
Computational algorithm:
Example 1: Point symmetry computation for the KdV equation
Example 2: Point symmetry computation for the flame model
Computation of Point Symmetries of DEs
[Shev., Ward, Interfaces and Free Boundaries (2007)]
ut + 6uux + uxxx = 0
u = u(x, t),
ut = ε2∆u+ u log u
u = u(x, y, t),
Example 3: Point symmetry classification for the nonlinear wave equation
utt = (c2(u)ux)x
Computation of Point Symmetries of DEs
[Ames et al (1981)], [Bluman, Shev., J. Math. An. App. (2007)]
A common jet:
Summary
Symmetries of PDEs:
• General applicability
• Construction of exact solutions (invariant & transformed)
• Useful results for many applications;
• Multiple useful extensions (approximate, nonlocal symmetries ,...)
• Relations with conservation laws(Noether’s theorem & beyond)
A local conservation law:∂
∂tΦ(x, t, u, ...) +
∂
∂xΨ(x, t, u, ...) = 0.
For a given PDE system, its conservation laws can be found algorithmically.
Example: Nonlinear diffusion equation ut = (L(u))xx
1+1 dim. (independent variables: ; dependent: )x, t u(x, t).
admits two local conservation laws:
Applications of conservation laws: • Direct physical meaning;• Analysis (existence, stability…);• Numerical methods;• Nonlocally related systems.
Conservation Laws
∂
∂t(u)− ∂
∂x
³(L(u))x
´= 0,
∂
∂t(xu)− ∂
∂x
³x(L(u))x − L(u)
´= 0
Potential equations:∂
∂tΦ(x, t, u, ...) +
∂
∂xΨ(x, t, u, ...) = 0 ⇒
½vx = Φ(x, t, u, ...),vt = −Ψ(x, t, u, ...)
Potential system: given system + potential equations.
Framework of Nonlocally Related PDE Systems
Example: Potential systems for the nonlinear diffusion equation.
Given system:
Potential system 1:
Ux, t ;u : ut = (L(u))xx
Potential system 2:
∂∂t (u)− ∂
∂x
³(L(u))x
´= 0 ⇒ UVx, t ;u, v :
½vx = u,vt = (L(u))x.
∂∂t (xu)− ∂
∂x
³x(L(u))x−L(u)
´= 0 ⇒ UWx, t ;u,w :
½wx = xu,wt = x(L(u))x −L(u).
Framework of nonlocally related systems:• Given system nonlocally related potential systems, subsystems;
• Solution sets are equivalent;
• Nonlocal relations analysis new results [Many examples];
• Systematic procedure.
Framework of Nonlocally Related PDE Systems
Applications of the framework:
• Additional (nonlocal) symmetries• Additional (nonlocal) conservation laws• Exact solutions• Non-invertible linearizations
• Generalizes to multi-dimensions
Example: Nonlocally related PDE systems for Planar Gas Dynamics
Framework of Nonlocally Related PDE Systems
Euler system Ex, t ; v, p, ρ:
⎧⎨⎩ ρt + (ρv)x = 0,ρ(vt + vvx) + px = 0,ρ(pt + vpx) +B(p, 1/ρ)vx = 0.
Gx, t, v, p, ρ, r = 0 :
⎧⎪⎪⎨⎪⎪⎩rx − ρ = 0,rt + ρv = 0,rx(vt + vvx) + px = 0,rx(pt + vpx) +B(p, 1/rx)vx = 0.
A potential system:
Local change of variables:
Exclude x…
Gy, t, x, v, p, ρ = 0 :
⎧⎪⎪⎨⎪⎪⎩q − xy = 0,v − xt = 0,vt + py = 0,pt +B(p, q)vy = 0,
Example: Nonlocally related PDE systems for Planar Gas Dynamics
Framework of Nonlocally Related PDE Systems
Euler system Ex, t ; v, p, ρ:
⎧⎨⎩ ρt + (ρv)x = 0,ρ(vt + vvx) + px = 0,ρ(pt + vpx) +B(p, 1/ρ)vx = 0.
Obtain the Lagrange form of gas dynamics equations:
where
q = 1/ρ, y =R xx0ρ(ξ)dξ.
Ly, t, v, p, q = 0 :
⎧⎨⎩ qt − vy = 0,vt + py = 0,pt +B(p, q)vy = 0.
Example: Nonlocally related PDE systems for Planar Gas Dynamics
Framework of Nonlocally Related PDE Systems
Euler system Ex, t ; v, p, ρ:
⎧⎨⎩ ρt + (ρv)x = 0,ρ(vt + vvx) + px = 0,ρ(pt + vpx) +B(p, 1/ρ)vx = 0.
EA1A2x,t; v,p,,1,2
EA2x,t; v,p,,2
Ly,t; p,q
Ex,t; v,p,
EA1A2A3x,t; v,p,,1,2,3
EA2A3x,t; v,p,,2,3
Ly,t; v,p,q
tmptmpEA1x,t; v,p,,1 LXy,t; v,p,q,x
Euler (E) and Lagrange (L) descriptions, as well as other equivalent descriptions, arise in a common mathematical framework.
Framework of Nonlocally Related PDE Systems
⎧⎨⎩ ρt + (ρv)x = 0,ρ(vt + vvx) + px = 0,ρ(pt + vpx) +B(p, 1/ρ)vx = 0.
Other physical objects related to nonlocally related PDE systems:
• Electromagnetic potentials
• Stream function and vorticity form of fluid dynamics equations
• Magnetic surfaces (flux function) in MHD:
Bt = curl(V ×B) ⇒ V ×B = grad Φ
Some References
Symmetries, conservation laws, nonlocal framework:
• G. Bluman, S. Kumei, “Symmetries and Differential Equations.” Springer: Applied Mathematical Sciences, Vol. 81 (1989).
• G. Bluman, A. Cheviakov, S. Anco, “Applications of Symmetry Methods to Partial Differential Equations.” Springer: Applied Mathematical Sciences, Vol. 168 (2010)
• G. Bluman, A. Cheviakov, S. Anco, Construction of Conservation Laws: How the Direct Method Generalizes Noether's Theorem (2009).
Symbolic symmetry computations:• A. Cheviakov, GeM software package for computation of symmetries and conservation laws of DEs, Comp. Phys. Comm. 176 (2007), 48-61.
Flame front model:• A. Cheviakov, M. Ward, A two-dimensional metastable flame-front and a degenerate spike-layer problem, Interfaces and Free Boundaries 9 (2007), 513 - 547.
Web: math.usask.ca/~shevyakov
Thank you for your attention!