symmetry methods for differential equations and their applications...
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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!