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.



X = [ξ, η]

(x, y)M


(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.



X = [ξ, η]

(x, y)M


(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


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


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


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


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:


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.)


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


−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


Astrophysical jets:• L ~ 103 - 106 light years;

• Self - collimated (cone angle <20o)



ρt + div ρV = 0


divB = 0

x∈Ω⊆R3

x V

P, ρ

B


Earth magnetosheath:• Deflects solar wind



ρt + div ρV = 0


divB = 0

x∈Ω⊆R3

x V

P, ρ

B


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,


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 ;


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 :


Computational algorithm:

Example 1: Point symmetry computation for the KdV equation

Example 2: Point symmetry computation for the flame model


[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


[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.


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


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,





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.





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.



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!

Symmetry Methods for Differential Equations and Their Applications...

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