Maple software package for computation of conservation laws of nonlinear self-adjoint systems of differential equations L. Galiakberova \Group Analysis of Mathematical Models in Natural and Engineering Sciences" research laboratory, Ufa State Aviation Technical University June 8, 2013 Methods for conservation laws construction 1. Noether theorem for Euler-Lagrange systems. Computer implementation: jets in Maple (M. Barakat, G. Hartjen). 2. Direct method for non-degenerate systems. Computer implementation: GeM in Maple (A. Cheviakov), ConLaw in Reduce (T. Wolf). 3. A-operator method (Yu. Chirkunov). Computer implementation: Maple implementation (K. Imamutdinova). 4. Ibragimov theorem for nonlinear self-adjoint equations. Computer implementation: CL SAS in Maple (A. Araslanov, L. Galiakberova). Conservation laws Consider differential system Fα¯(x; u; u(1); : : : ; u(k)) = 0; (F) with x = (x1; : : : ; xn), u = (u1; : : : ; um), α α α u(1) = fui g; ui = Di(u ); ::: u = fuα g; uα = D :::D (uα); (k) i1:::ik i1:::ik i1 ik @ α @ α @ Di = i + ui α + uij α + :::; @x @u @uj α = 1; : : : ; m; α¯ = 1;:::; m:¯ Def. Vector C = (C1;:::;Cn) with components i i C = C (x; u; u(1);::: ) is a conservation law of (F) if i Di(C )j(F) = 0: Ibragimov theorem System under consideration: Fα¯(x; u; u(1); : : : ; u(k)) = 0: (F) Def. Adjoint equations to (F): β¯ δ(v F ¯) F ∗(x; u; v; : : : ; u ; v ) ≡ β = 0; (F*) α (k) (k) δuα where v = (v1; : : : ; vm¯ ) | new dependent variables, δ/δuα | the Euler-Lagrange operator: 1 δ @ X @ = + (−1)sD :::D : δuα @uα i1 is @uα s=1 i1:::is Ibragimov theorem Def. (F) is nonlinearly self-adjoint if (F*)-equations are satisfied for all solutions of (F) upon a certain (nonzero) substitution α¯ α¯ v = ' (x; u; u(1); : : : ; u(r)); α¯ = 1;:::; m;¯ (Φ) i.e. the following equations hold: ∗ Fα(x; u; '; : : : ; u(k);'(k))j(F) = 0 or equivalently ∗ β¯ iβ¯ Fα(x; u; '; : : : ; u(k);'(k)) = µαFβ¯ + µα Di(Fβ¯) + :::: with certain multiplier functions µ. Ibragimov theorem Theorem. Any Lie point, Lie-Backlund or nonlocal symmetry @ @ X = ξi(x; u; u ;::: ) + ηα(x; u; u ;::: ) (1) @xi (1) @uα of nonlinear self-adjoint system (F) leads to a conservation law i Di(C )j(F) = 0 constructed as i i β¯ C = N (v Fβ¯)jv='; where 1 δ X δ N i = W α + D :::D (W α) ; i = 1; : : : ; n; δuα i1 is δuα i s=1 ii1:::is α α j α W = η − ξ uj : * Algorithm of the conservation law analysis of nonlinear self-adjoint systems Main stages of the algorithm 1. Construction of the adjoint system F ∗. 2. Calculation of substitution vα¯ = 'α¯(x; u; : : : ) that provides nonlinear self-adjointness for system F . 3. Construction of a conserved vector C corresponding to symmetry operator X. 4. Simplification of conserved vectors by addition trivial conserved vectors. Conserved vector C = (C1;:::;Cn) is trivial if satisfies one of the following conditions: i Di(C ) ≡ 0 (A) i C j(F) = 0 (B) C = CA + CB Algorithm of the conservation law analysis of nonlinear self-adjoint systems Main stages of the algorithm 1. Construction of the adjoint system F ∗. 2. Calculation of substitution vα¯ = 'α¯(x; u; : : : ) that provides nonlinear self-adjointness for system F . 3. Construction of a conserved vector C corresponding to symmetry operator X. 4. Simplification of conserved vectors by addition trivial conserved vectors. * Conserved vector C = (C1;:::;Cn) is trivial if satisfies one of the following conditions: i Di(C ) ≡ 0 (A) i C j(F) = 0 (B) C = CA + CB Algorithm of the conservation law analysis of nonlinear self-adjoint systems of differential equations Additional procedures I Verification if a vector satisfies the conservation equation for a given system: i α¯ iα¯ Di(C ) = µ Fα¯ + µ Di(Fα¯) + :::: 1 n I Verification if a conserved vector C = (C ;:::;C ) is trivial and satisfies one of the following conditions: i Di(C ) ≡ 0 (A) i C j(F) = 0: (B) Adjoint system construction. Automatization Kadomtsev-Petviashvili equation in the form of the system u − uu − u − w = 0; t x xxx y (KP) wx − uy = 0: Calculation of substitution that provides nonlinear self-adjointness for a system. Automatization Thomas equation: uxy + αuy + βux + γuxuy = 0; (Th) with constants α > 0, β > 0, γ 6= 0. Adjoint equation vxy − (α + γux)vy − (β + γuy)vx − 2γuxyv = 0: (ThA) Calculation of substitution that provides nonlinear self-adjointness for a system. Automatization Nonlinear wave equation with dissipation: ut + utt = (f(u)ux)x − (g(u)uy)y (WE) with arbitrary f(u), g(u). Adjoint equation: −vt + vtt = f(u)vxx + g(u)vyy (WEA) Construction of conserved vectors corresponding to a certain symmetry operator. Automatization Thomas equation (Th) is nonlinearly self-adjoint due to the substitution γu ' = B(x; y)e ;Bxy − αBy − βBx = 0: (ThP) @ Conserved vector corresponding to the symmetry operator X3 = @u : Construction of conserved vectors corresponding to a certain symmetry operator. Automatization After addition of the trivial conserved vector 1 1 D (Beγu); − D (Beγu) 2 y 2 x conserved vector C takes the form 1 γu 2 γu C = (γuy + β)Be ;C = (αB − Bx)e : Application. Model of atmospheric flows Atmospheric flows model: 1 Λ ≡ cos θ + sin θ + (AF) tθ tθθ sin θ t'' 1 1 + + 2k ' sin θ + k θ' cos θ + θθ' sin θ + ''' R0 sin θ cos θ 1 + " + + ''' sin θ θ' θθ' sin2 θ θ cos θ 1 2 cos θ − " + + ( − ) − '' ' = 0; sin θ θθ θθθ sin2 θ θ'' θ sin3 θ where = (t; θ; '); k, ", R0 | nonzero constant parameters. Equation (AF) is self-adjoint , i.e. the adjoint equation Λ∗ ≡ Λ∗(t; θ; '; ; v; : : : ) = 0 coincides with (AF) upon the substitution v = . Application. Model of atmospheric flows @ @ Symmetries X1 = @t and X2 = @' provide only trivial conserved vectors. @ For symmetry X3 = λ(t) @ with arbitrary function λ(t) the following nontrivial conserved vector was obtained for which the conservation equation is satisfied in the form 1 2 3 Dt(C ) + Dθ(C ) + D'(C ) = λ(t)Λ: Application. Model of atmospheric flows Symmetries @ @ @ X = 2R "t − "t + [(1 + 2kR ) cos θ − 2"R ] ; 4 0 @t @' 0 0 @ @ cos θ @ sin θ @ X5 = " sin h + " cos h − k sin θ + sin h ; @θ sin θ @' 2R0 @ @ cos θ @ sin θ @ X6 = " cos h − " sin h − k sin θ + cos h ; @θ sin θ @' 2R0 @ where h = ' + t , provide nontrivial conserved vectors with 2R0 conservation equations 1 2 3 Dt(C ) + Dθ(C ) + D'(C ) = ((2kR0 + 1) cos θ − 4"R0 )Λ; 1 2 3 Dt(C ) + Dθ(C ) + D'(C ) = −"( sin h + ' cos h)Dθ(Λ) − "( θ sin h + θ' cos h) + k + 1=(2R0) sin θ sin h Λ; 1 2 3 Dt(C ) + Dθ(C ) + D'(C ) = "( ' sin h − cos h)Dθ(Λ) − "( θ' sin h − θ cos h) − k + 1=(2R0) sin θ cos h Λ: Application. Nonlocal conservation laws Nonlinear heat equation ux ut = 2 1 + u x admits nonlocal symmetry @ @ X = −w + (1 + u2) ; @x @u u where w = u, w = x . Nonlocal conservation law: x t 1 + u2 1 2 C = (1 + u + wux)'; 2 ux ux C = 'x − w' 2 + (w'x − u') 2 ; 1 + u x 1 + u where ' = a1x + a2 with constants a1, a2. Conservation equation: 1 2 1 ux Dt(C ) + Dx(C ) = ut − 2 2'u 1 + u x Application. Approximate conservation laws Perturbed Korteweg-de Vries equation ut = uxxx + uux + "u admits approximate symmetry 9 @ @ @ X = 3t − "t2 + x(1 − 3"t) + (−2u + 6"tu + 3"x) : 2 @t @x @u Approximate conservation law: 3 C1 = u2 − 2" xu + tu2 ; 2 2 C2 = u2 − u3 − 2uu x 3 xx 2 3 2 + "(xu − 2ux + 2xuxx + 2tu − 3tux + 6tuuxx): Conservation equation: 1 2 Dt(C ) + Dx(C ) = 2u(ut − uxxx − uux − "u) − 2"(x + 3tu)(ut − uxxx − uux) Thank you for attention.