Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations

Transcription

1 Symbolic Computation of Conservation Laws of Nonlinear Partial Differential Equations Willy Hereman Department of Applied Mathematics and Statistics Colorado School of Mines Golden, Colorado whereman/ Applied Mathematics Seminar University of Manchester, Manchester, U.K. Tuesday, July 18, 2017, 11:00

2 Acknowledgements Douglas Poole (Chadron State College, Nebraska Ünal Göktaş (Turgut Özal University, Turkey Mark Hickman (Univ. of Canterbury, New Zealand Bernard Deconinck (Univ. of Washington, Seattle Several undergraduate and graduate students Research supported in part by NSF under Grant No. CCF This presentation was made in TeXpower

3 Outline Examples of conservation laws The Korteweg-de Vries equation Other scaling invariant quantities (KdV equation The Zakharov-Kuznetsov equation Methods for computing conservation laws Tools from the calculus of variations The variational derivative (testing exactness The homotopy operator (inverting D x and Div Application 1: The KdV equation Application 2: The Zakharov-Kuznetsov equation Demonstration of ConservationLawsMD.m Conclusions and future work

4 Additional examples Manakov-Santini system Camassa-Holm equation in (2+1 dimensions Khoklov-Zabolotskaya equation Shallow water wave model for atmosphere Kadomtsev-Petviashvili equation Potential Kadomtsev-Petviashvili equation Generalized Zakharov-Kuznetsov equation

5 Examples of Conservation Laws Example 1: Traffic Flow u = density of cars a b x Modeling the density of cars (Bressan, 2009 u(x, t density of cars on a highway (e.g., number of cars per 100 meters. s(u mean (equilibrium speed of the cars (depends on the density.

6 u = density of cars a b x Change in number of cars in segment [a, b] equals the difference between cars entering at a and leaving at b during time interval [t 1, t 2 ]: b t2 a b a ( u(x, t2 u(x, t 1 dx = ( t2 u t (x, t dt t 1 t 1 t2 dx = t 1 ( b a ( J(a, t J(b, t dt J x (x, t dx dt where J(x, t = u(x, ts(u(x, t is the traffic flow (e.g., in cars per hour at location x and time t.

7 Then, b a t2 t 1 (u t + J x dt dx = 0 holds (a, b, t 1, t 2 Yields the conservation law: u t + [s(u u] x = 0 or D t ρ + D x J = 0 ρ = u is the conserved density; J(u = s(u u is the associated flux. A simple Lighthill-Whitham-Richards model: ( s(u = s max 1 u, 0 u u max u max s max is posted road speed, u max is the jam density.

8 Example 2: Fluid and Gas Dynamics Euler equations for a compressible, non-viscous fluid: ρ t +.(ρ u = 0 (ρ u t +. ( u (ρu + p = 0 E t +. ( (E + p u = 0 or, in components ρ t +.(ρ u = 0 (ρ u i t +.(ρ u i u + p e i = 0 (i = 1, 2, 3 E t +. ( (E + p u = 0 Express conservation of mass, momentum, energy.

9 is the dyadic product. ρ is the mass density. u = u 1 e 1 + u 2 e 2 + u 3 e 3 is the velocity. p is the pressure p(ρ, e. E is the energy density per unit volume: E = 1 2 ρ u 2 + ρe. e is internal energy density per unit of mass (related to temperature.

10 Notation Computations on the Jet Space Independent variables x = (x, y, z Dependent variables u = (u (1, u (2,..., u (j,..., u (N In examples: u = (u, v, θ, h,... Partial derivatives u kx = k u x k, u kx ly = k+l u x k y l, etc. Examples: u xxxxx = u 5x = 5 u x 5 u xx yyyy = u 2x 4y = Differential functions 6 u x 2 y 4 Example: f = uvv x + x 2 u 3 xv x + u x v xx

11 Total derivatives: D t, D x, D y,... Example: Let f = uvv x + x 2 u 3 x v x + u x v xx Then D x f = f x + u f x u + u xx +v x f v + v xx f u x f + v xxx v x f v xx = 2xu 3 xv x + u x (vv x + u xx (3x 2 u 2 xv x + v xx +v x (uv x + v xx (uv + x 2 u 3 x + v xxx (u x = 2xu 3 xv x + vu x v x + 3x 2 u 2 xv x u xx + u xx v xx +uv 2 x + uvv xx + x 2 u 3 x v xx + u x v xxx

12 Conservation Laws for Nonlinear PDEs System of evolution equations of order M u t = F(u (M (x with u = (u, v, w,... and x = (x, y, z. Conservation law in (1+1-dimensions D t ρ + D x J = 0 where the dot means evaluated on the PDE. Conserved density ρ and flux J. P = if J vanishes at ±. ρ dx = constant in time

13 Conservation law in (2+1-dimensions D t ρ + J = D t ρ + D x J 1 + D y J 2 = 0 Conserved density ρ and flux J = (J 1, J 2. Conservation law in (3+1-dimensions D t ρ + J = D t ρ + D x J 1 + D y J 2 + D z J 3 = 0 Conserved density ρ and flux J = (J 1, J 2, J 3.

14 Reasons for Computing Conservation Laws Conservation of physical quantities (linear momentum, mass, energy, electric charge,.... Testing of complete integrability and application of Inverse Scattering Transform. Testing of numerical integrators. Study of quantitative and qualitative properties of PDEs (Hamiltonian structure, recursion operators,.... Verify the closure of a model.

15 Examples of PDEs with Conservation Laws Example 1: KdV Equation u t + 6u u x + 3 u x = 0 or u 3 t + 6uu x + u xxx = 0 shallow water waves, ion-acoustic waves in plasmas Diederik Korteweg Gustav de Vries

16 Dilation Symmetry u t + 6uu x + u xxx = 0 has dilation (scaling symmetry (x, t, u ( x λ, t λ 3, λ 2 u λ is an arbitrary parameter. Notion of weight: W (x = 1, thus, W (D x = 1. W (t = 3, hence, W (D t = 3. W (u = 2. Notion of rank (total weight of a monomial. Examples: Rank(u 3 = Rank(3u 2 x = 6. Rank(u 3 u xx = 10.

17 Key Observation: Scaling Invariance Every term in a density has the same fixed rank. Every term in a flux has some other fixed rank. The conservation law D t ρ + D x J = 0 is uniform in rank. Hence, Rank(ρ + Rank(D t = Rank(J + Rank(D x

18 First six (of infinitely many conservation laws: D t (u + D x (3u 2 + u xx = 0 ( D t u 2 ( + D x 4u 3 u 2 x + 2uu xx = 0 D t (u u2 x + D x ( 9 2 u4 6uu 2 x + 3u2 u xx u2 xx u xu xxx = 0 ( D t (u 4 2uu 2 x u2 xx + D 24 x 5 u5 18uu 2 x + 4u 3 u xx +2u 2 xu xx uu2 xx 4uu x u xxx 1 5 u2 xxx u xxu 4x = 0

19 ( D t u 5 5 u 2 u 2 x + uu 2 xx 1 14 u2 xxx ( + D x 5u 6 40u 3 u 2 x u xxxu 5x = 0 D t (u 6 10 u 3 u 2 x 5 6 u4 x + 3 u 2 u 2 xx u3 xx 3 7 uu2 xxx u2 4x ( + D 36 x 7 u7 75u 4 u 2 x u 4xu 6x = 0 Third conservation law: Gerald Whitham, 1965 Fourth and fifth: Norman Zabusky, Seventh (sixth thru tenth: Robert Miura, 1966

20 Robert Miura

21 Conservation law explicitly dependent on t and x: ( D t tu xu +D x (4tu 3 xu u x tu 2 x + 2tuu xx 1 3 xu xx = 0

22 First five: IBM 7094 computer with FORMAC (1966 storage space problem! IBM 7094 Computer

23 First eleven densities: Control Data Computer CDC-6600 computer (2.2 seconds large integers problem! Control Data CDC-6600

24 Other Scaling Invariant Quantities Generalized Symmetries: G(x, t, u (N is an Nth order generalized symmetry iff it leaves u t = F (x, t, u (M invariant for the replacement u u + ɛg, u ix u ix + ɛd i xg, within order ɛ: D t (u + ɛg = F (u + ɛg must hold up to order ɛ. Defining equation: D t G = F (u[g] F (u [G] is the Fréchet derivative of F (u in the

25 direction of G: F (u[g] = ɛ F (u + ɛg ɛ=0 = M i=0 (D i F xg u ix First 4 generalized symmetries of u t = 6uu x + u 3x G (1 G (2 G (3 G (4 = u x = 6uu x + u xxx = 30u 2 u x + 20u x u xx + 10uu xxx + u 5x = 140u 3 u x + 70u 3 x + 280uu x u xx + 70u 2 u xxx +70u xx u xxx + 42u x u 4x + 14uu 5x + u 7x Generalized symmetries are invariant under the scaling symmetry.

26 Recursion Operator: A recursion operator R connects symmetries G (j+s = RG (j, j = 1, 2,... s is seed (s = 1 in simplest case. Defining equation: Explicitly, D t R + [R, F (u] = 0 R t + R [F ] + R F (u F (u R = 0 where stands for composition, and R [F ] is the Fréchet derivative of R in the direction of F :

27 R [F ] = n i=0 (D i xf R u ix Recursion operator (KdV equation: R = D 2 x + 2uI + 2D x ud 1 x = D 2 x + 4uI + 2u x D 1 x For example, Ru x =(D 2 x + 2uI + 2D x ud 1 x u x =6uu x + u 3x R(6uu x + u 3x =(D 2 x + 2uI + 2D xud 1 x (6uu x + u 3x =30u 2 u x + 20u x u 2x + 10uu 3x + u 5x Recursion operator is invariant under the scaling symmetry.

28 Lax Pair: Key idea: Replace u t + 6uu x + u xxx = 0 with a compatible linear system (Lax pair: ψ xx + (u λ ψ = 0 ψ t + 4ψ xxx + 6uψ x + 3u x ψ = 0 ψ is eigenfunction; λ is constant eigenvalue (λ t = 0 (isospectral Lax Pair (L, M in Operator Form: Lψ = λψ and D t ψ = Mψ Require compatibility of both equations L t ψ + (LM MLψ = 0

29 Defining Equation: L t + [L, M] = O with commutator [L, M] = LM ML. Furthermore, L t ψ = [D t, L]ψ = D t (Lψ LD t ψ. Lax pair for the KdV equation: L = D 2 x + u I M = ( 4 D 3 x + 6uD x + 3u x I Lax pair is invariant under the scaling symmetry.

30 Example 2: The Zakharov-Kuznetsov Equation u t + αuu x + β(u xx + u yy x = 0 models ion-sound solitons in a low pressure uniform magnetized plasma. Conservation laws: ( D t (u + D α2 x u2 + βu xx + D y (βu xy = 0 D t (u 2 ( + D 2α x 3 u3 β(u 2 x u2 y + 2βu(u xx + u yy ( + D y 2βu x u y = 0

31 More conservation laws (ZK equation: ( D t (u 3 3β α (u2 x + u 2 y + D x 3u 2 ( α 4 u2 + βu xx 6βu(u 2 x + u 2 y + 3β2 α (u2 xx u 2 yy 6β2 + D y ( 3βu 2 u xy + 6β2 α u xy(u xx + u yy α (u x(u xxx + u xyy + u y (u xxy + u yyy = 0 ( D t tu 2 2 α xu + D x (t( 2α 3 u3 β(u 2 x u2 y + 2βu(u xx + u yy ( x(u 2 + 2β α u xx + 2β α u x + D y 2β(tu x u y + 1 α xu xy = 0

32 Methods for Computing Conservation Laws Use the Lax pair L and A, satisfying [L, A] = 0. If L = D x + U, A = D t + V then V x U t + [U, V ] = 0. ˆL = T LT 1 gives the densities, Â = T AT 1 gives the fluxes. Use Noether s theorem (Lagrangian formulation to generate conservation laws from symmetries (Ovsiannikov, Olver, Mahomed, Kara, etc.. Integrating factor methods (Anderson, Bluman, Anco, Cheviakov, Mason, Naz, etc. require solving ODEs (or PDEs.

33 Proposed Algorithmic Method Density is linear combination of scaling invariant terms (in the jet space with undetermined coefficients. Compute D t ρ with total derivative operator. Use variational derivative (Euler operator to express exactness. Solve a (parametrized linear system to find the undetermined coefficients. Use the homotopy operator to compute the flux (invert D x or Div.

34 Work with linearly independent pieces in finite dimensional spaces. Use linear algebra, calculus, and variational calculus (algorithmic. Implement the algorithm in Mathematica.

35 Tools from the Calculus of Variations Definition: A differential function f is a exact iff f = Div F. Special case (1D: f = D x F. Question: How can one test that f = Div F? Theorem (exactness test: f = Div F iff L u (j (x f 0, j = 1, 2,..., N. N is the number of dependent variables. The Euler operator annihilates divergences

36 Euler operator in 1D (variable u(x: M L u(x = ( D x k k=0 = u D x u kx u x + D 2 x u xx D 3 x u xxx + Euler operator in 2D (variable u(x, y: L u(x,y = M x k=0 M y ( D x k ( D y l l=0 = u D x + D 2 x u x D y u xx +D x D y u y +D 2 y u xy u kx ly u yy D 3 x u xxx

37 Question: How can one compute F = Div 1 f? Theorem (integration by parts: In 1D: If f is exact then F = D 1 x f = f dx = H u(x f In 2D: If f is a divergence then F = Div 1 f = (H (x u(x,y f, H(y u(x,y f The homotopy operator inverts total derivatives and divergences!

38 Homotopy Operator in 1D (variable x: H u(x f = 1 0 N (I u (jf[λu] dλ λ j=1 with integrand I u (jf = M x (j k=1 k 1 i=0 ix ( D x k (i+1 u (j f u (j kx (I u (jf[λu] means that in I u (jf one replaces u λu, u x λu x, etc. More general: u λ(u u 0 + u 0 u x λ(u x u x 0 + u x 0 etc.

39 Homotopy Operator in 2D (variables x and y: H (x u(x,y f = 1 0 N j=1 (I (x u (j f[λu] dλ λ H (y u(x,y f = 1 0 N j=1 (I (y u (j f[λu] dλ λ where for dependent variable u(x, y M x M y ( k 1 l i+j I u (x f = i u ix jy k=1 l=0 i=0 j=0 ( D x k i 1 ( D y l j ( k+l i j 1 k i 1 ( k+l k f u kx ly

40 Application 1: The KdV Equation u t + 6uu x + u xxx = 0 Step 1: Compute the dilation symmetry Set (x, t, u ( x λ, t λ a, λ b u = ( x, t, ũ Apply change of variables (chain rule λ (a+b ũ t +λ (2b+1 ũũ x +λ (b+3 ũ 3 x = 0 Solve a + b = 2b + 1 = b + 3. Solution: a = 3 and b = 2 (x, t, u ( x λ, t λ 3, λ2 u

41 Compute the density of selected rank, say, 6. Step 2: Determine the form of the density List powers of u, up to rank 6 : [u, u 2, u 3 ] Differentiate with respect to x to increase the rank u has weight 2 apply D 4 x u 2 has weight 4 apply D 2 x u 3 has weight 6 no derivatives needed

42 Apply the D x derivatives Remove total and highest derivative terms: D 4 x u {u 4x} empty list D 2 xu 2 {u 2 x, uu xx } {u 2 x } 2 since uu xx =(uu x x u x D 0 xu 3 {u 3 } {u 3 } Linearly combine the building blocks Candidate density: ρ = c 1 u c 2 u x

43 Step 3: Compute the coefficients c i Compute D t ρ = ρ t + ρ (u[u t ] = ρ t + M k=0 ρ u kx D k xu t = (3c 1 u 2 I + 2c 2 u x D x u t Substitute u t by (6uu x + u xxx E = D t ρ = (3c 1 u 2 I + 2c 2 u x D x (6uu x + u xxx = 18c 1 u 3 u x + 12c 2 u 3 x + 12c 2uu x u xx +3 c 1 u 2 u xxx + 2c 2 u x u 4x

44 Apply the Euler operator (variational derivative L u (x = δ m δu = ( D x k k=0 Here, E has order m = 4, thus L u (x E = E u D x E u x +D 2 x = 18(c 1 + 2c 2 u x u xx This term must vanish! E u xx D 3 x So, c 2 = 1 2 c 1. Set c 1 = 1 then c 2 = 1 2. u kx E u 3x +D 4 x E u 4x Hence, the final form density is ρ = u u x 2

45 Step 4: Compute the flux J Method 1: Integrate by parts (simple cases Now, E =18u 3 u x + 3u 2 u xxx 6u 3 x 6uu x u xx u x u xxxx Integration of D x J = E yields the flux J = 9 2 u4 6uu x 2 + 3u 2 u xx u2 xx u x u xxx

46 Method 2: Use the homotopy operator 1 J = D 1 x E = E dx = H u(x E = (I u E[λu] dλ λ with integrand I u E = M k 1 u ix ( D x k (i+1 E u kx k=1 i=0 0

47 Here M = 4, thus I u E = (ui( E u x + (u x I ud x ( E u xx +(u xx I u x D x + ud 2 x( E u xxx +(u xxx I u xx D x + u x D 2 x E ud3 x ( u 4x = (ui(18u u 2 x 6uu xx u xxxx +(u x I ud x ( 6uu x +(u xx I u x D x + ud 2 x (3u2 +(u xxx I u xx D x + u x D 2 x ud 3 x( u x = 18u 4 18uu 2 x + 9u 2 u xx + u 2 xx 2u x u xxx Note: correct terms but incorrect coefficients!

48 Finally, 1 J = H u(x E = (I u E[λu] dλ 0 λ 1 ( = 18λ 3 u 4 18λ 2 uu 2 x + 9λ 2 u 2 u xx + λu 2 xx 0 2λu x u xxx dλ = 9 2 u4 6uu 2 x + 3u 2 u xx u2 xx u x u xxx Final form of the flux: J = 9 2 u4 6uu 2 x + 3u2 u xx u2 xx u xu xxx

49 Application 2: Zakharov-Kuznetsov Equation u t + αuu x + β(u xx + u yy x = 0 Step 1: Compute the dilation invariance ZK equation is invariant under scaling symmetry (t, x, y, u ( t λ 3, x λ, y λ, λ2 u= ( t, x, ỹ, ũ λ is an arbitrary parameter. Hence, the weights of the variables are W (u = 2, W (D t = 3, W (D x = 1, W (D y = 1.

50 A conservation law is invariant under the scaling symmetry of the PDE. W (u = 2, W (D t = 3, W (D x = 1, W (D y = 1. For example, ( D t (u 3 3β α (u2 x +u 2 y + D x 3u 2 ( α 4 u2 +βu xx 6βu(u 2 x + u 2 y + 3β2 α (u2 xx u 2 xy 6β2 α (u x(u xxx +u xyy + u y (u xxy +u yyy ( + D y 3βu 2 u xy + 6β2 α u xy(u xx + u yy = 0 Rank (ρ = 6, Rank (J = 8. Rank (conservation law = 9.

51 Compute the density of selected rank, say, 6. Step 2: Construct the candidate density For example, construct a density of rank 6. Make a list of all terms with rank 6: {u 3, u 2 x, uu xx, u 2 y, uu yy, u x u y, uu xy, u 4x, u 3xy, u 2x2y, u x3y, u 4y } Remove divergences and divergence-equivalent terms. Candidate density of rank 6: ρ = c 1 u 3 + c 2 u 2 x + c 3 u 2 y + c 4 u x u y

52 Step 3: Compute the undetermined coefficients Compute D t ρ = ρ t + ρ (u[u t ] = ρ t + = M x k=0 M y l=0 ρ u kx ly D k x Dl y u t ( 3c 1 u 2 I + 2c 2 u x D x + 2c 3 u y D y + c 4 (u y D x + u x D y Substitute u t = (αuu x + β(u xx + u yy x. u t

53 E = D t ρ = 3c 1 u 2 (αuu x + β(u xx + u xy x + 2c 2 u x (αuu x + β(u xx + u yy x x + 2c 3 u y (αuu x + β(u xx + u yy x y + c 4 (u y (αuu x + β(u xx + u yy x x + u x (αuu x + β(u xx + u yy x y Apply the Euler operator (variational derivative M x M y L u(x,y E = ( D x k ( D y l E k=0 l=0 u kx ly ( = 2 (3c 1 β + c 3 αu x u yy + 2(3c 1 β + c 3 αu y u xy +2c 4 αu x u xy +c 4 αu y u xx +3(3c 1 β +c 2 αu x u xx 0

54 Solve a parameterized linear system for the c i : 3c 1 β + c 3 α = 0, c 4 α = 0, 3c 1 β + c 2 α = 0 Solution: c 1 = 1, c 2 = 3β α, c 3 = 3β α, c 4 = 0 Substitute the solution into the candidate density ρ = c 1 u 3 + c 2 u 2 x + c 3 u 2 y + c 4 u x u y Final density of rank 6: ρ = u 3 3β α (u2 x + u 2 y

55 Step 4: Compute the flux Use the homotopy operator to invert Div: ( J=Div 1 E = H (x u(x,y E, H(y u(x,y E where with I (x u E = H (x M x k=1 u(x,y E = 1 M y l=0 ( k 1 i=0 0 l j=0 (I u (x E[λu] dλ λ ( i+j i u ix jy ( D x k i 1 ( D y l j ( k+l i j 1 k i 1 ( k+l k E u kx ly Similar formulas for H (y u(x,y E and I(y u E.

56 Let A = αuu x + β(u xxx + u xyy so that E = 3u 2 A 6β α u xa x 6β α u ya y Then, ( J = H (x u(x,y E, H(y u(x,y E = ( 3α 4 u4 + βu 2 (3u xx + 2u yy 2βu(3u 2 x + u 2 y + 3β2 4α u(u 2x2y + u 4y β2 α u x( 7 2 u xyy + 6u xxx β2 α u y(4u xxy u yyy + β2 α (3u2 xx u2 xy u2 yy + 5β2 4α u xxu yy, βu 2 u xy 4βuu x u y 3β2 4α u(u x3y + u 3xy β2 4α u x(13u xxy + 3u yyy 5β2 4α u y(u xxx + 3u xyy + 9β2 4α u xy(u xx + u yy

57 However, Div 1 E is not unique. Indeed, J = J + K, where K = (D y θ, D x θ is a curl term. For example, θ =2βu 2 u y + β2 4α Shorter flux: ( 3u(u xxy +u yyy +10u x u xy +5u y (3u yy +u xx J = J K ( = 3u 2 ( α 4 u2 + βu xx 6βu(u 2 x + u 2 y + 3β2 α ( u 2 xx u 2 yy 6β2 α (u x(u xxx + u xyy + u y (u xxy + u yyy, 3βu 2 u xy + 6β2 α u xy(u xx + u yy

58 Software Demonstration Software packages in Mathematica Codes are available via the Internet: URL: whereman/

59 Additional Examples Manakov-Santini system u tx + u yy + (uu x x + v x u xy u xx v y = 0 v tx + v yy + uv xx + v x v xy v y v xx = 0 Conservation laws for Manakov-Santini system: ( D t fu x v x + D x (f(uu x v x u x v x v y u y v y ( f y(u t + uu x u x v y + D y f(u x v y + u y v x + u x vx 2 + f (u yu y yu x v x = 0 where f = f(t is arbitrary.

60 Conservation laws continued: ( D t (f(2u + vx 2 yu x v x + D x f (u 2 + uvx 2 + u y v vy 2 vxv 2 y y(uu x v x u x v x v y u y v y f y(v t + uv x v x v y + (f 2fxy 2 (u t + uu x u x v y + D y (f(vx 3 + 2v x v y u x v y(u x vx 2 + u x v y + u y v x +f (v y(2u + v y + vx 2 + (f y 2 2fx(u x v x + u y =0 where f = f(t is arbitrary. There are three additional conservation laws.

61 (2+1-dimensional Camassa-Holm equation (αu t + κu x u txx + 3βuu x 2u x u xx uu xxx x + u yy = 0 Interchange t with y (αu y + κu x u xxy + 3βuu x 2u x u xx uu xxx x + u tt = 0 Set v = u t to get u t = v v t = αu xy κu xx + u 3xy 3βu 2 x 3βuu xx + 2u 2 xx + 3u x u xxx + uu 4x

62 Conservation laws for the Camassa-Holm equation D t (fu + D x ( 1 α f(3β 2 u2 + κu 1 2 u2 x uu xx u tx + ( 1 2 f y 2 1 α fx(αu t+ κu x + 3βuu x 2u x u xx uu xxx ( u txx + D y ( 1 2 f y 2 1 α fxu y f yu = 0 ( D t (fyu + D 1 x α fy( 3β 2 u2 + κu 1 2 u2 x uu xx u tx + y( 1 6 f y 2 1 α fx(αu t + κu x + 3βuu x 2u x u xx uu xxx ( u txx + D y y( 1 6 f y 2 1 α fxu y + ( 1 α fx 1 2 f y 2 u = 0 where f = f(t is an arbitrary function.

63 Khoklov-Zabolotskaya equation describes e.g., sound waves in nonlinear media (u t uu x x u yy u zz = 0 Conservation law: ( D t (fu D 1 x 2 fu2 + (fx + g(u t uu x + D y ((fx + gu y (f y x + g y u + D z ((fx + gu z (f z x + g z u = 0 under the constraints f = 0 and g = f t where f = f(t, y, z and g = g(t, y, z.

64 Shallow water wave model (atmosphere u t + (u u + 2 Ω u + (θh 1 2 h θ = 0 θ t + u ( θ = 0 h t + (uh = 0 where u(x, y, t, θ(x, y, t and h(x, y, t. In components: u t + uu x + vu y 2 Ωv hθ x + θh x = 0 v t + uv x + vv y + 2 Ωu hθ y + θh y = 0 θ t + uθ x + vθ y = 0 h t + hu x + uh x + hv y + vh y = 0

65 First few conservation laws of SWW model: ρ (1 = h ρ (2 = h θ J (1 = h J (2 = h θ ρ (3 = h θ 2 J (3 = h θ 2 ρ (4 = h (u 2 + v 2 + hθ ρ (5 = θ (2Ω + v x u y J (5 = 1 2 θ J (4 = h u v 4Ωu 2uu y + 2uv x hθ y 4Ωv + 2vv x 2vu y + hθ x u v u v u (u2 + v 2 + 2hθ v (v 2 + u 2 + 2hθ

66 More general conservation laws for SWW model: ( ( D t (f(θh + D x f(θhu + D y f(θhv = 0 D t (g(θ(2ω + v x u x ( + D 1 x 2 g(θ(4ωu 2uu y + 2uv x hθ y ( + D 1 y 2 g(θ(4ωv 2u yv + 2vv x + hθ x = 0 for any functions f(θ and g(θ.

67 Kadomtsev-Petviashvili (KP equation (u t + αuu x + u xxx x + σ 2 u yy = 0 parameter α IR and σ 2 = ±1. Equation be written as a conservation law D t (u x + D x (αuu x + u xxx + D y (σ 2 u y = 0. Exchange y and t and set u t = v u t = v v t = 1 σ (u 2 xy + αu 2 x + αuu xx + u xxxx

68 Examples of conservation laws for KP equation (explicitly dependent on t, x, and y ( ( ( D t xu x +D x 3u 2 u xx 6xuu x +xu xxx +D y αxu y =0 ( ( D t yu x +D x (y(αuu x + u xxx +D y σ 2 (yu y u =0 ( tu ( D t + D α x 2 tu 2 + tu xx + σ2 y 2 4 t u t + σ2 y 2 4 t u xxx + ασ2 y 2 4 t uu x x tu t αx tuu x x tu xxx +D y ( x tu y + y2 u y 4 t yu 2 t = 0

69 More general conservation laws for KP equation: ( D t (fu + D x f( α 2 u2 + u xx +( σ2 2 f y 2 fx(u t + αuu x + u 3x +D y (( 1 2 f y 2 σ 2 fxu y f yu = 0 ( D t (fyu + D x fy( α 2 u2 + u xx +y( σ2 6 f y 2 fx(u t + αuu x + u 3x +D y (y( 1 6 f y 2 σ 2 fxu y +(σ 2 fx 1 2 f y 2 u =0 where f(t is arbitrary function.

70 Potential KP equation Replace u by u x and integrate with respect to x. u xt + αu x u xx + u xxxx + σ 2 u yy = 0 Examples of conservation laws (not explicitly dependent on x, y, t: ( ( ( D t u x + D α x 2 u2 x + u xxx + D y σ 2 u y = 0 ( D t (u 2 x + D 2α3 x u3 x u 2 xx + 2u x u xxx σ 2 u yy ( +D y 2σ 2 u x u y = 0

71 Conservation laws for pkp equation continued: ( ( D t u x u y + Dx αu 2 xu y + u t u y + 2u xxx u y 2u xx u xy +D y (σ 2 u 2 y 1 3 u3 x u t u x + u 2 xx = 0 ( D t (2αuu x u xx + 3uu 4x 3σ 2 u 2 y + D x 2αu t u 2 x + 3u 2 t 2αuu x u tx 3u tx u xx + 3u t u xxx + 3u x u t xx 3uu t xxx +D y ( 6σ 2 u t u y = 0 Various generalizations exist.

72 Generalized Zakharov-Kuznetsov equation u t + αu n u x + β(u xx + u yy x = 0 where n is rational, n 0. Conservation laws: ( D t (u + D α x n+1 un+1 + βu xx + D y ( βu xy = 0 ( D t u 2 ( + D 2α x n+2 un+2 β(u 2 x u 2 y + 2βu(u xx + u yy ( + D y 2βu x u y = 0

73 Third conservation law for gzk equation: ( D t u n+2 (n+1(n+2β (u 2 2α x + u 2 y + D x ( (n+2α 2(n+1 u2(n+1 + (n + 2βu n+1 u xx (n + 1(n + 2βu n (u 2 x + u 2 y + (n+1(n+2β2 2α (n+1(n+2β2 α (u x (u xxx + u xyy + u y (u xxy + u yyy (u 2 xx u 2 yy + D y ( (n + 2βu n+1 u xy + (n+1(n+2β2 α u xy (u xx + u yy = 0.

74 Conclusions and Future Work The power of Euler and homotopy operators: Testing exactness Integration by parts: D 1 x and Div 1 Integration of non-exact expressions Example: f = u x v + uv x + u 2 u xx fdx = uv + u 2 u xx dx Use other homotopy formulas (moving terms amongst the components of the flux; prevent curl terms

75 Broader class of PDEs (beyond evolution type Example: short pulse equation (nonlinear optics u xt = u + (u 3 xx = u + 6uu 2 x + 3u 2 u xx with non-polynomial conservation law ( D t 1 + 6u 2 x D x (3u u 2 x = 0 Continue the implementation in Mathematica Software: whereman

76 Thank You

77 Publications 1. D. Poole and W. Hereman, Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions, Journal of Symbolic Computation (2011, 26 pages, in press. 2. W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman, and B. M. Herbst, Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations, In: Advances of Nonlinear Waves and Symbolic Computation, Ed.: Z. Yan, Nova Science Publishers, New York (2009, Chapter 2, pp

78 3. W. Hereman, M. Colagrosso, R. Sayers, A. Ringler, B. Deconinck, M. Nivala, and M. S. Hickman, Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws. In: Differential Equations with Symbolic Computation, Eds.: D. Wang and Z. Zheng, Birkhäuser Verlag, Basel (2005, Chapter 15, pp W. Hereman, B. Deconinck, and L. D. Poole, Continuous and discrete homotopy operators: A theoretical approach made concrete, Math. Comput. Simul. 74(4-5, (2007.

79 5. W. Hereman, Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions, Int. J. Quan. Chem. 106(1, (2006.