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ASymbolicAlgorithmfortheComputationofConservationLaws fromLaxPairs Jacob Rezac Willy Hereman University of Delaware Colorado School of Mines NSF research award no. CCF-0830783 Motivation Conservation Law Construction We demonstrate a systematic method for the calculation of innitelymany conservation laws of a completely integrable partial dierential Manipulate Example Key Idea X and T so that [X, T ] = 0. The Korteweg-de Vries equation, ut + 6uux + uxxx = 0, equation (PDE). We're interested because conservation laws: • • • suggest the complete integrability of a PDE; for Assume a PDE has the Lax pair X = X0 + λX1 X0 with constant entries. o-diagonal and diagonal X1 and T = Pm i T λ i=0 i has a Lax pair [3] with are useful for numerical and physical applications; Step 1: Let P0 = X0 and recursively solve imply a Lax pair is strong [1]. Pn + [Γn+1 , X1 ] = Γn X0 − 0 Γn − n−1 X X= Pk Γn−k Hard to calculate; for the unknowns Unwieldy (see gure). Pn and Γn . The diagonal entries of of We extend a result from Drinfel'd and Sokolov in [2] to calculate Pn are conserved densities, ρn . Step 2: Compute Jn = Preliminaries to nd the uxes Consider the PDE 1 P1 = i 2 0 −1 u 0 −u 0 0 u 1 P3 = i 8 , T on the jet space of (ρn )t dx are said to be a Jn . Z J1 = 2 . u + uxx 0 0 −u2 . Z (ρ1 )t dx = − 2 6uux + uxx dx = −3u − uxx and, similarly, J2 = −4u + 150 2 ux − 2uuxx . Similar results are seen, for example, with the modied Korteweg-de Lax pair if Vries equation, the sin- and sinh-Gordon equations, and the nonlinear Xt − Tx + [X, T ] = 0 Schrodinger equation. Further work is needed to adjust the algorithm on (1). If (1) is such that functions ρ(x, t) and J(x, t) to more general PDEs. 100 satisfy References ρt − Jx = 0, then the PDE is said to admit a conservation law with density ux 0 −i The uxes are (1) u(x, t) i 0 ρ3 = u2 3 and −λ ρ1 = u Step 3: Reduce to simplest form. ut = f (u, ux , uxx , . . .). X and Cleaning up, Z conservation laws given a Lax pair. Matrix functions Ti λi The recurrence relationship gives, e.g., k=0 • T = of the form i=0 Problems with Conservation Laws: • T 3 P ρ and [1] S.Y. Sakovich. On zero-curvature representations of evolution equations. 50 Journal of Physics A: Mathematical and General, 28(10):2861-2869, 1995. J. Lax pairs are gauge invariant. Given a nonsingular matrix X̃ = GXG −1 −1 + Gx G and are Lax pairs for the same PDE as X −1 T̃ = GT G and T. [2] V.G. Drinfel'd and V.V. Sokolov. Lie algebras and equations of Korteweg- G, de Vries type. Journal of Mathematical Sciences, 30(2):1975-2036, 1985. −1 + Gt G 2 Number of terms in 4 nth 6 8 10 conservation law for the Korteweg-de Vries equation (blue) and number of seconds to simultaneously compute these rst n conservation laws using unoptimized Mathematica code (red). Translated from original Russian. [3] M.J. Ablowitz and H. Segur. form. Solitons and the Inverse Scattering Trans- SIAM, Philadelphia, PA, 1981.