Computation of densities and fluxes of nonlinear differential‐difference equations

Direct methods to find conserved densities and fluxes of differential‐difference equations are presented and illustrated for the Kac‐van Moerbeke (KvM) and modified Volterra lattices. A Miura map which connects both lattices is explicitly constructed based on homotopic deformation. The map is used recursively to compute conserved densities of the KvM lattice. The algorithms presented could be implemented in computer algebra systems and could be used to investigate the integrability of semi‐discrete lattices.

[1]  B. Fuchssteiner,et al.  Mastersymmetries and Multi-Hamiltonian Formulations for Some Integrable Lattice Systems , 1989 .

[2]  D. Levi,et al.  SIDE III—Symmetries and Integrability of Difference Equations , 2000 .

[3]  S. Manakov.  Complete integrability and stochastization of discrete dynamical systems , 1974 .

[4]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[5]  J. Satsuma,et al.  Symmetries and Integrability of Difference Equations , 1999 .

[6]  R. Yamilov.  CONSTRUCTION SCHEME FOR DISCRETE MIURA TRANSFORMATIONS , 1994 .

[7]  Alexey Borisovich Shabat,et al.  Symmetry Approach to the Integrability Problem , 2000 .

[8]  A. Shabat,et al.  To a transformation theory of two-dimensional integrable systems , 1997 .

[9]  Y. Suris.  The Problem of Integrable Discretization: Hamiltonian Approach , 2003 .

[10]  R. I. Yamilov,et al.  Master symmetries for differential-difference equations of the Volterra type , 1994 .

[11]  Elizabeth L. Mansfield,et al.  Towards approximations which preserve integrals , 2001, ISSAC '01.

[12]  Willy Hereman,et al.  Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations , 1997, J. Symb. Comput..

[13]  Y. Suris.  On some integrable systems related to the Toda lattice , 1996, solv-int/9605010.

[14]  Peter J. Olver,et al.  Symmetries and Integrability of Difference Equations , 1999 .

[15]  W. Hereman,et al.  Integrability Tests for Nonlinear Evolution Equations , 1999, solv-int/9904022.

[16]  Decio Levi,et al.  Conditions for the existence of higher symmetries of evolutionary equations on the lattice , 1997 .

[17]  Joshua A. Leslie,et al.  The Geometrical Study of Differential Equations , 2001 .

[18]  Alexey Borisovich Shabat,et al.  The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems , 1987 .

[19]  S. I. Svinolupov,et al.  The multi-field Schrödinger lattices , 1991 .

[20]  Willy Hereman,et al.  Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations , 1999, Adv. Comput. Math..

[21]  Willy Hereman,et al.  Review of symbolic software for lie symmetry analysis , 1997 .

[22]  V. Sokolov,et al.  The Symmetry Approach to Classification of Integrable Equations , 1991 .

[23]  S. Ulam,et al.  Studies of nonlinear problems i , 1955 .

[24]  Robert M. Miura,et al.  Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation , 1968 .

[25]  W. Hereman,et al.  Computation of conserved densities for systems of nonlinear differential-difference equations1 , 1997, solv-int/9704016.

[26]  Simon Ruijsenaars,et al.  Relativistic Toda systems , 1989 .

[27]  Vito Volterra,et al.  Leçons sur la théorie mathématique de la lutte pour la vie , 1931 .

[28]  R. Hirota,et al.  N -Soliton Solutions of Nonlinear Network Equations Describing a Volterra System , 1976 .

[29]  Michael D. Colagrosso,et al.  Algorithmic integrability tests for nonlinear differential and lattice equations , 1998, solv-int/9803005.

[30]  Mark Kac,et al.  On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices , 1975 .

[31]  J. Moser,et al.  Three integrable Hamiltonian systems connected with isospectral deformations , 1975 .

[32]  D. Levi,et al.  Extension of the spectral-transform method for solving nonlinear differential difference equations , 1978 .

[33]  Ayse Humeyra Bilge.  Classification of Integrable Evolution Equations of the Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ , 1996 .

[34]  G. Teschl.  Jacobi Operators and Completely Integrable Nonlinear Lattices , 1999 .

[35]  W. Hereman,et al.  Computation of conservation laws for nonlinear lattices 1 , 1998 .

[36]  Morikazu Toda,et al.  Theory Of Nonlinear Lattices , 1981 .

[37]  Peter E. Hydon,et al.  Conservation laws of partial difference equations with two independent variables , 2001 .

[38]  Y. Suris.  New integrable systems related to the relativistic Toda lattice , 1996, solv-int/9605006.

[39]  W. Hereman,et al.  Algorithmic Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations , 1998, solv-int/9802004.

[40]  S. I. Svinolupov,et al.  Multi-component Volterra and Toda type integrable equations , 1999 .

[41]  Willy Hereman,et al.  Computation of conservation laws for nonlinear lattices , 1998, solv-int/9801023.

[42]  Y. Suris.  Miura transformations for Toda--type integrable systems, with applications to the problem of integrable discretizations , 1999, solv-int/9902003.

[43]  Elizabeth L. Mansfield,et al.  A Variational Complex for Difference Equations , 2004, Found. Comput. Math..

[44]  A. Shabat,et al.  Lattice representations of integrable systems , 1988 .

[45]  Yuri B. Suris,et al.  INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES , 1997, solv-int/9709005.