Study On Symbolic Computation Of Conservation Laws For Nonlinear Differential-Difference Equations

In the field of contemporary nonlinear science, the integrability of nonlinear equations attracts much attention of researchers. With the well-known Wu's elimination method and the computer algebraic system Maple, we study various algebraic algorithms of conservation laws to nonlinear differential-difference equation(DDE) and automated derivation. Our main work includes the following three parts.In PartⅠ, we focus on four typical algorithms for constructing infinite conservation laws to the family of discrete isospectral evolution equations in (1+1)-dimensions. These methods contain Ricatti equation method, characteristic function formal solution method, multi-system trace identity method and B(a|¨)cklund transform method.PartⅡis devoted to studying integrability of nonlinear DDEs from the view of conservation law starting from scaling symmetry. The "divide and conquer" strategy is used to improve the key steps of the undetermined coefficient algorithm for constructing polynomial conservation laws of nonlinear DDEs, so the calculation complexity problem caused by the dramatic increase of redundancy is solved. Furthermore, Wu's elimination method is applied to solve the obtained nonlinear algebraic equations thus to get a more efficient algebraic algorithm. In addition, the discrete Euler operator and homotopy operator of order zero are preliminarily studied.PartⅢis devoted to giving the corresponding implementation software package CLawDDEs based on the improved algebraic algorithm in Maple. As long as the system of first order polynomial DDEs in (1+1)-dimensions is given, no matter whether a single equation or the coupled equation, CLawDDEs can automatically output the scaling symmetries of the variables and a series of possible polynomial conservation laws for different rank. For exponential function equations and triangular function equations etc., CLawDDEs can also be utilized to construct the conservation laws after proper variable substitution. For parameterized nonlinear DDEs, the software is able to automatically filter out the parameter constraints to guarantee integrability so as to get some new inte-grable systems.