Generalized Hierarchy Of Semi-discrete Equations With Variable Coefficients And Its Inverse Scattering Transform

For special systems that are spatially discrete,but continuous in relation to time,we call them nonlinear semi-discrete systems.It is difficult to derive and solve nonlinear semi-discrete equations.Although some scholars have solved part of the semi-discrete problems by some methods,but there are still a lot of questions to be studied.In paper,based on the classic Toda chain,two hierarchies of generalized semi-discrete equations with variable coefficients are derived,the derived hierarchies of equations appear as a "superposition" of the equations,not only the original hierarchy of isospectral Toda chain equations and nonisospectral Toda chain equations can be included,but also has a series of variable-coefficient functions.The derived two hierarchies of equations are finally solved by carrying the direct scattering analysis,determining the time-dependence of scattering data,solving the GLM equation,and reconstructing the potentials.Firstly,the spectral problem of semi-discrete matrix is generalized by introducing variable-coefficient functions,then based on the generalized spectral problem of semi-discrete matrix,a new hierarchy of generalized isospectral hierarchy of semi-discrete equations with variable coefficients is derived.then,the inverse scattering method is extended to the derived generalized isospectral hierarchy of variable-coefficient semi-discrete equations,exact solution and soliton solutions are obtained.In this case of N=1,2,3 the single-soliton,double-soliton and three-soliton solutions are simulated numerically.Secondly,by introducing variable coefficients and time-dependent spectral parameter,the same semi-discrete matrix spectral problem is generalized.Then based on the generalized semi-discrete matrix spectral problem,a new generalized nonisospectral hierarchy of semi-discrete equations with variable coefficients is derived.Finally,theinverse scattering method is extended to the derived generalized nonisospetral hierarchy of variable-coefficient semi-discrete equations and exact solutions and N-soliton solution are obtained.