A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries

Abstract We have known that the Tu scheme is a powerful tool for generating isospectral integrable hierarchies of evolution equations. However, it is not clear that how to apply it to generate nonisospectral integrable hierarchies along with the time-evolution of the spectral parameter as the polynomial form in the spectral parameter λ . In the paper, we propose a method for generating isospectral and nonisospectral integrable hierarchies by improved the Tu scheme. As one of applications, a spectral–nonisospectral problem is introduced for which a kind of isospectral plus nonisospectral integrable hierarchy is presented, including the generalized KdV equations and the cylinder-KdV equation with variable coefficients. One of the generalized KdV equations generalizes the variable-coefficient integrable equation given by Li Yishen. Therefore, we choose it to discuss and obtain its various symmetries with the help of Lie symmetry analysis which looks complicated, but another two sets of infinite symmetries are produced by the infinite conserved densities of the integrable equation. As the second application, an isospectral–nonisospectral AKNS-Kaup-Newell soliton hierarchy (briefly, AKNS-K-N-SH) is derived from an appropriate spectral problem whose K symmetries and a set of new τ symmetries are generated by applying Lie group analysis. As a reduction of the AKNS-K-N-SH, a generalized sine-Gordon equation is singled out.