Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

Huanhuan Lu,Yufeng Zhang

doi:10.3390/sym13101797

Abstract

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

Highlights

Isospectral-NonisospectralIn the development of the past few decades, the nonlinear evolution equations have attracted increasing attentions from many researchers with their significant roles in describing the nonlinear dynamic behaviors in various fields
While a group of scholars devoted themselves to the study of solving equation, there were a group of scholars who devoted themselves to the derivation of integrable hierarchies of equations
By utilizing the second loop algebra, another line spectral problem is introduced, for which the isospectral-nonisospectral super integrable hierarchy containing five even superfunctions and four odd superfunctions is obtained in the frame of zero-curvature equation and Tu scheme

Summary

Introduction

In the development of the past few decades, the nonlinear evolution equations have attracted increasing attentions from many researchers with their significant roles in describing the nonlinear dynamic behaviors in various fields. We found that the Tu scheme is usually utilized to generate isospectral integrable hierarchies of equations by choosing proper loop algebras. How to use it to generate nonisospectral integrable hierarchies of equations will be a problem worth considering. By employing the first loop algebra, an isospectral-nonisospectral super integrable hierarchy containing two even superfunctions and two odd superfunctions is derived under the case of λt = ∑ k j (t)λ− j. By utilizing the second loop algebra, another line spectral problem is introduced, for which the isospectral-nonisospectral super integrable hierarchy containing five even superfunctions and four odd superfunctions is obtained in the frame of zero-curvature equation and Tu scheme.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies

Comparison and Discussion

Conclusions