A method for generating isospectral and nonisospectral hierarchies of equations as well as symmetries. (English) Zbl 1436.37083
The aim of this paper is to apply an improved Tu scheme [G. Tu, J. Math. Phys. 30, No. 2, 330–338 (1989; Zbl 0678.70015)] to obtain two isospectral- and nonisospectral hierarchies of evolution equations.
An isospectral and nonisospectral problem related to isospectral and nonisospectral integrable hierarchies is presented. This includes the generalized Korteweg-de Vries (KdV) equations and the cylinder-KdV equation with variable coefficients. One of the generalized KdV equations generalizes a known variable-coefficient integrable equation. The authors obtain for this equation infinite sets of symmetries and conserved densities.
As a second application, an isospectral and nonisospectral AKNS-Kaup-Newell soliton hierarchy is derived from a proper spectral problem whose symmetries and \(\tau\)-symmetries are found by applying Lie group analysis. A special reduction gives a generalized sine-Gordon equation.
An isospectral and nonisospectral problem related to isospectral and nonisospectral integrable hierarchies is presented. This includes the generalized Korteweg-de Vries (KdV) equations and the cylinder-KdV equation with variable coefficients. One of the generalized KdV equations generalizes a known variable-coefficient integrable equation. The authors obtain for this equation infinite sets of symmetries and conserved densities.
As a second application, an isospectral and nonisospectral AKNS-Kaup-Newell soliton hierarchy is derived from a proper spectral problem whose symmetries and \(\tau\)-symmetries are found by applying Lie group analysis. A special reduction gives a generalized sine-Gordon equation.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K06 | General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 35Q51 | Soliton equations |
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
Citations:
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