A hierarchy of nonlinear evolution equations and finite-dimensional involutive systems

Authors

Zhijun Qiao, The University of Texas Rio Grande ValleyFollow

Document Type

Article

Publication Date

6-1994

Abstract

A spectral problem and an associated hierarchy of nonlinear evolution equations are presented in this article. In particular, the reductions of the two representative equations in this hierarchy are given: one is the nonlinear evolution equation rl= - ar,- 2icu/3] r2] r which looks like the nonlinear Schrijdinger equation, the other is the generalized derivative nonlinear Schrijdinger equation rt= $ar,,- ialr12r- a/3(lr12r),- a j3I r I 2r,-2iap21r14r which is just a combination of the nonlinear Schrijdinger equation and two different derivative nonlinear Schrodinger equations [D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 789 (1978); M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 21, 1006 (1980)]. The spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system under a constraint between the potentials and the spectral functions. At the end of this article, the involutive solutions of the hierarchy of nonlinear evolution equations are obtained. Particularly, the involutive solutions of the reductions of the two representative equations are developed.

Comments

©1994 American Institute of Physics. Original published version available at https://doi.org/10.1063/1.530882

Recommended Citation

Qiao, Zhijun. 1994. “A Hierarchy of Nonlinear Evolution Equations and Finite‐dimensional Involutive Systems.” Journal of Mathematical Physics 35 (6): 2971–77. https://doi.org/10.1063/1.530882.

Publication Title

Journal of Mathematical Physics

DOI

10.1063/1.530882