Exterior differential expression of the (1+1)-dimensional nonlinear evolution equation with Lax integrability

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:435

Article

Su, Chuan-Qi^1,2  Gao, Yi-Tian^1,2  Yu, Xin^1,2  Xue, Long^1,2,3  Shen, Yu-Jia^1,2 

[1] Beijing Univ Aeronaut & Astronaut, Minist Educ, Key Lab Fluid Mech, Beijing 100191, Peoples R China

[2] Beijing Univ Aeronaut & Astronaut, Natl Lab Computat Fluid Dynam, Beijing 100191, Peoples R China

[3] Aviat Univ Air Force, Flight Training Base, Fuxin 123100, Liaoning, Peoples R China

关键词: (1+1)-Dimensional nonlinear evolution equation with Lax integrability;    Exterior differential;    Complete integrability;    Frobenius theorem;    Darboux transformation;   

DOI  :  10.1016/j.jmaa.2015.10.036

来源: Elsevier

【 摘 要 】

A (1 + 1)-dimensional nonlinear evolution equation with Lax integrability is investigated in this paper with the aid of differential forms and exterior differentials. Equivalent definition Of the Lax integrability is given. Relation between the Lax integrability and complete integrability is discussed. Geometric interpretation of the Lax equation is presented. Gauge transformation and Darboux transformation are restated in terms of the differential forms and exterior differentials. If lambda I-S is a Darboux transformation between two Lax equations, the system that the matrix S should satisfy is given, where lambda is the spectral parameter and I is the identity matrix. The completely integrable condition of that system is obtained according to the Frobenius theorem, and it is shown that S = H Lambda H-1 satisfies that completely integrable condition with H and Lambda as two matrices. (C) 015 Elsevier Inc. All rights reserved.

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