Journal
SYMMETRY-BASEL
Volume 15, Issue 9, Pages -Publisher
MDPI
DOI: 10.3390/sym15091623
Keywords
R-matrix; Lie group; Poisson bracket; integrable system; 05.45.Yv; 02.30.Jr; 02.30.Ik
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Based on the R-matrix theory, this paper investigates a series of generalized integrable systems and their corresponding solution types by abstracting Lax pairs, exhibiting zero-curvature equations, and using nonisospectral techniques.
Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey-Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a Backlund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions.
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