Abstract
In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term.
Highlights
Painlevé analysis is an important method for testing Painlevé integrable property [1,2,3,4,5,6,7]of nonlinear partial differential equations (PDEs)
Painlevé property or Painlevé integrability for nonlinear PDEs means that the solutions of the given
The so-called WTC method of Painlevé analysis proposed by Weiss, Tabor, and Carnevale [2] is an effective approach for Painlevé test of nonlinear PDEs
Summary
Painlevé analysis is an important method for testing Painlevé integrable property [1,2,3,4,5,6,7]. It is worth mentioning that one of the advantages of the Painlevé analysis method [1,2,3] is to provide a useful tool for the reduction or linearization of nonlinear PDEs. The Lax integrability and multiple soliton solutions of Equations (1) and (2) were obtained in [8,18,19]. The Bäcklund transformations, two reductions, and some exact solutions of Equations (1) and (2) have been obtained by using the Painlevé truncation technique This is due to our consideration of balancing v xx − 4wx and 2vv x rather than the highest-order derivative term v xx and the highest-order nonlinear term 2vv x in form for Equation (1) in the process of using the Painlevé test to deal with the leading term analysis
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