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Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients. (English) Zbl 1304.35601

Summary: We first propose a way for generating Lie algebras from which we get a few kinds of reduced Lie algebras, denoted by \(R^{6}\), \(R^{8}\) and \(R_{1}^{6}\), \(R_{2}^{6}\), respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra \(R^{6}\) whose compatibility gives rise to an integrable hierarchy with 4-potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra \(R_{1}^{6}\) to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again, via using the Lie algebra \(R_{2}^{6}\), we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.

MSC:

35Q51 Soliton equations
35C05 Solutions to PDEs in closed form
35C07 Traveling wave solutions
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
17B80 Applications of Lie algebras and superalgebras to integrable systems
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