On generating (2 + 1)-dimensional hierarchies of evolution equations

Abstract Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1 + 1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2 + 1 dimensions is obtained by the Hamiltonian operator of the above (1 + 1)-dimensional case and the TAH scheme. The new hierarchy of 2 + 1 dimensions can be reduced to a coupled (2 + 1)-dimensional nonlinear equation and furthermore it can be reduced to the (2 + 1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2 + 1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2 + 1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.

[1]  Xing-Biao Hu,et al.  A powerful approach to generate new integrable systems , 1994 .

[2]  W. Ma.  An approach for constructing nonisospectral hierarchies of evolution equations , 1992 .

[3]  Franco Magri,et al.  A Simple model of the integrable Hamiltonian equation , 1978 .

[4]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[5]  Engui Fan,et al.  Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure , 2001 .

[6]  Gui‐zhang Tu,et al.  The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems , 1989 .

[7]  Xing-Biao Hu,et al.  An approach to generate superextensions of integrable systems , 1997 .

[8]  M. Ablowitz,et al.  Integrable systems and reductions of the self-dual Yang–Mills equations , 2003 .

[9]  M. Ablowitz,et al.  A self-dual Yang-Mills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions , 1993 .

[10]  R. Andrushkiw,et al.  A trace identity and its application to integrable systems of 1 + 2 dimensions , 1991 .

[11]  Wenxiu Ma,et al.  Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras , 2006 .

[12]  E. Fan,et al.  A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations , 2001 .

[13]  R. Hirota.  Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告) , 1976 .