Two (2+1)-dimensional expanding dynamical systems associated to the mKP hierarchy

An isospectral problem with a parameter ? is presented, for which a modified KdV (mKdV) hierarchy is easily derived from the Tu scheme. Then two different expanding integrable models are obtained, respectively. The main purpose for this focuses on deriving a (2+1)-dimensional mKdV hierarchy (called the mKP hierarchy) and the corresponding different (2+1)-dimensional expanding models (including the linear and nonlinear), whose Hamiltonian structures are obtained by employing an identity developed by us in the paper. As the reduced consequences, the linear and nonlinear (2+1)-dimensional expanding models of the mKP equation are generated.

[1]  Zhang Yu-feng.  Lie Algebras for Constructing Nonlinear Integrable Couplings , 2011 .

[2]  Zhenya Yan,et al.  An improved algebra method and its applications in nonlinear wave equations , 2004 .

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

[4]  Yufeng Zhang,et al.  A direct method for integrable couplings of TD hierarchy , 2002 .

[5]  Yufeng Zhang,et al.  An extended trace identity and applications , 2008 .

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

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

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

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

[10]  A pair of finite-dimensional integrable systems possessing the common non-dynamical r-matrix , 2002 .

[11]  Yufeng Zhang,et al.  On generating (2 + 1)-dimensional hierarchies of evolution equations , 2014, Commun. Nonlinear Sci. Numer. Simul..

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

[13]  Jing Gao,et al.  Two (2 + 1)-dimensional hierarchies of evolution equations and their Hamiltonian structures , 2014, Appl. Math. Comput..

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

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

[16]  M. Błaszak.  Miura map and Bi-Hamiltonian formulation for restricted flows of the KdV hierarchy , 1993 .

[17]  Yufeng Zhang,et al.  Four Lie algebras associated with R6 and their applications , 2010 .

[18]  Zhang Yu.  Lie Algebras for Constructing Nonlinear Integrable Couplings , 2011 .

[19]  H. Tam,et al.  Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang—Mills Equations , 2014 .

[20]  Blazej M. Szablikowski,et al.  Classical R-matrix theory for bi-Hamiltonian field systems , 2009, 0902.1511.

[21]  W. Ma.  Integrable couplings of soliton equations by perturbations I: A general theory and application to the KDV hierarchy , 1999, solv-int/9912004.

[22]  Xianguo Geng,et al.  On quasi-periodic solutions of the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation , 1999 .