An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation

A new integrable hierarchy of evolution equations is obtained by making use of a Lie algebra and Tu-Ma scheme, from which a new generalized Broer-Kaup (gBK) equation is produced. Then two kinds of Darboux transformations, the bilinear presentation, the bilinear Backlund transformation and the new Lax pair of the gBK equation are generated, respectively, by employing the Bell polynomials.

[1]  Engui Fan,et al.  Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik–Novikov–Veselov equation , 2009 .

[2]  Wen-Xiu Ma,et al.  Computers and Mathematics with Applications Linear Superposition Principle Applying to Hirota Bilinear Equations , 2022 .

[3]  J. Nimmo,et al.  On the combinatorics of the Hirota D-operators , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  Johan Springael,et al.  Classical Darboux transformations and the KP hierarchy , 2001 .

[5]  Johan Springael,et al.  Soliton Equations and Simple Combinatorics , 2008 .

[6]  Yi Zhang,et al.  Hirota bilinear equations with linear subspaces of solutions , 2012, Appl. Math. Comput..

[7]  Z. Qiao.  Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions , 2004 .

[8]  Engui Fan,et al.  Super extension of Bell polynomials with applications to supersymmetric equations , 2012 .

[9]  Kwok Wing Chow,et al.  Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation , 2011 .

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

[11]  Z. Qiao.  A new integrable equation with cuspons and W/M-shape-peaks solitons , 2006 .

[12]  Engui Fan,et al.  New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation , 2011 .

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

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

[15]  Yishen Li,et al.  Darboux transformations of classical Boussinesq system and its multi-soliton solutions , 2001 .

[16]  Wen-Xiu Ma,et al.  Darboux transformations of classical Boussinesq system and its new solutions , 2000 .

[17]  Engui Fan,et al.  Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation , 2000 .

[18]  Shengtai Li,et al.  A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions , 2002 .

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

[20]  Wen-Xiu Ma,et al.  Nonlinear continuous integrable Hamiltonian couplings , 2011, Appl. Math. Comput..

[21]  Wenxiu Ma.  Darboux Transformations for a Lax Integrable System in 2n Dimensions , 1996, solv-int/9605002.

[22]  Engui Fan,et al.  The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials , 2011 .