Hon-Wah Tam

The Hirota-Satsuma coupled KdV equation and a coupled Ito system revisited

The well-known Hirota-Satsuma coupled KdV equation and a coupled Ito system are reviewed. A new type of soliton solutions to these two systems under constant boundary condition at infinity is found. The so-called generalized Hirota-Satsuma coupled KdV system is also considered. Starting from its bilinear forms, we obtain a Backlund transformation and the corresponding nonlinear superposition formulae. As a result, soliton solutions first obtained by Satsuma and Hirota can be rederived. Moreover, rational solutions are also given.

Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov

Abstract We consider the Leznov lattice in this paper. By a dependent variable transformation, the Leznov lattice is transformed into a quadri-linear form. This form is further transformed into a bilinear form by the introduction of an auxiliary variable. We present a Backlund transformation and a nonlinear superposition formula for the Leznov lattice. As an application of the results, soliton solutions and lump solutions are derived. Besides, starting from the bilinear BT, a Lax pair for the Leznov lattice is obtained.

Lax pairs and Bäcklund transformations for a coupled Ramani equation and its related system

Abstract A coupled Ramani equation and its related system are proposed. By dependent variable transformation, they are transformed into bilinear equations. Lax pairs and Backlund transformations are presented for these two systems. Soliton solutions and rational solutions to the systems could be obtained.

Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: a bilinear approach

The Kupershmidt five-field lattice is considered in this paper. By a dependent variable transformation, the Kupershmidt lattice is transformed into a bilinear form by the introduction of three auxiliary variables. We present a Backlund transformation and a nonlinear superposition formula for the Kupershmidt lattice. As an application of the results, soliton solutions are derived.

Soliton solutions to the Jimbo–Miwa equations and the Fordy–Gibbons–Jimbo–Miwa equation

Abstract This Letter considers two Jimbo–Miwa equations and the Fordy–Gibbons–Jimbo–Miwa equation in their bilinear form. Three-soliton solutions to these equations are explicitly derived by the Hirota method with the assistance of Mathematica.

Four Lie algebras associated with R6 and their applications

The first part in the paper reads that a three-dimensional Lie algebra is first introduced, whose corresponding loop algebra is constructed, for which isospectral problems are established. By employing zero curvature equations, a modified Kaup–Newell (mKN) soliton hierarchy of evolution equations is obtained. The corresponding hereditary operator and Hamiltonian structure are worked out, respectively. Then two types of enlarging semisimple Lie algebras isomorphic to the linear space R6 are followed to construct, one of them is a complex Lie algebra. Their corresponding loop algebras are also given so that two types of new isospectral problems are introduced to generate two kinds of integrable couplings of the above mKN hierarchy. The hereditary operators, Hamiltonian structures of the hierarchies are produced again, respectively. The exact computing formulas of the constant γ appearing in the trace identity and the variational identity are derived under the semisimple algebras. The second part of this pap...

New integrable differential-difference systems: Lax pairs, bilinear forms and soliton solutions

Two new integrable differential-difference systems with their Lax pairs are proposed. By the dependent variable transformations, these integrable lattices can be transformed into bilinear equations. With the assistance of Mathematica, three-soliton solutions are explicitly obtained. We have also shown that these lattices can be obtained from a special case of the coupled bilinear equations under reduction. Furthermore a bilinear Backlund transformation and the corresponding nonlinear superposition formula concerning the coupled bilinear equations are presented. Besides, it is also illustrated that the y-flow of these coupled bilinear equations can be transformed into a lattice previously derived by the authors. Starting from the corresponding bilinear Backlund transformation, its corresponding Lax pair is obtained.

Three kinds of coupling integrable couplings of the Korteweg–de Vries hierarchy of evolution equations

We introduce three kinds of column-vector Lie algebras Ls(s=1,2,3). By making invertible linear transformations we get the corresponding three induced Lie algebras. According to the defined loop algebras Ls of the Lie algebras Ls(s=1,2,3), we establish three various isospectral problems. Then by applying Tu scheme, we obtain three different coupling integrable couplings of the Korteweg–de Vries (KdV) hierarchy and further reduce them to three kinds of explicit coupling integrable couplings of the KdV equation. One of the coupling integrable couplings of the KdV hierarchy of evolution equations possesses Hamiltonian structure obtained by using the quadratic-form identity and it is Liouville integrable.

An integrable system and associated integrable models as well as Hamiltonian structures

Starting from an existed Lie algebra introduces a new Lie algebra A1 = {e1, e2, e3} so that two isospectral Lax matrices are established. By employing the Tu scheme an integrable equation hierarchy denoted by IEH is obtained from which a few reduced evolution equations are presented. One of them is the mKdV equation. The elliptic variable solutions and three kinds of Darboux transformations for one coupled equation which is from the IEH are worked out, respectively. Finally, we take use of the Lie algebra A1 to generate eight higher-dimensional Lie algebras from which the linear integrable couplings, the nonlinear integrable couplings, and the bi-integrable couplings of the IEH are engendered, whose Hamiltonian structures are also obtained by the variational identity. Then further reduce one coupled integrable equation to get a nonlinear generalized mKdV equation.

Nonisospectral negative Volterra flows and mixed Volterra flows: Lax pairs, infinitely many conservation laws and integrable time discretization

In this paper, by means of the discrete zero curvature representation, nonisospectral negative Volterra flows and mixed Volterra flows are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for the two nonisospectral flows and obtain the formulae of the corresponding conserved densities and associated fluxes. Integrable time discretizations for several isospectral equations of the two flows are also presented.