Xing-Biao Hu

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.

Construction of dKP and BKP equations with self-consistent sources

A new procedure is firstly proposed to construct soliton equations with self-consistent sources (SESCSs) in bilinear forms, starting from the Gram-type determinant solution or Gram-type Pfaffian solution of soliton equations without sources; the soliton solutions of SESCSs can then be given. This procedure is applied to the 2D Toda lattice equation, the discrete KP equation and the BKP equation.

New type of Kadomtsev–Petviashvili equation with self-consistent sources and its bilinear Bäcklund transformation

A new type of the KP equation with self-consistent sources (KPESCS) first found by Melnikov (1983 Lett. Math. Phys. 7 129?36) is re-constructed via source generation procedure. A new feature of the obtained KPESCS is that we allow y-dependence of the arbitrary constants in the determinantal solution for the KP equation while applying the source generation procedure. We also propose a new idea of commutativity of source generation procedure and B?cklund transformations to generate a BT for the new KPESCS which indicates the integrability of the KPESCS.

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.

Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation

The molecule solution of an equation related to the lattice Boussinesq equation is derived with the help of determinantal identities. It is shown that this equation can for certain sequences be used as a numerical convergence acceleration algorithm. Numerical examples with applications of this algorithm are presented.

A q-difference version of the ϵ-algorithm

In this paper, a q-difference version of the -algorithm is proposed. By using determinant identities the solutions of an initial value problem thus arisen can be expressed as ratios of Hankel determinants. It is shown that in numerical analysis this algorithm can be used to compute the approximation limt→∞f(t), and in the field of integrable systems it could be viewed as the q-difference version of the modified Toda molecule equation.

A special integrable differential-difference equation and its related systems: Bilinear forms soliton solutions and Lax pairs

In this paper, a special integrable differential-difference equation and its related systems are studied. First of all, by using dependent variable transformations, this special lattice is transformed into two bilinear forms. As a result, the corresponding soliton solutions are obtained. A coupled set of bilinear equations is proposed and related to the same special lattice in a certain way. We also derive the t -flow and z -flow of the coupled bilinear equations. Lax pairs for the t -flow and the z -flow are given. Furthermore, a bilinear Backlund transformation and the corresponding nonlinear superposition formula for the coupled bilinear equations are presented. Soliton solutions to the coupled bilinear equations are derived.

Integrable discretizations of the (2+1)-dimensional sinh-Gordon equation

In this paper, we propose two semi-discrete equations and one fully discrete equation and study them by Hirotas bilinear method. These equations have continuum limits into a system which admits the (2+1)-dimensional generalization of the sinh-Gordon equation. As a result, two integrable semi-discrete versions and one fully discrete version for the sinh-Gordon equation are found. Backlund transformations, nonlinear superposition formulae, determinant solution and Lax pairs for these discrete versions are presented.

Pfaffianization of the three-dimensional three-wave equation

A pfaffianized version of the three-dimensional three-wave equation is found using Hirota and Ohtas pfaffianization procedure. In addition, n-lump solutions to the pfaffianized system are presented.

Multistep –algorithm, Shanks’ transformation, and the Lotka–Volterra system by Hirota’s method

In this paper, we propose a multistep extension of the Shanks sequence transformation. It is defined as a ratio of determinants. Then, we show that this transformation can be recursively implemented by a multistep extension of the ε–algorithm of Wynn. Some of their properties are specified. Thereafter, the multistep ε–algorithm and the multistep Shanks transformation are proved to be related to an extended discrete Lotka–Volterra system. These results are obtained by using Hirota’s bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations. 1. The scenery Let (Sn) be a sequence of numbers converging to S. If its convergence is slow, it can be transformed, by a sequence transformation, into a set of new sequences {(T (n) k )}, depending on two indexes k and n, and converging, under certain assumptions, faster to the same limit, that is such that lim n→∞ T (n) k − S Sn − S = 0, or lim k→∞ T (n) k − S Sk − S = 0, or both. A well–known example of such a transformation is the Richardson extrapolation process, which gives rise to the Romberg’s method for accelerating the convergence of the trapezoidal rule for approximating a definite integral. Let us mention that sequence transformations can also be applied to diverging power sequences, thus leading, in some situations, to interesting results such as analytic continuation (this is the case of the ε–algorithm which, applied to the partial sum of a divergent power series, computes its Padé approximants). In many sequence transformations, the terms of the new sequences can be expressed as ratios of determinants, and there exists, in each particular case, a (usually nonlinear) recursive algorithm for avoiding the computation of these determinants and implementing the transformation under consideration [11, 39, 44, 43, 45]. The most well–known transformation of this type is due to Shanks [37, 38]. It can be implemented via the ε–algorithm of Wynn [46]. Recently, a new recursive algorithm for accelerating the convergence of sequences was derived by He, Hu, Sun Received by the editor December 21, 2010 and, in revised form, March 14, 2011. 2010 Mathematics Subject Classification. Primary 65B05, 39A14, 37K10.