Integrability of auto-Bäcklund transformations and solutions of a torqued ABS equation

An auto-Bäcklund transformation for the quad equation Q1₁ is considered as a discrete equation, called H2^a, which is a so called torqued version of H2. The equations H2^a and Q1₁ compose a consistent cube, from which an auto-Bäcklund transformation and a Lax pair for H2^a are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund transformation for H2^a we derive a seed solution and a one-soliton solution. From this solution it is seen that H2^a is a semi-autonomous lattice equation, as the spacing parameter q depends on m but it disappears from the plane wave factor.     

Wronskian Form of N-Solitonic Solution for a Variable-Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.     

On an Auto-Bäcklund Transformation for (2+1)-Dimensional Variable Coefficient Generalized KP Equations and Exact Solutions

By the application of the extended homogeneous balance method, we derive an auto-Bäcklund transformation (BT) for (2+1)-dimensional variable coefficient generalized KP equations. Based on the BT, in which there are two homogeneity equations to be solved, we obtain some exact solutions containing single solitary waves.     

Detecting identical entanglement pure states for two qubits

In this paper, investigation is made on a Kadomtsev–Petviashvili-based system, which can be seen in fluid dynamics, biology and plasma physics. Based on the Hirota method, bilinear form and Bäcklund transformation (BT) are derived. N-soliton solutions in terms of the Wronskian are constructed, and it can be verified that the N-soliton solutions in terms of the Wronskian satisfy the bilinear form and Bäcklund transformation. Through the N-soliton solutions in terms of the Wronskian, we graphically obtain the kink-dark-like solitons and parallel solitons, which keep their shapes and velocities unchanged during the propagation.     

<Emphasis Type="Italic">N</Emphasis>-Solitonic Solution in Terms of Wronskian Determinant for a Perturbed Variable-Coefficient Korteweg-de Vries Equation

With the help of symbolic computation, we first derive the bilinear form and an auto-Bäcklund transformation for a perturbed variable-coefficient Korteweg-de Vries equation in this paper. We also construct the N-solitonic solution in the Wronskian form and give the corresponding proof via the Wronskian technique. Furthermore, the authors verify that the (N?1)- and N-solitonic solutions indeed satisfy the auto-Bäcklund transformation.     

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Bäcklund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via the Bäcklund transformation (BT) and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the G_N(t) for the original diagonal one.     

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation

In this Letter, the Bäcklund transformation for the (2+1)-Calogero equation is presented in the bilinear form. Furthermore, a nonlinear superposition formula related to the transformation is proved rigorously. By the way, the Wronskian determinant solution is also derived and verified completely.     

COMMUTATIVITY OF PFAFFIANIZATION AND BÄCKLUND TRANSFORMATIONS: THE LEZNOV LATTICE

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice introduced by J. Zhao etc. [1]. This implies that Pfaffianization and Bäcklund transformation is commutative for the Leznov lattice. Moreover, since the two-dimensional Toda lattice constitutes the Leznov lattice, it is obvious that the commutativity is also valid for it.     

Bäcklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for (2+1)-Dimensional Breaking Soliton Equations with Symbolic Computation

In this paper, two types of the (2+1)-dimensional breaking soliton equations are investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinear forms and Bäcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilinear equations. Via the Wronskian technique, it is proved that theBäcklund transformations obtained are the ones between the (N-1)- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonic properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

Bilinear forms through the binary Bell polynomials,N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-Bäcklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and Bäcklund transformations are different from those in the existing literature. All of our results are dependent on the water-wave dispersive power.     

The exact solutions to the complex KdV equation

In this Letter, Wronskian solutions for the complex KdV equation are obtained by Hirota's bilinear method. Moreover, starting from the bilinear Bäcklund transformation, multi-soliton solutions are presented for the same equation. At the same time, it is also shown that these two kinds of solutions are equivalent.     

Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications

In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schrödinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Bäcklund transformation transforms between (N − 1)- and N-soliton solutions.     

PROLONGATION STRUCTURE APPROACH TO THE GROUP-THEORETICAL TECHNIQUE FOR GENERATING STATIONARY AXISYMMETRIC SPACE-TIMES

The group-theoreti cal technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)xc(t) structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem(HRP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.     

A Bäcklund Transformation of the Restricted mKdV Flow with a Rosochatius Deformation

A Bäcklund transformation of the restricted mKdV flow with a Rosochatius deformation is constructed. Its Lax representation and thus N invariants in involution are presented. Such Bäcklund transformation is a Rosochatius deformation of that of the restricted mKdV flow.     

Bäcklund Transformat ion and Mult isoliton-Like Solutions for (2+1)-Dimensional Dispersive Long Wave Equations

A Bäcklund transformation of the (2+1)-dimensional dispersive long wave equations is derived by using the developed homogeneous balance method. by means of the Bäcklund transformation, the new multisoliton-like solution and other two types of exact solutions to these equations are constructed.     

A variational approach to bäcklund and infinitesimal bäcklund transformation for Kadomtsev-Petviashvili equation

A new form of the Bäcklund transformation for the Kadomtsev-Petviashvili equation is obtained through the variational formalism. At no stage we use the equation of motion for the deduction of the Bäcklund transformation. So we follow the method of Steudel for the derivation of the infinite number of conservation laws using Noether's theorem in three dimensions and the corresponding infinitesimal Bäcklund transformation.     

New Exact Solution of (N+1)-Dimensional Burgers System

In this letter, using a Bäcklund transformation and the new variable separation approach, we find a new general solution of the (N+1)-dimensional Burgers system. The form of the universal formula obtained from many (2+1)-dimensional system is extended.     

Double Bäcklund transformations and fission solutions of axisymmetric gravitational equations

The results of Kyriakopoulos are extended to a double-complex form, and a double Bäcklund transformation of the Ernst equation is derived concretely. By using the noncommutative relation between this double Bäcklund transformation and the dual mapping, a fission-type generation process of new solutions is discussed.