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-Bäcklund transformation for a variable-coefficient 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-Bäcklund transformation through the Wronskian technique.     

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.     

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.     

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.     

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.     

Quasi-periodic wave solutions,soliton solutions,and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition.     

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.     

New Bilinear Bäcklund Transformation and Higher Order Rogue Waves with Controllable Center of a Generalized (3+1)-Dimensional Nonlinear Wave Equation

In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.     

Bcklund Transformations and Solutions of a Generalized Kadomtsev-Petviashvili Equation

In this paper,the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials.The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function,respectively.And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution.In the end,the bilinear Bcklund transformations are derived.     

<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.     

Lie Symmetry Analysis,Bäcklund Transformations and Exact Solutions to (2+1)-Dimensional Burgers' Types of Equations

This paper is concerned with the (2+1)-dimensional Burgers' and heat types of equations. All of the geometic vector fields of the equations are obtained, an optimal system of the equation is presented. Especially, the Bäcklund transformations (BTs) for the Burgers' equations are constructed based on the symmetry. Then, all of the symmetry reductions are provided in terms of the optimal system method, and the exact explicit solutions are investigated by the symmetry reductions and Bäcklund transformations.     

On the Quasi-Periodic Wave Solutions and Asymptotic Analysis to a (3+1)-Dimensional Generalized Kadomtsev—Petviashvili Equation

In this paper, a (3+1)-dimensional generalized Kadomtsev—Petviashvili (GKP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the generalized Bell's polynomials, we succinctly construct the Hirota's bilinear equation to the GKP equation. By virtue of multidimensional Riemann theta functions, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta function periodic waves (quasi-periodic waves) for the (3+1)-dimensional GKP equation. Interestingly, the one-periodic waves are well-known cnoidal waves, which are considered as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional that they have two independent spatial periods in two independent horizontal directions. Finally, we analyze asymptotic behavior of the multiperiodic periodic waves, and rigorously present the relationships between the periodic waves and soliton solutions by a limiting procedure.     

Bäcklund transformations as nonlinear Dirac equations

It is pointed out that the Bäcklund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations. The generalisations are inhomogenisations of the Cauchy Riemann equations (or their hyperbolic analogue), whose condensed form makes the transformations easy to remember, which suggests ways to generalise to more than 2 dimensions, and which suggest that complex analysis techniques may be helpful in understanding the transformations.     

Wronskian and Grammian solutions for (3+1)-dimensional Jimbo–Miwa equation

In this paper, based on the Hirota's bilinear method, the Wronskian and Grammian technique, as well as several properties of determinant, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented, which guarantees that the Wronskian determinant and the Grammian determinant solve the (3+1)-dimensional Jimbo-Miwa equation in the bilinear form, and then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions. At last, with the aid of Maple, some of these special exact solutions are shown graphically.     

Soliton Solutions for Nonisospectral BKP Equation

The soliton solutions for the nonisospeetral BKP equation are derived through Hirota method and Pfaffian technique. We also derive the bilinear Baeklund transformations for the isospectral and nonisospeetral BKP equation and find solutions with the help of the obtained bilinear Baeklund transformations.     

Solutions of the nonlocal(2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy

The(2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are presented.Solutions in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced hierarchies.The advantage of our method is that we start from the known solutions of the unreduced bilinear equations,and obtain solitons and multiple-pole solutions for the variety of classical and nonlocal reductions.Dynamical behaviors of some obtained solutions are illustrated.It is remarkable that for some real nonlocal equations,amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions.     

In oceanography,acoustics and hydrodynamics: An extended coupled (2+1)-dimensional Burgers system

In oceanography, acoustics and hydrodynamics, people pay attention to the Burgers-type equations for different wave processes, one of which is an extended coupled (2+1)-dimensional Burgers system hereby under investigation. Based on the scaling transformation, Bell polynomials, Hirota operators and symbolic computation, we structure out two hetero-Bäcklund transformations, each of which to a solvable linear partial differential equation, and construct two sets of the bilinear forms, with the relevant one- and two-soliton solutions. Results rely on the coefficients in the original system.     

The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vries equation

In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is investigated. With the help of symbolic computation, the N-soliton solution is derived through the Hirota method. Then the bilinear Bäcklund transformations and Lax pairs are presented. At last, we show some interactions of solitary waves.     

Soliton Solutions,B/icklund Transformations and Lax Pair for a （3 ＋ 1）-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids

Under investigation in this paper is a （3 q- 1）-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, Backlund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.     

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.