Wronskian Technique Research Articles

Classification of solutions for the (2+1)-dimensional Fokas–Lenells equations based on bilinear method and Wronskian technique

Classification of solutions for the (2+1)-dimensional Fokas–Lenells equations based on bilinear method and Wronskian technique

Wronskian solutions and [formula omitted]–soliton solutions for the Hirota–Satsuma equation

By employing the Hirota method and Wronskian technique, we firstly give the bilinear form, N-soliton solutions and the Wronskian solutions of the Hirota–Satsuma equation. Explicit one- and two-soliton solutions are given for the Hirota–Satsuma equation. The solutions of good Boussinesq equation are obtained through the Miura transformation. The two solitons have the degenerated forms of antisoliton-antikink and W-shape type.

Wronskian solution, Bäcklund transformation and Painlevé analysis to a (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Abstract The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution.

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Nonautonomous dynamics of local and nonlocal Fokas–Lenells models

We investigate the dynamics of solutions to the local and nonlocal nonautonomous Fokas–Lenells (FL) models. Solutions to the local and nonlocal reductions are presented using the Wronskian technique by imposing constraints on the double Wronskian solutions to the unreduced nonautonomous FL system. Classifications of the soliton solutions are shown in relation to the distribution of the eigenvalues. Moreover, dynamics of these solutions are presented, along with the impacts of the variable coefficient . Notably, the coefficient influences the top traces and velocities of the found solutions in both the local and nonlocal cases, enriching this study.

Solutions to nonlocal nonisospectral (2 + 1)-dimensional breaking soliton equations

In this paper, we use the double Wronskian reduction technique to consider nonlocal reductions of a nonisospectral (2+1)-dimensional breaking Ablowitz--Kaup--Newell--Segur equation. Various types of solutions, including soliton-like solutions and Jordan-block solutions, for the resulting nonlocal equations are derived. Dynamics of these obtained solutions are analyzed and illustrated.

Bilinear forms, N-soliton solutions, breathers and lumps for a (2+1)-dimensional generalized breaking soliton system

In this paper, a ([Formula: see text])-dimensional generalized breaking soliton system, for the interactions of the Riemann wave with a long wave, is investigated. Via the Hirota method, bilinear forms different from those in the existing literatures are derived. [Formula: see text]-soliton solutions are constructed via the Wronskian technique. Solitons with the crest curves being curvilineal are constructed, whose shape changes with the propagation. Parallel solitons have been obtained. Directions of the soliton propagation change, and speeds of the solitons are different: The higher the amplitude of the soliton is, the faster the soliton propagates. Breathers are constructed. Solutions consisting of a lump and two solitons are derived: Two solitons propagate in the same direction and the lump occurs in the region of the interaction between the two solitons.

Decomposition of 2-Soliton Solutions for the Good Boussinesq Equations

We consider decompositions of two-soliton solutions for the good Boussinesq equation obtained by the Hirota method and the Wronskian technique. The explicit forms of the components are used to study the dynamics of 2-soliton solutions. An interpretation in the context of eigenvalue problems arising from KdV type equations and transport equations is considered. Numerical examples are included.

Double Wronskian Solutions for a Generalized Nonautonomous Nonlinear Equation in a Nonlinear Inhomogeneous Fiber**Supported by the Fundamental Research Funds for the Central Universities under Grant No. 2018MS132

A generalized nonautonomous nonlinear equation, which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber, is investigated. N-soliton solutions for such an equation are constructed and verified with the Wronskian technique. Collisions among the three solitons are discussed and illustrated, and effects of the coefficients σ1(x, t), σ2(x, t), σ3(x,t) and v(x, t) on the collisions are graphically analyzed, where σ1(x, t), σ2(x, t), σ3(x, t) and v(x, t) are the first-, second-, third-order dispersion parameters and an inhomogeneous parameter related to the phase modulation and gain(loss), respectively. The head-on collisions among the three solitons are observed, where the collisions are elastc. When σ1(x, t) is chosen as the function of x, amplitudes of the solitons do not alter, but the speed of one of the solitons changes. σ2(x, t) is found to affect the amplitudes and speeds of the two of the solitons. It reveals that the collision features of the solitons alter with σ3(x, t) = −1.8x. Additionally, traveling directions of the three solitons are observed to be parallel when we change the value of v(x, t).

Generalized Wronskian solutions of modified Boussinesq equation

The generalized Wronskian solutions whose elements satisfy matrix equation of linear problem for the modified Boussinesq equation are obtained through Wronskian technique. Furthermore, rational solutions, Matveev solutions, complexiton solutions and interaction solutions are derived by taking special cases in general solutions.

WRONSKIAN SOLUTIONS OF (2+1) DIMENSIONAL NON-LOCAL ITO EQUATION

In this work, the Wronskian determinant technique is performed to(2+1)-dimensional non-local Ito equation in the bilinear form. First, we obtainsome su¢ cient conditions in order to show Wronskian determinant solves the(2+1)-dimensional non-local Ito equation. Second, rational solutions, solitonsolutions, positon solutions, negaton solutions and their interaction solutionswere deduced by using the Wronskian formulations

Wronskian and linear superposition solutions to generalized KP and BKP equations

We aim to find exact solutions to some generalized KP- and BKP-type equations by the Wronskian technique and the linear superposition principle. By applying Wronskian identities of the bilinear KP hierarchy, three Wronskian formulations are first furnished, with all generating functions for matrix entries satisfying a system of combined linear partial differential equations. Second, we apply the linear superposition principle to exponential traveling wave solutions of the introduced generalized KP and BKP equations. Each exponential wave in the Wronskian and N-wave solutions satisfies the corresponding nonlinear dispersion relation.

Wronskian and Grammian solutions for a [formula omitted]-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, a (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation is investigated. Based on the Hirota method and auxiliary variable, Bäcklund transformation is obtained. Under certain conditions, the Hirota bilinear forms can be reduced to the Plücker and Jacobi identities via the Wronskian and Grammian solutions. Through the Wronskian technique and Pfaffian derivative formulae, N-soliton solutions in the Wronskian and Grammian are given. Graphically presented, soliton collisions are elastic both in the Wronskian and Grammian, and after each collision, the solitons keep their amplitudes unchanged except for the phase shifts.

Multiple-soliton solutions and a generalized double Wronskian determinant to the (2+1)-dimensional nonlinear Schr\xf6dinger equations

A (2+1)-dimensional nonlinear Schrödinger equation is mainly discussed. Based on the Hirota direct method and the Wronskian technique, multiple-soliton solutions and a generalized double Wronskian determinant are obtained, respectively.

Solitons for a (2+1)-dimensional Sawada–Kotera equation via the Wronskian technique

Under investigation in this letter is a (2+1)-dimensional Sawada–Kotera equation. Solitons are obtained by virtue of the Wronskian technique. Via the fourth- and sixth-order Plücker relations for the Wronskian, we give a proof for the N -soliton solutions. Interactions between/among the two/three solitons are investigated, and it seems that those interactions are elastic.

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.

The triple Wronskian solutions to the variable-coefficient Manakov model

In this paper, the variable-coefficient Manakov model whose bilinear form exists is mainly discussed. Based on the Wronskian technique, the triple Wronskian form solutions are obtained and the interactions between the two solitons are investigated.

Soliton Solutions for Nonlinear Evolution Equations with Symbolic Computation

The study of exact solutions of nonlinear evolution equations plays an important role in soliton theory. In this paper, we concentrate on proposing to a new Wronskian condition for the Levi equations by utilizing the Wronskian technique, which will lead to a way of constructing new double Wronskian representation of solutions for the Levi equations. Furthermore, some soliton solutions from the particular form of Wronskian determinant solutions are obtained with the aid of the symbolic computation system Maple.

Double Wronskian Solution and Soliton Properties of the Nonisospectral BKP Equation**Supported by the Research Committee of The Hong Kong Polytechnic University under Grant No. G-YM37, the AMSS-PolyU Joint Research Institute for Engineering and Management Mathematics under Grant No. 1-ZVA8, National Natural Science Foundation of China under Grant Nos. 11271362 and 11375030, Beijing Natural Science Fund Project and Beijing City Board

Based on the Wronskian technique and Lax pair, double Wronskian solution of the nonisospectral BKP equation is presented explicitly. The speed and dynamical influence of the one soliton are discussed. Soliton resonances of two soliton are shown by means of density distributions. Soliton properties are also investigated in the inhomogeneous media.

New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.