Wronskian Form Research Articles

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

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 structures of planar symplectic ensembles

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently discovered that the limiting correlation kernel of the symplectic Ginibre ensemble in the vicinity of the real line can be expressed in a unified form of a Wronskian. We derive scaling limits for variations of the symplectic Ginibre ensemble and obtain such Wronskian structures for the associated universality classes. These include almost-Hermitian bulk/edge scaling limits of the elliptic symplectic Ginibre ensemble and edge scaling limits of the symplectic Ginibre ensemble with boundary confinement. Our proofs follow from the generalised Christoffel–Darboux formula for the former and from the Laplace method for the latter. Based on such a unified integrable structure of Wronskian form, we also provide an intimate relation between the function in the argument of the Wronskian in the symplectic symmetry class and the kernel in the complex symmetry class which form determinantal point processes in the plane.

Exact solutions of a (3+1)-dimensional nonlinear evolution equation based on its Wronskian form

In this paper, the Hirota bilinear method is applied to investigate the exact solutions of a (3+1)-dimensional nonlinear evolution equation. The soliton, breather and lump solutions satisfying specific Wronskian conditions are obtained. Especially, starting from the 2Mth-order Wronskian determinant solution, we obtain the determinant expression of arbitrary Mth-order lump solution through the elementary transformation and long wave limit method. Some images are used as examples to illustrate their dynamics. These results can be helpful to understand the propagation processes of nonlinear waves in some nonlinear physical systems, such as fluid mechanics, nonlinear optics and so on.

Formula omitted]-lump solution, soliton solution and rational solution to a (3+1)-dimensional nonlinear model

In the previous study, the one-lump solution is given to the dimensionally reduced forms of a (3+1)-dimensional nonlinear model via the positive quadratic function method. The main work of this paper is to construct the M-lump solution and the Wronskian solution to this (3+1)-dimensional nonlinear model. Firstly, the M-lump solution is constructed by using the long wave limit method. As an example, the three-dimensional plots of one-, two- and three-lump solutions and the corresponding density plots have been shown through selecting appropriate parameters. What is more, their motion process is analyzed systematically. Secondly, a sufficient condition of Wronskian solution is given by using the properties of determinant and Plücker relation. Based on the Wronskian form, we obtain the soliton solution and the rational solution by selecting the elements in the determinant which satisfy the linear partial differential systems. Finally, several specific examples are presented.

On (2+1)-dimensional mixed AKNS hierarchy

In this paper we revisit the (2+1)-dimensional mixed AKNS hierarchy recently studied by Gürses and Pekcan (2019, 2021). We present a unified bilinear form for the whole hierarchy and solutions in double Wronskian form. Local and nonlocal reductions of the hierarchy are considered and explicit N-soliton solutions for the reduced equations are obtained by means of a reduction technique on double Wronskians.

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.

Degeneration of trigonal curves and solutions of the KP-hierarchy

It is known that soliton solutions of the KP-hierarchy correspond to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to be some intermediate solutions between solitons and rational solutions. They are considered as certain limits of solitons.The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects solutions of the KP-hierarchy to the defining equations of algebraic curves. We define a class of solutions in the Wronskian form which contain soliton solutions as a subclass and prove that, using the Sato Grassmannian, the degenerate trigonal solutions are connected to those solutions by certain gauge transformations.

An integral containing a Bessel Function and a Modified Bessel Function of the First Kind

Here we discuss the calculation of an integral containing the Bessel function J0(r) and the modified Bessel function of the first kind I1(r). The calculus is based on a function of J0(r), I1(r) and of their derivatives, having a Wronskian form. The method here described could be useful for training the students in the manipulation of such integrals.

Solitons for a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.

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.

Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction

We have investigated the femtosecond soliton propagation in inhomogeneous fiber, which is described by the modified inhomogeneous Hirota equation with variable coefficient (MIH-vc). With the aid of AKNS method, corresponding Lax pair is constructed. By virtue of the Darboux transformation method and symbolic computation, the analytic one- and two-soliton solutions are explicitly obtained. Using obtained solutions, we graphically discuss the features of femtosecond solitons in modified inhomogeneous Hirota system by changing the profile of variable coefficients. We analyze various form of group velocity dispersion, third order dispersion and nonlinearity parameter for periodic amplification system, exponentially distributed system, parabolic solitons, periodic exponentially modulated system, which will be observable in the future experiments. These results are potentially useful in future experiments and soliton control for long-distance optical communication. Finally, the soliton solutions of the MIH-vc equation in double Wronskian form is constructed and further verified using the Wronskian technique by substitute in bilinear equations.

Solutions to the modified Korteweg–de Vries equation

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, arXiv:nlin.SI/0603008]. In the present paper, we review solutions to the modified Korteweg–de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.

Wronskian Form Solutions for a Variable Coefficient Kadomtsev—Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relationship between the solution of a generalized variable coefficient Kadomtsev—Petviashvili (vKP) equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.

New breather solutions of the model describing few-optical-cycle solitons beyond the slowly varying envelope approximation

The modified Korteweg–de Vries-sine-Gordon equation provides a general (1 + 1)-dimensional model describing few-cycle-pulse soliton propagation in two-component nonlinear media. For this model, we give its solutions in Wronskian form, which is convenient to describe interactions between solitons, breathers and their limit forms. Among these solutions, we get a stationary breather solution. Besides, the limit breather solution can provide two identical breathers traveling along the trajectories that are similar to logarithm curves rather than straight lines. As a reduction, we can also obtain solutions to the modified Korteweg–de Vries equation and the sine-Gordon equation.

Soliton solution to BKP equation in Wronskian form

We find out a group of identities, named Wronskian identities of bilinear KP hierarchy here, as well as two useful properties of D-operator. This makes it possible to easily search for some new Wronskian solutions for PDE which owns bilinear form, and to simplify the process of the proof. As an application of this new method, we propose the first Wronskian condition for the BKP I equation and for the BKP II equation, respectively. In addition, we also discuss several generalized BKP equations and obtain their Wronskian conditions.

New interaction solutions of (3+1)-dimensional Zakharov–Kuznetsov equation

Based on forms and structures of Wronskian determinant, a novel Wronskian form expansion method is presented to find new interaction solutions for (3+1)-dimensional Zakharov–Kuznetsov equation and the functions used in Wronskian determinants that do not satisfy linear partial differential equations. Results have shown that this method is a powerful tool not only to construct a number of traveling wave solutions but also to construct interaction solutions between two different kinds of functions for nonlinear wave equations in mathematical physics and engineering.

Bäcklund transformation and Wronskian solitons for the [formula omitted]-dimensional Nizhnik–Novikov–Veselov equations

Korteweg–de Vries-type equations are seen to describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, an isotropic extension of which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equations. Hereby, based on the Hirota bilinear method and symbolic computation, we derive the bilinear form and Bäcklund transformation for such an extension. N-soliton solutions in the Wronskian form are given, and it can be verified that the Bäcklund transformation can connect the (N−1)- and N-soliton solutions. Solitonic propagation and collision are discussed: the larger-amplitude soliton moves faster and then overtakes the smaller one. After the collisions, the solitons keep their original shapes and velocities invariant except for the phase shift. Collisions among the two and three solitons are all elastic.

Construction of exact solutions to the Jimbo–Miwa equation through Bäcklund transformation and symbolic computation

Based on a simple transformation, and with the aid of symbolic computation, a Bäcklund transformation relating the Jimbo–Miwa equation and a system of linear partial differential equations is obtained, which enables us to construct exact solutions of the Jimbo–Miwa equation through the Wronskian determinants of independent solutions of the linear system. Particularly, explicit Wronskian form N-soliton solutions for the Jimbo–Miwa equation are presented. Moreover, the introduced transformation also helps to construct bi-soliton-like solutions of the Jimbo–Miwa equation. Due to the arbitrary functions they contain, the bi-soliton-like solutions can represent various waves such as classical cross-line bi-solitons, curved bi-solitons and bi-soliton-like breathers.

New Wronskian Form of the N-Soliton Solution to a (2 + 1)-Dimensional Breaking Soliton Equation

Soliton phenomena exist in nonlinear science and the financial field. By using the Wronskian technique, a new Wronskian condition is proposed for a (2 + 1)-dimensional breaking soliton equation. Moreover, with the help of the bilinear transformation, a new Wronskian form of the N-soliton solution is obtained for the (2 + 1)-dimensional breaking soliton equation.