Double Wronskian Solutions Research Articles

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.

The Fokas–Lenells equations: Bilinear approach

AbstractIn this paper, the Fokas–Lenells (FL) equations are investigated via bilinear approach. We bilinearize the unreduced FL system, derive double Wronskian solutions, and then, by means of a reduction technique we obtain variety of solutions of the reduced equations. This enables us to have a full profile of solutions of the classical and nonlocal FL equations. Some obtained solutions are illustrated based on asymptotic analysis. As a notable new result, we obtain solutions to the FL equation, which are related to real discrete eigenvalues and not reported before in the analytic approaches. These solutions behave like (multi)periodic waves or solitary waves with algebraic decay. In addition, we also obtain solutions to the two‐dimensional massive Thirring model from those of the FL equation.

Solitons to the derivative nonlinear Schrödinger equation: Double Wronskians and reductions

In this paper, we derive solutions to the derivative nonlinear Schrödinger equation, which are associated to real and complex discrete eigenvalues of the Kaup–Newell spectral problem. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup–Newell equations and their reductions by means of bilinear method and a reduction technique. The reduced equations include the derivative nonlinear Schrödinger equation and its nonlocal version. Some obtained solutions allow not only periodic behavior, but also solitons on periodic background. Dynamics are illustrated.

New double Wronskian exact solutions for a generalized (2+1)-dimensional nonlinear system with variable coefficients

New generalized (2+1)-dimensional Boussinesq system with variable coefficients has been introduced. A double Wronskian solutions has been formulated to the new system under certain constraints on the variable coefficients. Hirota differential operator and its properties have been employed to transform the system into a bilinear form. Moreover, in order to better understand the dynamic behavior, the characteristics of the double Wronskian solutions are discussed through some diverting graphics under different parameters choices.

Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions

In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation

In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

Double Wronskian solutions to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation

In this paper double Wronskian solutions to the Broer–Kaup–Kupershmidt (BKK) equation are investigated by using of Hirota method and binary bell polynomials. By introducing different auxiliary functions, the bilinear equations for the BKK equation in different forms are obtained respectively. Based on the distinct Hirota bilinear equations, two groups of double Wronskian solutions to the BKK equation are constructed.

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

All possible nonlocal versions of the derivative nonlinear Schrodinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the Kaup–Newell equation and the Gerdjikov–Ivanov equation which are gauge equivalent to each other. Their solutions are obtained by composing constraint conditions on the double Wronskian solution of the Chen–Lee–Liu equation and the nonlocal analogues of the gauge transformations among them. Through the Jordan decomposition theorem, those solutions of the reduced equations from the Chen–Lee–Liu equation can be written as canonical form within real field.

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).

Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction

In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schrödinger hierarchy from the known ones of the Ablowitz–Kaup–Newell–Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new. The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions.

Fusion and fission phenomena for the soliton interactions in a plasma

Investigation in this paper is given to a generalized (3 + 1) -dimensional variable-coefficient Kadomtsev-Petviashvili equation for a plasma. Via the bilinear form, the singular and double Wronskian soliton solutions are derived, respectively, under the different variable-coefficient constraints. Interactions between the two solitons are depicted, where the soliton fusion and fission phenomena are respectively pictured out, both for the velocity-unvarying and velocity-varying two solitons. Soliton velocity is related to the variable coefficients h(t), l (t), q(t), m(t) and n(t), while the soliton amplitude is not affected by them, where h(t), l(t) and q(t) are the perturbed effects, m(t) and n(t) stand for the disturbed wave velocities along the transverse spatial coordinates.

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.

Binary Bell polynomials, Hirota bilinear approach to Levi equation

Combining the binary Bell polynomials and Hirota method, we obtained two kinds of equivalent bilinear equations for the Levi equation. Then, we got the double Wronskian solutions of the Levi equation by virtue of one of the bilinear equations. Furthermore, we constructed the bilinear Bäcklund transformation and the Lax pair. Finally, we also derived the Darboux transformation and the infinite conservation laws of the Levi equation.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

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.

Nonautonomous Matter-Wave Solitons in a Bose–Einstein Condensate with an External Potential

Nonautonomous matter-wave solitons in a Bose–Einstein condensate with an external potential are reported. Via the non-isospectral Ablowitz–Kaup–Newell–Segur system, the Gross–Pitaevskii equation is found to have the double Wronskian solutions. The verification of such solutions is finished through some double Wronskian identities. With the zero-potential Lax pair, the double Wronskian solutions give the nonautonomous N-soliton solutions that contain 2N parameters. For characterizing the asymptotic behavior of the bright two-soliton solutions, the explicit expressions of asymptotic solitons are given. Effects of the linear and harmonic potentials on the bound states between two matter-wave solitons are discussed.

New Exact Solutions of the (2 + 1)-Dimensional AKNS Equation

N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.

Deduction of Double Wronskian Solution and Rational-like Solutions for Generalized Non-autonomous Schrodinger Equation

In this paper, we have presented the bilinear form, a generalized double Wronskian solution of a non-autonomous Schrodinger equa tion. Furthermore, we found rational-like solutions by taking special case in general solutions.

Double Wronskian solutions of a nonlinear Schrödinger equation in an averaged dispersion-managed fiber system

A nonlinear Schrödinger equation in the presence of chirp and loss terms, which describes the optical solitons in an averaged dispersion-managed (DM) fiber system with fiber losses, is studied via symbolic computation. N-soliton solutions are constructed and verified with the Wronskian technique. Analytic one-, two- and three-soliton solutions are discussed. The soliton has a linear frequency chirp. Soliton width increases exponentially while soliton amplitude, energy and speed decrease exponentially along the DM fiber. As the chirp-loss coefficient increases, soliton width gets wider, while soliton amplitude, energy and speed become smaller. Interactions between the two solitons and among the three solitons are discussed and illustrated. For the larger initial soliton separations, interactions result in some smaller envelopes, which soon disappear due to the chirp-loss effect. When the soliton separations almost reaches zero, solitons interact quasi-periodically along the DM fiber, while each soliton undergoes broadening and decaying.

Fully resonant soliton interactions in the Whitham–Broer–Kaup system based on the double Wronskian solutions

With the aim of exploring whether the (1+1)-dimensional coupled nonlinear evolution equations admit abundant soliton interactions, like the cases in the Kadomtsev–Petviashvili II equation, we in this paper study the double Wronskian solutions to the Whitham–Broer–Kaup (WBK) system. We give the parametric condition for two double Wronskians to generate the non-singular, non-trivial and irreducible soliton solutions. Via the asymptotic analysis of two double Wronskians, we show that the soliton solutions of the WBK system is in general linearly combined of fully resonant (M,N)- and (M−1,N+1)-soliton configurations. It turns out that the WBK system can exhibit various complex soliton structures which are different pairwise combinations of elastic, confluent and divergent interactions. From a combinatorial viewpoint, we also explain that the asymptotic solitons of a [(M,N),(M−1,N+1)]-soliton solution are identified by a pair of Grassmannian permutations.