N-soliton Solutions Research Articles

Generalized -dressing method for coupled nonlocal NLS equation

In this paper, we investigate a general integrable coupled nonlocal nonlinear Schrödinger (cnNLS) equation with reverse space and reverse time. First, we introduce a 4-component nonlocal nonlinear Schrödinger (nNLS) equation. Based on two 3 × 3 matrices -problems, we obtain the N-soliton solutions of the 4-component nNLS equation by constructing two spectral transformation matrices R and with some specific scattering data and . We have the symmetry condition for R and . We express the determinant in the N-soliton solution in the form of sums with the help of Cauchy matrix properties to facilitate follow-up reduction. The general nonlocal reduction of the 4-component nNLS equation to the cnNLS equation is discussed in detail. After obtaining the 1-soliton solution of cnNLS equation, we analyse the nonsingular region of the single soliton solution, with spectral parameters k 1 and λ 1 are conjugate. We also draw some typical images of 1-soliton solution. The 2-soliton solutions for cnNLS equation are derived and their asymptotic behaviours are discussed. After that we discuss the dynamic behaviour of the two waves in 2-soliton solution under the condition of different spectral parameters values.

On the asymptotic stability of N-soliton solution for the short pulse equation with weighted Sobolev initial data

In this work, we are devoted to studying the Cauchy problem of the short pulse (SP) equation with weighted Sobolev initial data. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in solitonic region of space-time, (yt)=ξ for any fixed ξ=ξ0∈(0,+∞), and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the SP equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum with residual error up to O(t−1). Combining with previous results presented by Yang and Fan, that is the long time asymptotic expansion of the solution q(x,t) in solitonic region ξ=yt∈(−∞,0), our results show that the soliton resolution conjecture of the SP equation is fully solved in all space-time solitonic regions. Moreover, considering properties of the scattering map of SP equation, we derive an asymptotic stability result for a class of initial values.

Darboux transformation, Infinite conservation laws and Exact solutions for nonlocal Hirota equation with variable coefficient

Abstract This article presents the construction of a nonlocal Hirota equation with variable coefficient and its Darboux transformation. Using zero-seed solutions, 1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation, along with the expression for N-soliton solutions. The influence of coefficient functions on the solutions is investigated by choosing different coefficient functions, and the dynamics of the solutions are analyzed. For the first time, this article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations. The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

Localized waves and interaction solutions to an integrable variable coefficients Jimbo-Miwa equation

In this paper, the reduced variable coefficients Jimbo-Miwa (vcJM) equation is studied. Firstly, the integrability of the reduced vcJM equation is verified by Painlevé analysis. Based on the Hirota bilinear method and the long wave limit method, the N-soliton solutions, rational and semirational solutions of the vcJM equation are obtained. By choosing different parameters and coefficient functions, some of different kinds of local waves, including of solition, breather wave and lumps, of the equation are obtained. Furthermore, the interaction solutions between different local waves are obtained. The dynamical behavior of the interaction between different local waves is studied by modifying the time parameters and the process is displayed by figures.

A coupled complex mKdV equation and its N-soliton solutions via the Riemann–Hilbert approach

This paper concerns the initial value problem of a coupled complex mKdV (CCMKDV) equations ut+uxxx+6(|u|2+|v|2)ux+6u(|v|2)x=0,vt+vxxx+6(|u|2+|v|2)vx+6v(|u|2)x=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{aligned} &u_{t}+u_{xxx}+6 \\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2} \\bigr)u_{x}+6u\\bigl( \\vert v \\vert ^{2} \\bigr)_{x}=0, \\\\ &v_{t}+v_{xxx}+6\\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2}\\bigr)v_{x}+6v \\bigl( \\vert u \\vert ^{2}\\bigr)_{x}=0, \\end{aligned} $$\\end{document} proposed by Yang (Nonlinear Waves in Integrable and Nonintegrable Systems, 2010), which is associated with a 4 times 4 scattering problem. Based on matrix spectral analysis, a fourth-order matrix Riemann–Hilbert problem is formulated. By solving a specific nonregular Riemann–Hilbert problem with zeros, we present the N-soliton solutions for the CCMKDV system. Moreover, the single-soliton solutions are displayed graphically.

Inverse scattering transform for the integrable fractional derivative nonlinear Schrödinger equation

In this paper, we propose an integrable fractional derivative nonlinear Schrödinger (fDNLS) equation with the aid of the completeness of the squared eigenfunctions for the Kaup–Newell system. Then we further construct an appropriate Riemann–Hilbert problem, from which we obtain a reconstruction formula of the solution of the fDNLS equation. The fractional N-soliton solution is carried out explicitly by means of determinants and the fractional one-soliton solution is verified rigorously.

Stability of kink, anti kink and dark soliton solution of nonlocal Kundu–Eckhaus equation

We study the modulational instability and soliton solution for nonlocal Kundu–Eckhaus (KE) equation which is distinctly different from KE equation. KE equation can describe the dynamical behavior of optical pulse envelopes in birefringent fiber. Although, the present equation has novel space, which induce new physical effects and inspire novel physical applications. Therefore, through the Jacobi elliptic function method, the elliptic function solution is derived for the nonlocal KE. Deng et al. (2022) found the N-soliton solution for nonlocal KE equation by the use of Hirota bilinear method. Though, the present problem of the solution is new which is Jacobi elliptic function solution. It is found that the solution exhibits a variety of forms of soliton as dark, kink and antikink soliton due to distinct choices of values of focusing and defocusing parameters. Importantly, with the use of the Jacobi elliptic solution, the role of focusing and defocusing parameters of cubic, quintic nonlinearity and self frequency shift on optical soliton in birefringent fiber for nonlocal KE equation is discussed. The linear stability analysis is used to construct the gain of nonlocal Kundu–Eckhaus equation and using it to analysis the stability/instability of various forms of soliton.

Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Nonlinear coherent structures of nucleus-acoustic wave excitations in multi-nuclei quantum plasmas with ultra-relativistically degenerate electrons and positrons

The propagation dynamics of the nucleus-acoustic waves (NAW) in a quantum plasma composed of nondegenerate inertial light nuclei, stationary heavy nuclei, and ultra-relativistically degenerate electrons and positrons has been theoretically investigated within the framework of the Boussinesq equation, which is valid for a bi-directional propagation of a small but finite amplitude limit. The N-soliton solution of the Boussinesq equation is derived using Hirota's method. It is found that positive potential structures exist in the sonic and supersonic regimes, whereas negative potential structures are found to be present in the subsonic regime. Pertinent plasma properties are analyzed for one-, two-, and three-soliton solutions in terms of different parameters. In addition to the typical solitary wave solutions, our findings indicate that the nonlinear NAW has breather structures. The three- and four-soliton solutions are used to construct the elastic interaction solutions of the breather–soliton and breather–breather, respectively. The findings are discussed in the context of ultra-relativistic astrophysical plasmas.

Multiple soliton solutions and symmetry analysis of a nonlocal coupled KP system

A nonlocal coupled Kadomtsev–Petviashivili (ncKP) system with shifted parity () and delayed time reversal () symmetries is generated from the local coupled Kadomtsev–Petviashivili (cKP) system. By introducing new dependent variables which have determined parities under the action of , the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the cKP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.

Localized waves and their novel interaction solutions for a dimensionally reduced (2 + 1)-dimensional Kudryashov Sinelshchikov equation

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

Line solitons, lumps, and lump chains in the (2+1)-dimensional generalization of the Korteweg–de Vries equation

This article focuses on the exploration of novel soliton molecules for the (2+1)-dimensional Korteweg–de Vries equation. Specifically, Hirota bilinear form is derived through Bell polynomial method, and the hybrid solution comprising one lump soliton and an arbitrary number of line solitons or/and lump chains is derived through the introduction of a long wave limit and new constraint conditions between the parameters of the N-soliton solutions and velocity resonance. The paper presents both analytical and graphical demonstrations of a range of interactions, showcasing the propagation of nonlinear localized waves. The findings of this study could contribute to a better understanding of physical phenomena associated with the propagation of nonlinear localized waves.

Interaction solutions for the (2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation in incompressible fluid

This paper aims to search for the solutions of the (2+1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation. Lump solutions, breather solutions, mixed solutions with solitons, and lump-breather solutions can be obtained from the N-soliton solution formula by using the long-wave limit approach and the conjugate complex method. We use both specific circumstances and general higher-order forms of the hybrid solutions as examples. With the help of maple software, we create density and 3D graphs to summarize the dynamic properties of these solutions. Additionally, it is possible to observe how the solutions’ trajectory, velocity, and shape vary over time.

Resonance Soliton, Breather and Interaction Solutions of the Modified Kadomtsev–Petviashvili-II Equation

In this paper, we investigate the modified Kadomtsev–Petviashvili-II (mKP-II) equation, which has important applications in fluid dynamics, plasma physics and electrodynamics. By utilizing the Hirota bilinear method, the N-soliton solutions of the mKP-II equation are obtained. The resonance Y-type soliton, and the interaction between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by imposing some constraints to the parameters of the N-soliton solutions. Moreover, the novel type of double opening resonance Y-type soliton solutions are obtained by selecting some appropriate parameters in 3-soliton solutions. By making some conjugate assumptions in the parameters, the multiple breathers are presented. Furthermore, the hybrid solutions consisting of multiple breathers and resonance Y-type solitons are investigated. The dynamics of these hybrid solutions are analyzed using both numerical simulations and graphical methods.

Soliton solution, breather solution and rational wave solution for a generalized nonlinear Schrödinger equation with Darboux transformation

In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons.

Higher-order breather, lump and hybrid solutions of (2 + 1)-dimensional coupled nonlinear evolution equations with time-dependent coefficients

This paper investigates a class of (2 + 1)-dimensional coupled nonlinear evolution equation with time-dependent coefficients in an inhomogeneous medium via the Hirota bilinear method. Combining the long wave limit method and complex conjugate transform, the higher-order breather and lump solutions are initially constructed. Furthermore, hybrid solutions among N-soliton, lump and breather solutions are derived by linear constraints on the parameters. Meanwhile, the dynamic evolution behaviour of some special concrete solutions under different time-dependent coefficients is presented visually in the form of images.

Rogue wave solutions of (3+1)-dimensional Kadomtsev-Petviashvili equation by a direct limit method

On the bases of N-soliton solutions of Hirota’s bilinear method, high-order rogue wave solutions can be derived by a direct limit method. In this paper, a (3+1)-dimensional Kadomtsev-Petviashvili equation is taken to illustrate the process of obtaining rogue waves, that is, based on the long-wave limit method, rogue wave solutions are generated by reconstructing the phase parameters of N-solitons. Besides the fundamental pattern of rogue waves, the triangle or pentagon patterns are also obtained. Moreover, the different patterns of these solutions are determined by newly introduced parameters. In the end, the general form of N-order rogue wave solutions are proposed.

The Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation, soliton interactions and position shift

In this paper, we are concerned with the Riemann–Hilbert approach for the higher-order Gerdjikov–Ivanov equation with nonzero boundary conditions. A uniformization variable will be introduced to make it forms a one-to-one mapping with the spectral parameter. The analyticity, symmetric and asymptotic behaviors of the eigenfunctions and scattering datas will be calculated in detail and the Riemann–Hilbert problem will be constructed to obtain the expression for the N-soliton solutions. In addition, the case of Jost solutions and scattering coefficients near the branch points will also be analyzed. On the other hand, when the boundary conditions turn to zero, the formula for the position shifts and phase shifts of solitons resulting from soliton interactions will be solved.

Symmetry reduction and exact solutions of the (3+1)-dimensional nKdV-nCBS equation

In this paper, the exact solutions of the (3+1)-dimensional nKdV-nCBS equation are investigated by various approaches. Firstly, according to Lie group theory, the (3+1)-dimensional nKdV-nCBS equation is reduced to two (1+1)-dimensional partial differential equations. Secondly, three different types of exact solutions are obtained by the homoclinic test approach, including the singular kinked-type periodic solitary wave solutions, the kinked solitary wave solutions and the kinked-cross periodic wave solutions. The diagrams of the solutions for the specified parameters are given. Furthermore, taking one reduced equation as an example, the N-soliton solutions are derived. And under some constraints, higher-order soliton solution can degenerate into lower-order soliton solution, which is a specific degeneration phenomenon of this equation.

Direct reduction approach and soliton solutions for the integrable space–time shifted nonlocal Sasa-Satsuma equation

This paper investigates the space–time shifted nonlocal Sasa-Satsuma equation which is constructed by imposing a space–time shifted symmetry constraint on the two-component Sasa-Satsuma system. By applying the direct reduction approach, specific parameter values are determined to obtain N-soliton solutions of the shifted nonlocal equation. The resulting one-soliton and two-soliton solutions are analyzed both theoretically and graphically. Additionally, the asymptotic behavior of two-soliton solution with two purely imaginary eigenvalues is studied. The findings shed light on the mathematical properties and dynamic behaviors of shifted nonlocal integrable systems.