Two-soliton Solutions Research Articles

Breather, soliton, multiple-pole, and interaction solutions to the Hirota–Satsuma equation

In this paper, we investigate serval types of localized waves of the Hirota–Satsuma equation by using the Hirota method. By means of two identities, the bilinear form of the Hirota–Satsuma equation is proposed. Then, the one-, two-, and three-soliton solutions are given explicitly and analyzed. The one-soliton solution could present the type of soliton or antisoliton, which depends on the sign of the wave number. The soliton–soliton, soliton–antisoliton, and antisoliton–antisoliton interactions are analyzed through graphics in the two-soliton solution case. The interactions among three solitons/antisolitons are considered in two cases. On basis of the soliton solutions, we give the conditions of obtaining breather solutions and interaction solutions between breathers and solitons. Multiple-pole solutions are derived by taking the limit of wave numbers and appropriate phase parameters. The dynamics of double- and triple-pole solutions are analyzed.

Soliton dynamics in (2+1) dimensional Heisenberg spin chain with Dzyaloshinskii–Moriya interaction in nanowire systems

In this study, we delve into the theoretical framework of the (2+1)-dimensional Kadomtsev–Petviashvili equation coupled with Dzyaloshinskii–Moriya (DM) interaction, elucidated from the Landau-Lifshitz equation, a prominent model in the realm of ferromagnetic dynamics. Employing the binary Bell polynomial technique along with the Hirota bilinear form, we construct one- and two-soliton solutions for the considered physical system. Through the manipulation of parameters present in these solutions, we explore the various nonlinear dynamical phenomena characterizing ferromagnetic nano-wires with DM interaction. Our investigations elucidate that solitons can be effectively manipulated in the nanowire system through judicious selection of system parameters. Moreover, our findings illustrate that soliton profiles exhibit compression, amplification, shifting, and changes in orientation during evolution. These insights hold significant implications for experimentalists aiming to analyze the propagation of solitons in ferromagnetic nanowire systems.

Nonlinear evolution of disturbances in higher time-derivative theories

We investigate the evolution of localized initial value profiles when propagated in integrable versions of higher time-derivative theories. In contrast to the standard cases in nonlinear integrable systems, where these profiles evolve into a specific number of N-soliton solutions as dictated by the conservation laws, in the higher time-derivative theories the theoretical prediction is that the initial profiles can settle into either two-soliton solutions or into any number of N-soliton solutions. In the latter case this implies that the solutions exhibit oscillations that spread in time but remain finite. We confirm these analytical predictions by explicitly solving the associated Cauchy problem numerically with multiple initial profiles for various higher time-derivative versions of integrable modified Korteweg-de Vries equations. In the case with the theoretical possibility of a decay into two-soliton solutions, the emergence of underlying singularities may prevent the profiles from fully developing or may be accompanied by oscillatory, chargeless standing waves at the origin.

Dark-bright solitons and the role of their interaction in breather generation

Abstract We study fundamental dark-bright solitons and their interaction of vector nonlinear Schrödinger
equations in both focusing and defocusing regimes. Classification of possible types soliton solutions is
given. There are two types of solitons in the defocusing case and four types of solitons in the focusing
case. The number of possible variations of the two-soliton solutions depends on this classification.
We demonstrate that only special types of two-soliton solutions in the focusing regime can generate
breathers of scalar nonlinear Schrödinger equation. The cases of solitons with equal and unequal
velocities in the superposition are considered. Numerical simulations confirmed the validity of our
exact solutions.

[formula omitted]-dressing approach and [formula omitted]-soliton solutions of the general reverse-space nonlocal nonlinear Schrödinger equation

Using the ∂̄-dressing method, we study the general reverse-space nonlocal nonlinear Schrödinger (nNLS) equation. Beginning with a 3 × 3 matrix ∂̄-problem, the associated spatial and time spectral problems are obtained through two linear constraint equations. Furthermore, the gauge equivalence between the Heisenberg chain equation and the general reverse-space nNLS equation is established. By employing a recursive operator, a hierarchy for the general reverse-space nNLS equation is proposed. Moreover, by selecting a suitable spectral transformation matrix, the N-soliton solutions of the general reverse-space nNLS equation are calculated, yielding the explicit one-soliton and two-soliton solutions.

Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media.

Solitons with varying amplitudes and/or varying velocities in a variable-coefficient mixed spectral generalized Sasa–Satsuma equation revealed through the Riemann–Hilbert method

The integrable Sasa–Satsuma (SS) equation, a higher-order Nonlinear Schrödinger (NLS)-type model, has important applications in the field of optics. In this study, we present a generalized SS equation with variable coefficients derived from a mixed spectral problem, abbreviated as vcmsgSSE, and construct its N-soliton solution using the Riemann–Hilbert (RH) method. First, we provide the Lax pair associated with the vcmsgSSE and perform a spectral analysis on it, which allows us to derive a solvable RH problem. Then, by solving this RH problem we construct an explicit expression for N-soliton solution of the vcmsgSSE. Finally, under a special reduction, we obtain the specific one-soliton solution and two-soliton solution of the vcmsgSSE, and use them to illustrate graphically the N-soliton solution with varying amplitude and/or varying velocity corresponds to the physical background of non-uniform medium assumed by the vcmsgSSE.

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.

Localized stem structures in quasi-resonant two-soliton solutions for the asymmetric Nizhnik–Novikov–Veselov system

Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2 + 1)-dimensional two-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. a12 ≈ 0), and the other by strongly quasi-resonant collisions (i.e. a12 ≈ +∞). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system.

Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10−8) of the conservation laws as well as the tiny numerical variations (less than 1 × 10−9) of the amplitudes and the propagating velocities of the solitons.

Integrability, and stability aspects for the non-autonomous perturbed Gardner KP equation: Solitons, breathers, [formula omitted]-type resonance and soliton interactions

This article examines the soliton-type solutions, their interactions, and the integrable properties of a non-autonomous perturbed Gardner KP (NPGKP) equation. For the NPGKP equation under consideration, a bilinear structure, and a Bäcklund transformation are designed explicitly, which claim the integrability of the system under some constraints. The stability of the obtained solutions is discussed using modulation instability. The bilinear form demonstrates the dynamic characteristics of multiple solitons, breathers, and their interactions in response to external impulses. Furthermore, it allows for determining the solitons’ amplitudes and velocities. The two-soliton solution yields a first-order breather solution. At the same time, the analytical investigation focuses on the interaction between a single breather and a single-soliton within the multi-soliton solution. This investigation also identifies the resonance of Y-shaped solitons and examines the dynamical characteristics of the interaction between resonant Y-shaped solitons and M-fissionable pulses. The multi-shock solutions and the collisions between shocks are analyzed in the presence of external influences. Graphical representations of the relationships between different sorts of achieved solutions are provided explicitly.

Darboux transformation-based LPNN generating novel localized wave solutions

Darboux transformation method is one of the most essential and important methods for solving localized wave solutions of integrable systems. In this work, we introduce the core idea of Darboux transformation of integrable systems into the Lax pairs informed neural networks (LPNNs), which we proposed earlier. By fully utilizing the data-driven solutions, spectral parameter and spectral function obtained from LPNNs, we present the novel Darboux transformation-based LPNN (DT-LPNN). The notable feature of DT-LPNN lies in its ability to solve data-driven localized wave solutions and spectral problems with high precision, and it also can employ Darboux transformation formulas of integrable systems and non-trivial seed solutions to discover novel localized wave solutions that were previously unobserved and unreported. The numerical results indicate that, by utilizing the single-soliton solutions as the non-trivial seed solutions, we obtain novel localized wave solutions for the Kraenkel–Manna–Merle (KMM) system by employing DT-LPNN, in which solution u changes from original bright single-soliton on zero background wave to new dark single-soliton dynamic behavior on a variable non-zero background wave. Moreover, by treating the two-soliton solutions as the non-trivial seed solutions, DT-LPNN generates novel localized wave solutions for the KMM system that exhibit completely different dynamic behaviors from prior two-soliton solutions. DT-LPNN combines the Darboux transformation theory of integrable systems with deep neural networks, offering a new approach for generating novel localized wave solutions using non-trivial seed solutions.

Bifurcation, Traveling Wave Solutions and Dynamical Analysis in the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-Dimensional Extended Vakhnenko–Parkes Equation

This paper is concerned with the bifurcation of the traveling wave solutions, as well as the dynamical behaviors and physical property of the soliton solutions of the (2+1)-dimensional extended Vakhnenko–Parkes (eVP) equation. Firstly, based on the traveling wave transformation, the planar dynamical system corresponding to the (2+1)-dimensional eVP equation is derived, and then the singularity type and trajectory map of this system are obtained and analyzed. Based on the bifurcation of this system, the analytical expression for the periodic wave solution is given and shown graphically. Secondly, the N-soliton solutions are obtained via the bilinear method, and some important physical quantities and asymptotic analysis of one-soliton and two-soliton solutions are discussed. The results obtained in this paper might be useful for understanding the propagation of high-frequency waves.

Elastic overtaking collisions of large-amplitude ion-acoustic solitons

Overtaking collisions of large-amplitude solitons are investigated via fluid simulations for a plasma consisting of cold ions and Boltzmann-distributed electrons. To achieve this, a new fluid simulation code is presented. In addition, a novel approach to construct soliton initial conditions is developed. Using these ideas, initial conditions are combined that allows the simulation of overtaking collisions. It is shown that, in the small-amplitude regime, simulation results agree well with the two-soliton solution obtained from reductive perturbation theory. Interestingly, in the large amplitude regime, both the slow and fast solitons re-emerge after the collision with no significant change, showing that the collisions remain elastic. A comparison between reductive perturbation analysis and the simulations show that the only significant effect of higher order nonlinearities on overtaking collisions is a reduction in the magnitude of the phase shifts of both solitons.

Nondegenerate Soliton Solutions of (2+1)-Dimensional Multi-Component Maccari System

Abstract For a multi-component Maccari system with two spatial dimensions, nondegenerate one-soliton and two-soliton solutions are obtained with the bilinear method. It can be seen by drawing the spatial graphs of nondegenerate solitons that the real component of the system shows a cross-shaped structure, while the two solitons of the complex component show a multi-solitoff structure. At the same time, the asymptotic analysis of the interaction behavior of the two solitons is conducted, and it is found that under partially nondegenerate conditions, the real and complex components of the system experience elastic collision and inelastic collision, respectively.

[formula omitted]-soliton solutions and their dynamic analysis to the generalized complex mKdV equation

A new generalized complex modified Korteweg–de Vries (mKdV) equation is studied by using Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a 3 × 3 matrix spectral problem for the generalized complex mKdV equation. Then, we can formulate the Riemann-Hilbert problem via the spectral analysis of the x-part of the Lax pair. According to the symmetry properties of the potential matrix, we find two cases of zero structures for the Riemann-Hilbert problem. By solving the particular Riemann-Hilbert problem and using the inverse scattering transformation, we obtain the unified formulas of the N-soliton solutions for the generalized complex mKdV equation. In addition, the dynamical behaviors of the single-soliton solution and the two-soliton solution are analyzed by choosing appropriate parameters.

An integrable generalization of the Fokas–Lenells equation: Darboux transformation, reduction and explicit soliton solutions

Abstract Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly, the explicit one- and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

Generalized molecule-like structure in atomic nuclei

Starting from a |ϕ|4 field theory with universal pn interacting angular and distance coordinates (θ,r) to depict nuclear long-range correlations, we have derived a cubic Schrödinger equation for NpNnpn interacting virtual bosons, which are believed to dominate the generic evolution of nuclear structure from spherical multi-nucleon shell-model states to deformed collective rotational states. It is shown that two-soliton solution in θ space manifests two opposite pn-type Ising-like chains existing in ground states and their gradual alignment responsible for nuclear rotational excitations. Of particular prominence is two-soliton solution with respect to r: the mutual relative motion of valence protons and neutrons around valence-nucleon mass centers considered as the dynamic equilibrium positions of a non-linear Schrödinger breather. And the resulting non-vanishing root mean square 〈r2〉 divides half the valence nucleons belonging to a chain into four local mass centers and proposes two chain-corresponding molecule-like tetrads outside inert cores pairwise hidden in ground states. Such solitons turn out to be an analogue of opposite axion fluids around a local mass center in the galactic dark-matter halo outside the soliton core (Garnier et al. (2021) [67]) and the present study facilitates making the latter more clear. The generalized molecule-like structure arouses a requisite to the addition of cluster-interior and cluster-to-core electric dipole moments in particular to pear-shaped nuclei, whose variants Schiff moments play an important role in measuring diamagnetic atomic electric dipole moments and therefore understanding CP-violating new physics beyond the standard model. We put additional parts in evidence by befitting the NpNn scheme verified already by electric quadrupole moments in same nuclei.

The bilinear neural network method for solving Benney–Luke equation

Benney–Luke equation, the estimation of water wave propagation on the water’s surface, is significantly important in studying the tension of water waves in physics. This paper focuses on the Bilinear Neural Network method (BNNM) to find the solutions to the Benney–Luke equation. Using the Hirota bilinear operator, the Benney–Luke equation is transformed into a bilinear expression and establishes models of neurons. The method construction is straightforward, well-organized, and effective for finding various exact analytic solutions of the Benny–Luke equation consisting of one-soliton solutions, two-soliton solutions, one-soliton hyperbolic wave solutions, two-soliton hyperbolic wave solutions, mixed-soliton hyperbolic solutions, and mixed-multi hyperbolic solutions by choosing the appropriate test functions. The solutions are rigorously depicted in three dimensions to observe the mutable behavior.

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.