Multi-soliton Solutions Research Articles

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

A direct approach for solving the cubic Szegö equation

AbstractWe study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Exact Solutions of Boussinesq Equations By Hirota Direct Method

Boussinesq Equations (BSQ) are the focus of this article. First, we provide a basic overview of Hirota's D operator, which is used to build multi-soliton solutions for equations involving nonlinear evolution. After that, some details regarding fourth-order BSQ are provided, and we use Hirota's direct method to find a one-solution solution. Hirota's bilinear approach is also used to solve a class of sixth-order Boussinesq-like equations with nonlinear evolution. The outcomes verified that this approach requires complete integrability.

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.

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

New type of solutions for the modified Korteweg–de Vries equation

In this letter we report a new type of multi-soliton solutions for the modified Korteweg–de Vries (mKdV) equation. These solutions contain τ functions of the trigonometric solitons and classical solitons simultaneously. A new bilinear form of the mKdV equation is introduced to derive these solutions. The obtained solutions display as solitons living on a periodic background, which are analyzed and illustrated.

The multi-soliton solutions of another two-component Camassa–Holm equation with Darboux transformation approach

In this paper, we develop another approach to construct the multi-soliton solutions of a two-component Camassa–Holm equation in terms of Wronskians with help of a reciprocal transformation and a gauge transformation. Its kink solution, loop solution and smooth soliton solution are presented. Then with the non-trivial limiting procedure, the solution of Camassa–Holm equation is also derived from that of two-component Camassa–Holm equation.

Interactions among lump optical solitons for coupled nonlinear Schrödinger equation with variable coefficient via bilinear method

In this paper, a non-autonomous (3+1) dimensional coupled nonlinear Schrödinger equation (NLSE) with variable coefficients in optical fiber communication is analyzed. By means of bilinear technique and symbolic computations, new multi-soliton solutions of the coupled model in different trigonometric and lump functions are given. Then, in terms of perturbed waves, considering the steady state solution and the small perturbation on the three directions x, y, z and the time t, the soliton transmission are also considered. The behaviour of interaction among lump periodic soliton is studied and optical soliton solutions are reached. This study has certain significance for the analysis of other nonlinear dispersion systems and the application of optical physics. The results are presented through graphs generated by using Maple. The important feature of the proposed study is to show different behaviour of the soliton at each component. The behaviour of solitons, their interactions, and their transformations are all governed by the fundamental concept of energy conservation in all three examples. We demonstrate the efficiency of our suggested methodology for analyzing the NLSE equations using the numerical simulations and analytical tools, yielding fresh insights into their behaviour and solutions. Our findings help to develop mathematical tools for investigating nonlinear partial differential equation (NLPDEs) and provide new insights on the dynamics of NLSE equations, which have implications for many domains of physics and applied mathematics

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

The major contribution in this paper is to inquire into some new exact solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE) which plays a major role in area of the incompressible fluid. Taking advantage of the Cole-Hopf transform, we extract its bilinear form. Then two different kinds of the multi-lump solutions are probed by applying the new homoclinic approach. Secondly, the Y-shape soliton solutions are explored via assigning the resonance conditions to the N-soliton solutions. Additionally, the complex multi kink soliton solutions (CMKSSs) are investigated through the Hirota bilinear method. Lastly, some other wave solutions including the kink and anti-kink solitary wave solutions are developed with the aid of two efficacious approaches, namely the variational method and Kudryashov method. In the meantime, the profiles of the accomplished solutions are displayed graphically via Maple.

Dynamical property of interaction solutions to the Chafee-Infante equation via NMSE method

In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

Nonlinear Waves in (3 + 1)-Dimensions: Multi-Soliton Solutions of a Partial Differential Equation

Nonlinear Waves in (3 + 1)-Dimensions: Multi-Soliton Solutions of a Partial Differential Equation

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

Analyzing sensitivity and multi-soliton solutions in the Estevez–Mansfield–Clarkson equation: Insights into dynamics of bifurcation and chaos

In this investigation, an analysis of the Estevez–Mansfield–Clarkson equation, a model equation employed in the examination of shape formation in liquid drops, optics, and mathematical physics, is undertaken. Firstly, multiple wave solitons, including 1-soliton, 2-soliton, and 3-soliton structures, are successfully generated through the utilization of a multiple exp-function technique. Subsequently, the conversion of the partial differential equation into an ordinary differential equation is executed. The extraction of various traveling wave patterns, such as kink, anti-kink, periodic, and exponential functions, is then carried out using the new auxiliary equation method. The outcomes are visually represented through 3-dimensional, 2-dimensional, and density plots, employing Mathematica software. Following this, an investigation into the qualitative dynamics of the equation is conducted, examining aspects such as bifurcation and chaos. Critical points are identified for bifurcation, and the dynamical system undergoes an outward force, resulting in the identification of chaotic patterns. Furthermore, the model’s sensitivity across different initial values is explored. These solutions hold immense significance in the domains of nonlinear fiber optics and telecommunications that help in deepening our knowledge about the basic physical model.

Study of multi solitons, breather soliton structures in the earth's magnetotail region

The investigation of wave packets with varying magnitudes, which predominantly carry energy across different physical mediums, proves to be highly intriguing and unveils numerous nonlinear characteristics within Earth's magnetotail region and magnetosphere, as confirmed by extensive space observations. Nonlinear features in wave dynamics are elucidated through diverse nonlinear structures, among which breather structures hold prominence and manifest across various wave fields. We have considered an unmagnetized collisionless homogeneous hydrodynamical governing equations set in an electron-ion plasma comprising hot and cold ions. In this study, we have explored 1-soliton, 2-soliton, and breather structures in the framework of the Gardner equation (GE). We have derived the GE from the normalized set of governing equations employing the reductive perturbation technique (RPT). The GE, an extension of the Korteweg-de Vries (KdV) equation, encompasses the combined effects of quadratic and cubic nonlinearities. By employing the Hirota bilinear method (HBM), we have obtained multi-soliton solutions and breather soliton solutions. We have noticed the variations in electron temperature ratios and density concentration ratios substantially impact and reshape breather soliton structures. Such alterations in breather structures, influenced by different electron temperatures and concentration ratios, hold potential significance in understanding energy transport within Earth's magnetotail region. The alteration of shapes and soliton structures may also be investigated in the dynamics of wave fields upon interaction such as hydrodynamics, optics, optic fibers, signal processing, etc.

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.

Riemann-Hilbert approach and multi-soliton solutions of nonlocal Newell-type long wave-short wave equation

This article’s purpose is to investigate the inverse scattering transform of the nonlocal long wave-short wave (LW-SW) equation and its multi-soliton solutions via Riemann-Hilbert (RH) approach. By using spectral analysis to the Lax pair of LW-SW equation, the RH problem can be constructed. However, we consider spectral analysis from the time part rather than the usual space part, since it is hard to obtain the analyticity of the space part. Then the RH problem can be solved and the formula of the soliton solutions can be given. We provide several special soliton solutions including Y-shaped solitons, V-shaped solitons, bound-state solitons and mixed four-soliton solutions. Compared with the local case, the solutions of nonlocal LW-SW equation exhibit distinct characteristics that (i) these soliton solutions are strictly symmetric with respect to x = 0 under special parameter conditions, (ii) the mixed four-soliton solution, which combines Y-type and bound-state solitons, is novel.

Plenty of novel soliton molecules and modulation instability in the coherently coupled nonlinear Schrödinger equation

We investigate the coherently coupled nonlinear Schrödinger (NLS) equation, which can describe the propagation of two orthogonally polarized electromagnetic waves in a nonlinear fiber. The multi-soliton solutions are derived by using the Hirota bilinear method. Multi-breather solutions are presented by assuming the relationship between the parameters of the multi-solitons. Based on multi-soliton solutions and multi-breather solutions, some new interaction structures consisting of soliton molecules, breather molecules and soliton-breather molecules are found by applying the velocity resonance condition. The interactions among different types of soliton molecules, which can be observed in marine and oceanic waters, are investigated through numerical simulation. Ultimately, baseband modulation instability of the coherently coupled NLS equation is analyzed. It is of great significance for the study of soliton formation and rogue waves in the coherently coupled NLS equation.