Multiple Soliton Solutions Research Articles

Multiple soliton, soliton molecules and the other diverse wave solutions to the (2+1)-dimensional Kadomtsev–Petviashvili equation

The central purpose of this paper is to explore the nonlinear dynamics of the (2+1)-dimensional Kadomtsev–Petviashvili equation (KPE). The multiple soliton solutions (MSSs) are constructed via applying the Hirota method. Then the soliton molecules on the ([Formula: see text]-, ([Formula: see text]- and ([Formula: see text]-planes are extracted via imposing the velocity resonance conditions to the MSSs. Eventually, two effective techniques, the sub-equation approach (SEA) and the variational approach (VA), are employed to probe some other diverse wave solutions, which are the bright wave, dark wave, singular wave and the singular periodic wave solutions. The dynamics of the extracted solutions are unveiled graphically to exhibit the physical attributes. The attained solutions in this paper can enlarge the exact solutions of the (2+1)-dimensional KPE and enable us to understand the nonlinear dynamic behaviors better.

New (3+1)-dimensional integrable generalized KdV equation: Painlevé property, multiple soliton/shock solutions, and a class of lump solutions

The present work aims to examine a newly proposed (3+1)-dimensional integrable generalized Korteweg-de Vries (gKdV) equation. By employing the Weiss- Tabor-Carnevale technique in conjunction with Kruskal ansatz, we establish the com- plete integrability of the suggested model by demonstrating its ability to satisfy the Painlev´e property. The bilinear form of the (3+1)-dimensional gKdV equation is em- ployed to construct multiple soliton solutions. By manipulating the various values of the corresponding parameters, we generate a category of lump solutions that exhibit localization in all dimensions and algebraic decay.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

This research mainly concerned with some new solutions of the (3 + 1)‐dimensional nonlinear evolution equation (NEE). First, we extract the resonant multiple soliton solutions (RMSSs) by taking advantage of the linear superposition principle (LSP) and weight algorithm (WA). Then the non‐singular complexiton and singular complexiton solutions are developed by introducing pairs of the conjugate parameters. Besides, the complex multiple‐soliton solutions are also explored with the aid of the bilinear approach. The graph descriptions of the attained solutions are paraded to show the dynamical properties. The outcomes of this work are all new and are desirous to bring some new perspective to the investigation of the complexiton solutions to the other partial differential equations (PDEs).

On the Painlevé integrability and nonlinear structures to a (3 + 1)-dimensional Boussinesq-type equation in fluid mediums: Lumps and multiple soliton/shock solutions

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers.

Abundant soliton solutions of the modified KdV-KP equation

The precise solutions of the (2+1)-dimensional modified KdV-KP equation are examined in this study. Three reliable methods, namely, the modified auxiliary equation scheme, extended rational sine–cosine methodology, and extended rational sinh–cosh technique, are utilized for the first time to extract multiple soliton solutions for the governing model. These types of solutions have applications in plasma, solid state, neuronal, biological production, and diffusion processes. Graphical simulations of the obtained solutions are included in the form of 3 dimensional surface plots to reveal their physical structure and dynamical features.

Dynamic behavior of positron acoustic multiple-solitons in an electron–positron-ion plasma

We explore the dynamic characteristics of positron acoustic multiple-solitons in an unmagnetized plasma containing mobile cold positrons, Kappa-distributed superthermal hot electrons and positrons, and stationary positive ions. This study investigates the overtaking collisional effects, various parametric impacts, and phase shifts in the electron–positron-ion (e-p-i) plasma. Through the reductive perturbation technique, we derived the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. The multiple-soliton solutions (MSS) are then obtained using the simplified Hirota method and Cole-Holf transformation. The investigation shows that the amplitudes and widths of multi-solitons decrease with the increasing hot positron concentration but increase with the increasing hot electron index parameter. The results are expected to help us understand the dynamics of waves propagation in the pulsar magnetosphere, the active galactic nuclei, the quasar’s relativistic jet, the inner region of the accretion disk surrounding a black hole, etc.

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques.

The aim of the present study is to identify multiple soliton solutions to the nonlinear coupled Broer-Kaup-Kupershmidt (BKK) system, including beta, conformable, local-fractional, and M-truncated derivatives. The coupled Broer-Kaup-Kupershmidt system is employed for modelling nonlinear wave evolution in mathematical models of fluid dynamics, plasmic, optical, dispersive, and nonlinear long-gravity waves. The travelling wave solutions to the above model are found using the Unified and generalised Bernoulli sub-ODE techniques. By modifying certain parameter values, we may create bright soliton, squeezed bell-shaped wave, expanded v-shaped soliton, W-shaped wave, singular soliton, and periodic solutions. The four distinct kinds of derivatives are compared quite effectively using 2D line graphs. Also, contour plots and 3D graphics are given by using Mathematica 10. Lastly, any pair of propagating wave solutions has symmetrical geometrical forms.

A Hamiltonian equation produces a variety of Painlevé integrable equations: solutions of distinct physical structures

PurposeThe purpose of this paper is to investigate a variety of Painlevé integrable equations derived from a Hamiltonian equation.Design/methodology/approachThe newly developed Painlevé integrable equations have been handled by using Hirota’s direct method. The authors obtain multiple soliton solutions and other kinds of solutions for these six models.FindingsThe developed Hamiltonian models exhibit complete integrability in analogy with the original equation.Research limitations/implicationsThe present study is to address these two main motivations: the study of the integrability features and solitons and other useful solutions for the developed equations.Practical implicationsThe work introduces six Painlevé-integrable equations developed from a Hamiltonian model.Social implicationsThe work presents useful algorithms for constructing new integrable equations and for handling these equations.Originality/valueThe paper presents an original work with newly developed integrable equations and shows useful findings.

Non-singular complexiton, singular complexiton and the breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation

This work aims to construct some novel exact solutions of the (2 + 1)-dimensional Sawada-Kotera equation (SKE). First, the resonant multiple soliton solutions (RMSSs) are discussed by employing the weight algorithm (WA) and linear superposition principle (LSP). Then, based on the RMSSs, the non-singular complexiton and singular complexiton solutions are developed by taking pairs of the conjugate parameters. Finally, the breather wave solutions (BWSs) of the sin-cosh type are explored through the symbolic computation and test function method. The dynamics of the corresponding extracted solutions are displayed graphically for showing the physical properties. The achievements of this article can provide some new insights to the (2 + 1)-dimensional SKE.

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

Multiple soliton solutions of the Korteweg-de Vries in the frame of the time-dependent variable coefficient

Multiple soliton solutions of the Korteweg-de Vries in the frame of the time-dependent variable coefficient

Soliton molecules, Y-type soliton and complex multiple soliton solutions to the extended (3+1)-dimensional Jimbo-Miwa equation

Abstract The central purpose of this paper is extracting some novel and interesting soliton solutions of the extended (3+1)-dimensional Jimbo-Miwa equation(JME) which acts as an extension of the classic (3+1)-dimensional JME for the plasma and optics. First, we study the N-soltion solutions that is developed by the Hirota bilinear method (HBM). Then, the soliton molecules and Y-type soliton solutions are constructed via imposing the novel resonance conditions to the N-soltion solutions. In addition, we also explore the complex multiple soliton solutions via the HBM. The dynamic properties of the N-soltion, soliton molecules, Y-type soliton as well as the complex multiple soliton solutions are presented graphically. The developed soliton solutions of this research are all new and can enable us apprehend the nonlinear dynamic behaviors of the extended (3+1)-dimensional JME better.

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

The coupled space-shifted nonlocal nonlinear Schrödinger equation: Multiple bright-dark double-pole solitons, multiple negaton-type solitons, and their associated mixed solitons

We study the multiple bright-dark double-pole solitons, multiple negaton-type solitons, and their associated mixed solitons of the coupled space-shifted nonlocal nonlinear Schrödinger equation. These three distinct classes of soliton waveforms are derived from multiple soliton solutions through three different long-wave limit procedures with specific restrictions of the parameters of the solutions. The multiple bright-dark double-pole soliton solutions are symmetric about the point (x02,0), where x0 is the space-shifting parameter of the coupled space-shifted nonlocal nonlinear Schrödinger equation. The space-shifting parameter x0 only affects half of the multiple bright-dark negaton-type solitons, while it has no impact on the other half. The mixed soliton solutions are composed of multiple double-pole solitons and negaton-type solitons. The unique properties of these three distinct classes of soliton waveforms are examined by performing the long-time asymptotic analysis for them.

Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach

Multiple optical soliton solutions for wave propagation in nonlinear low-pass electrical transmission lines under analytical approach

Soliton solutions of a (2+1)-dimensional nonlinear time-fractional Bogoyavlenskii equation model

In this analysis, we propose a mathematical approach named the extended (ℵ, ℜ) expansion scheme to integrate nonlinear fractional and classical evolution models. We utilize the technique to the time fractional Bogoyavlenskii equation, which signifies the (2 + 1)-dimensional interaction of propagating Riemann waves along a definite axis and a wave normal to it. Additionally, we integrate the model using the new extended (G′/G) expansion scheme. Consequently, we derive exact wave solutions, including singular and multiple periodic soliton solutions, kink waves, anti-kink waves, bell-shaped waves, lump waves, rogue waves, periodic lump waves, and interactions between lump and kink wave profiles. The properties of the fractional parameter on the achieved outcomes are also analyzed. We have created 3D plots, 3D plots with contour lines, and 2D plots of our attained solutions using the computational software, Maple. These systems can also represent various solutions for other fractional models in the domains of nonlinear science and engineering.

Data-driven wave solutions of (2+1)-dimensional nonlinear dispersive long wave equation by deep learning

In this paper, we introduce a deep learning framework to solve the (2+1)-dimensional nonlinear dispersive long wave equation. Using it, we study the data-driven solitary wave solutions, periodic wave solutions, kink wave solutions and various multiple soliton solutions for the equations with initial–boundary value conditions. On the one hand, we obtain the data-driven traveling wave solutions of the equations, which have high accuracy compared with corresponding exact traveling wave solutions. On the other hand, we set more complicated initial–boundary value problems of the equations, whose exact traveling wave solution cannot be easy to obtain, but our network still works well and gives its numerical solution. The results show that our deep learning algorithm is able to accurately capture the dynamical behavior of traveling waves of the (2+1)-dimensional nonlinear dispersive long wave equation.

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable (3+1)-dimensional combined potential Kadomtsev--Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation (pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new (3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new (3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined pKP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

On the Painlevé integrability of three-extensions to Mikhailov–Novikov–Wang equations: Multiple solitons, shocks, and other physical solutions

The current work examines three (1 + 1)-dimensional Mikhailov–Novikov–Wang (MNW) equations. The Painlevé criteria are employed for testing the integrability of the evolution equations. Using the simplified Hirota's approach, multiple soliton solutions for the family of the MNW equation are derived. Significant physical solutions, such as shock waves, periodic solutions, and many others, are also obtained for each equation under consideration. The current investigation provides insights into the integrability features of these evolution equations. The obtained outcomes will contribute to comprehending and studying many enigmatic phenomena that consistently manifest in nature and various nonlinear media, including optical fiber, fluid mechanics, and plasma physics.

Bäcklund transformation for analyzing a cylindrical Korteweg–de Vries equation and investigating multiple soliton solutions in a plasma

The presence of the geometrical curvature makes the planar Korteweg–de Vries (KdV) equation inadequate to describe the propagation of nonlinear waves. In many scientific disciplines including plasma physics, nonlinear optics, oceanography, and communications, the cylindrical KdV (CKdV) equation becomes the appropriate choice for modeling these waves. Motivated by these applications, the Bäcklund transformation is used to analyze and find an analytical solution to the CKdV equation in the present investigation. For the first time, the multi-soliton solutions, including single-, two-, and three-soliton solutions, are investigated, and a general scheme is given to find N-soliton solutions of the CKdV equation in the context of plasma physics. Numerous researchers may find the given solutions helpful in understanding the mechanism of the generation of multi-solitons in their laboratory experiments and may also engender interest in the space physics community to look for these structures in the data coming from a variety of satellites roaming in space.