Multiple Soliton Solutions Research Articles

Painlevé analysis for a new (3 +1 )-dimensional KP equation: Multiple-soliton and lump solutions

The current work proposes a new (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation ((3 + 1)-KPE). We verify the integrability of this equation using the Painlevé analysis (PA). The bilinear formula is applied to the extended KPE to explore multiple-soliton solutions. Also, we formally establish a class of lump solutions using distinct values of the parameters.

A generalized nonlinear fifth-order KdV-type equation with multiple soliton solutions: Painlevé analysis and Hirota Bilinear technique

In present work, we formulate a new generalized nonlinear KdV-type equation of fifth-order using the recursion operator. This equation generalizes the Sawada-Kotera equation and the Lax equation that study the vibrations in mechanical engineering, nonlinear waves in shallow water, and other sciences. To determine the integrability, we use Painlevé analysis and construct solutions for multiple solitons by employing the Hirota bilinear technique to the established equation. It produces a bilinear form for the driven equation and utilizes the Lagrange interpolation to create a dependent variable transformation. We construct the solutions for multiple solitons and show the graphics for these built solutions. The mathematical software program Mathematica employs symbolic computation to obtain the multiple solitons and various dynamical behavior of the solutions for newly generated equation The Sawada-Kotera equation and Lax equation have various applications in mechanical engineering, plasma physics, nonlinear water waves, soliton theory, mathematical physics, and other nonlinear fields.

Multiple soliton solutions and other solutions of a variable-coefficient Korteweg–de Vries equation

In this paper, we construct multiple soliton solutions of a time-dependent variable-coefficient Korteweg–de Vries equation in the form of bright, dark and singular soliton solutions. Furthermore, the periodic solutions are derived. These will be achieved via the aid of solitary wave ansatz methods. In addition, other exact solutions of a time-dependent variable-coefficient Korteweg–de Vries equation are attained using symmetry reductions. In one special case, the power series method is employed to derive a solution of the aforesaid equation. We also examine the conservation laws of the aforementioned equation using the variational approach. A brief physical representation of the derived conserved vectors will be conversed.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

This study retrieves some new exact solutions to the 2 + 1 -dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced exp − φ ξ -expansion method is imposed to the 2 + 1 -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

New dynamical behaviors for a new extension of the Shallow water model

The aim of this work, is to construct some novel solutions for a new extension of the shallow water model in (3+1)-dimensions. Based on two methods namely; simplified Hirota’s method and a long-wave method a class of solutions are reported. Multiple soliton solutions in complex form, breather wave, and mixed breather–soliton solutions are constructed by using simplified Hirota’s method. To explore rational solutions and construct a class of physical interaction phenomena, a long-wave method is applied to N-soliton solutions. To our knowledge, the addressed solutions in this article are novel. Moreover, to better realize the physical meaning of obtained solutions, all solutions are plotted in three-dimension. The obtained solutions may be used in a wide variety of physical scenarios relevant to the environment, including modeling of floods and tsunamis as well as flow in rivers and open channels.

Symmetry Reductions, Cte Method and Interaction Solutions for Sharma-Tasso-Olver-Burgers Equation

In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. An nonauto-Bäcklund (BT) theorem that includes the over-determined equations and the consistent condition is obtained by the CTE method. By using the nonauto-BT theorem, the interactions between one-soliton and the cnoidal wave, and between one-soliton and the multiple resonant soliton solutions, are constructed. The dynamics of these novel interaction solutions are shown both in analytical and graphical forms. The results are potentially useful for explaining ocean phenomena.

Electron acoustic singular solitons interaction in the Earth’s magnetotail region

Electron acoustic potential structures are generated in the presence of a large-amplitude electric field in Earth’s magnetotail region, Earth’s magnetosphere according to the numerous space observations by satellites. In this article, we study the electron-acoustic solitary waves (EASWs) in an unmagnetized electron–ion plasma consisting of cold electrons and isothermal ions with two different temperatures. Nonlinear evolution equations are derived from the hydrodynamic model of collisionless, unmagnetized electron–ion plasma. Exact analytic solutions are obtained from derived Korteweg-de Vries (KdV) equations and modified Korteweg-de Vries (MKdV) equations separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significant multi singular soliton interaction has been studied for KdV singular solitons and MKdV singular solitons in the critical parameter set.

Singular solitons interaction of dust ion acoustic waves in the framework of Korteweg de Vries and Modified Korteweg‐de Vries equations with (r,q) distributed electrons

AbstractTwo nonlinear evolution equations (NLEEs), namely, Korteweg‐de Vries (KdV) and modified Korteweg‐de Vries (MKdV) are derived from the dust hydrodynamic model of collisionless, unmagnetized dusty plasma with electrons following double spectral velocity distribution, that is, (r,q) distribution. Regular as well as singular analytic solutions of derived KdV equations and MKdV equations are obtained separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significance of multi singular soliton interaction has been studied for KdV singular solitons and for MKdV singular solitons in the critical parameter set.

Solitons and symmetry reduction solutions of a nonlocal two‐mode Korteweg–de Vries equation

The two‐mode Korteweg–de Vries (TMKdV) equation describes propagation of two different waves modes simultaneously. By using consistent correlated bang (CCB) method, a nonlocal form of the TMKdV equation is constructed and converted into two local equations with its dependent variables having definite parity. Multiple soliton solutions of the nonlocal TMKdV equation are obtained by using known multiple soliton solutions of the TMKdV equation, and some singular travelling wave solutions are also generated in a similar way. The classical Lie symmetry method is also carried on the nonlocal TMKdV equation to get its symmetry reduction solutions.

A novel and efficient method for obtaining Hirota’s bilinear form for the nonlinear evolution equation in (n+1) dimensions

Bilinearization of nonlinear partial differential equations (PDEs) is essential in the Hirota method, which is a widely used and robust mathematical tool for finding soliton solutions of nonlinear PDEs in a variety of fields, including nonlinear dynamics, mathematical physics, and engineering sciences. We present a novel systematic computational approach for determining the bilinear form of a class of nonlinear PDEs in this article. It can be easily implemented in symbolic system software like Mathematica, Matlab, and Maple because of its simplicity. The proven results are obtained by using a developed method in Mathematica and applying a logarithmic transformation to the dependent variable. Finally, the findings validate the implemented technique’s competence, productivity, and dependability. The approach is a useful, authentic, and simple mathematical tool for calculating multiple soliton solutions to nonlinear evolution equations encountered in nonlinear sciences, plasma physics, ocean engineering, applied mathematics, and fluid dynamics. • This paper introduces a new systematic computational approach for constructing bilinear forms for nonlinear PDEs. • The given approach can handle a class of (n+1)-dimensional nonlinear PDEs. • Various examples of well-known nonlinear evolution equations up to the (4+1)-dimension are shown. • The results of various nonlinear equations are provided using the proposed method.

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

Integrable (3+1)-dimensional Ito equation: variety of lump solutions and multiple-soliton solutions

Solitons, rogue waves and interaction behaviors of a third-order nonlinear Schrödinger equation

In this paper, we investigate a third-order nonlinear Schrödinger equation (NLSE) which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The higher-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detailed calculations.

On multiple soliton solutions of the extended (3+1)-dimensional Jimbo–Miwa equations

In this paper, we deal with a set of equations, namely the extended (3 + 1)-dimensional Jimbo–Miwa (JM) equations. We focus on the calculations of soliton solutions to the aforementioned equations using the multiple exponential function method (MEF). As a result, we derive up the one to three-wave soliton solution to the both equations, while presenting the profiles obtained through the constant values related to those solutions. It is noteworthily to point out that, all the calculations in this paper are made with software package Maple. Also, the profiles of the waves depend on the arbitrary values of autocephalous parameters.

Novel multiple soliton and front wave solutions for the 3D-Vakhnenko–Parkes equation

The multiple exp-function technique, as an extension to Hirota’s perturbation algorithm, is employed to find N-soliton solutions of the 3D Vakhnenko–Parkes (3D-VP) equation. This is a straightforward and effective alternative method. Consequently, one- two- and three-soliton solutions besides the front wave solutions and their wave numbers are obtained, which are completely distinct from the other published literature. 3D and density plots were investigated via the help of the Maple package program by suitably choosing the free parameters in the solutions. Further, the physical meaning for these solutions is clarified. We mention that the propagation of high-frequency waves in a relaxing medium can be investigated by discussing the dynamical behavior of the obtained solutions for the 3D-VP equation.

Generalized fifth-order nonlinear evolution equation for the Sawada-Kotera, Lax, and Caudrey-Dodd-Gibbon equations in plasma physics: Painlevé analysis and multi-soliton solutions

This research aims to investigate a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and other nonlinear sciences. The Painlevé analysis is used to determine the integrability of the equation, and the simplified Hirota technique is applied to construct multiple soliton solutions with an investigation of the dispersion relation and phase shift of the equation. We utilize a linear combination approach to construct a system of equations to obtain a general logarithmic transformation for the dependent variable. We generate one-soliton, two-soliton, and three-soliton wave solutions using the simplified Hirota method and showcase the dynamics of these solutions graphically through interaction between one, two, and three solitons. We investigate the impact of the system’s parameters on the solitons and periodic waves. The SK, Lax, and CDG equations have a wide range of applications in nonlinear dynamics, plasma physics, oceanography, soliton theory, fluid dynamics, and other sciences.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods: the Lie symmetry analysis, the generalized Riccati equation mapping approach, and the modified Kudryashov approach, we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation (NCDR) with power-law nonlinearity and density-dependent diffusion. Lie symmetry analysis is one of the powerful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables. By the Lie group technique, we obtain one-parameter invariant transformations, determining equations and corresponding vectors for the considered convection-diffusion-reaction equation. By treating the parameters of the governing equation as constants, the applied methods yield a variety of new closed-form solutions, including inverse function solutions, periodic solutions, exponential function solutions, dark solitons, singular solitons, combo bright-singular solitons, and the combine of bright-dark solitons and dark-bright solitons. Moreover, using the Bäcklund transformation of the generalized Riccati equation and modified Kudryashov method, we can construct multiple solitons and other solutions of the considered equation. The obtained new solutions of this work demonstrate that the used approaches are powerful and effective in dealing with nonlinear equations, and that these solutions are required to explain many biological and physical phenomena. Comparing our obtained solutions of this paper with the ones obtained in the literature, we see that our solutions are new and not reported elsewhere. These newly formed soliton solutions will be more beneficial in the various disciplines of ocean engineering, plasma physics, and nonlinear sciences.

Comment on ‘two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions’

Comment on ‘two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions’

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

Characteristics of the new multiple rogue wave solutions to the fractional generalized CBS-BK equation

IntroductionThe multiple Exp-function scheme is employed for searching the multiple soliton solutions for the fractional generalized Calogero-Bogoyavlenskii-Schiff-Bogoyavlensky- Konopelchenko equation. ObjectivesMoreover, the Hirota bilinear technique is utilized to detecting the lump and interaction with two stripe soliton solutions. MethodsThe multiple Exp-function scheme and also, the semi-inverse variational principle will be used for the considered equation. ResultsWe have obtained more than twelve sets of solutions including a combination of two positive functions as polynomial and two exponential functions. The graphs for various fractional-order α are designed to contain three dimensional, density, and y-curves plots. Then, the classes of rogue waves-type solutions to the fractional generalized Calogero-Bogoyavlenskii-Schiff-Bogoyavlensky- Konopelchenko equation within the frame of the bilinear equation, is found. ConclusionFinally, a direct method which is called the semi-inverse variational principle method was used to obtain solitary waves of this considered model. These results can help us better understand interesting physical phenomena and mechanism. The dynamical structures of these gained lump and its interaction soliton solutions are analyzed and indicated in graphs by choosing suitable amounts. The existence conditions are employed to discuss the available got solutions.