Multiple Soliton Solutions Research Articles

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G′/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

Multiple soliton solutions and other travelling wave solutions to new structured (2+1)-dimensional integro-partial differential equation using efficient technique

The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In this study finds numerous classes of exact solutions for a new structured (2 + 1)-dimensional Ito integro-differential equation using the help of the Mathematica software. The Improved Modified Extended Tanh Function Scheme (IMETFS) is utilised to address the aforementioned equation analytically. Bright, dark, and singular soliton solutions are produced. Additionally, periodic, exponential, rational, singular periodic, and Weierstrass elliptic doubly periodic results are achieved. The method employed includes the nonlinear evolution equations that arise in a variety of real-world situations, and it is efficient, applicable, and simple to handle. For certain obtained solutions, specific options of free constants are presented in 3D, 2D, and contour graphical depictions.

Multiple solitons and breathers on periodic backgrounds in the complex modified Korteweg–de Vries equation

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

Multiple Soliton Solutions of Generalized Reaction Duffing Model Arising in Various Mechanical Systems

Multiple Soliton Solutions of Generalized Reaction Duffing Model Arising in Various Mechanical Systems

A novel discretization of the Yajima-Oikawa equation: Cauchy matrix approach

The Cauchy matrix approach, rooted in the Sylvester equation, plays a crucial role in defining the τ functions of nonlinear evolution equations. In this paper, the Cauchy matrix approach is employed to introduce a novel integrable semi-discrete counterpart of the one-dimensional Yajima-Oikawa (YO) system. This new system is linked with the differential-difference Kadomtsev-Petviashvili (KP) equation with self-consistent sources (SCS). Based on the Cauchy matrix approach of the KP system, we systematically construct multiple soliton and multiple pole solutions. Furthermore, we analyze and illustrate various examples of soliton solutions for further understanding.

Painlevé integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

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.

On simulations of 3D fractional WBBM model through mathematical and graphical analysis with the assists of fractionality and unrestricted parameters

This study retrieves some novel exact solutions to the family of 3D space–time fractional Wazwaz–Benjamin–Bona–Mahony (WBBM) equations in the context of diverse nonlinear physical phenomena resulting from water wave mechanics. The family of WBBM equations is transformed for this purpose using a space and time fractional transformation into an ordinary differential equation (ODE). The ODE then uses a strong method, namely the Unified Method. Consequently, lump solutions, dark-bright soliton, singular and multiple soliton solutions, and periodic solutions are investigated. The disparities between the current study's conclusions and previously acquired solutions via other approaches are examined. All wave solutions produced are determined to be novel in terms of fractionality, unrestricted parameters, and implemented technique sense. The impact of unrestricted parameters and fractionality on the obtained solutions are visually presented, along with physical explanations. It is observed that the wave portents are varied with the increase of unrestricted parameters as well as fractionality. We dynamically showed that the appropriate transformation and the applied Unified approach more proficient in the study of water wave dynamics and might be used in future researches to clarify the many physical phenomena. The novelty of this work validate that the proposed method seem simple and useful tools for obtaining the solutions in PDEs and it is expected to use in mathematical physics and optical engineering.

Exploring the chaotic structure and soliton solutions for (3 + 1)-dimensional generalized Kadomtsev–Petviashvili model

The study of the Kadomtsev–Petviashvili (KP) model is widely used for simulating several scientific phenomena, including the evolution of water wave surfaces, the processes of soliton diffusion, and the electromagnetic field of transmission. In current study, we explore some multiple soliton solutions of the (3+1)-dimensional generalized KP model via applying modified Sardar sub-equation approach (MSSEA). By extracting the novel soliton solutions, we can effectively obtain singular, dark, combo, periodic and plane wave solutions through a multiple physical regions. We also investigate the chaotic structure of governing model using the chaos theory. The behavior of the collected solutions is visually depicted to demonstrate the physical properties of the proposed model. The solutions obtained in this paper can expand the existing solutions of the (3+1)-dimensional KP model and enhance our understanding of the nonlinear dynamic behaviors. This approach allows for consistent and effective treatment of the computation process for nonlinear KP model.

Multiple soliton solutions of some conformable fractional nonlinear models using Sine–Cosine method

Multiple soliton solutions of some conformable fractional nonlinear models using Sine–Cosine method

Resonant multiple soliton and rogue type multiple lump wave solutions of the modified KdV–KP equation

Resonant multiple soliton and rogue type multiple lump wave solutions of the modified KdV–KP equation

Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3 + 1)-dimensional kadomtsev-petviashvili equation

Some new exact solutions of the generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation (KPE) are explored in this study. Firstly, the resonant multiple soltion solutions (RMSs) are discussed via employing the linear superposition principle and weight algorithm. Then, by introducing pairs of the conjugate parameters to the RMSs, the complexiton solutions including the non-singular complexiton and singular complexiton solutions are extracted. In addition, the complex multiple kink soliton solutions are also probed by employing the bilinear approach. Finally, we investigate the rational wave solutions via the test function method and symbolic computation. By choosing the appropriate parameters, the graph descriptions of the derived solutions are presented to show the dynamical properties. The outcomes of this work are desirous to bring some new perspective to the study of the complexiton, complex solutions and rational wave solutions to the other PDEs.

Soliton solutions of nonlinear stochastic Fitz-Hugh Nagumo equation

This study deals with the stochastic Fitz-Hugh Nagumo (FHN) equation and its multiple soliton solutions. The underlying model has numerous applications in neuroscience that express the pulse behavior of neurons. In general, different kinds of noise affect neurons, e.g., oscillations in the opening and closing of ion stations within cell membranes and the fluctuations of the different conductivities in the system. This fluctuation creates a sequence of stochastic excitation. Various applications are branching Brownian motion process, flame propagation, nuclear reactor theory, autocatalytic chemical reaction, mobility in neurons, population growth in the open environment, and liquid environment. So, need of the hour to consider the FHN equation under the impact of noise. The ϕ6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^6$$\\end{document}-model expansion method is used to extract the analytical solutions that give a dynamic attitude of the transmission for the nerve impulses of a nervous system. The different constraint conditions for the existence of these solutions are also discussed. The solutions of this model are represented in hyperbolic, trigonometric, and rational forms. The 2 and 3-dimensional behavior of these solutions are depicted by choosing the different values of parameters. The impact of noise on the physical system is analyzed and its real-world applications are discussed. The spikes in the solutions are controlled through the Borel function. These important results will open a new horizon of research for the young researchers.

Multiple soliton and traveling wave solutions for the negative-order-KdV-CBS model

The (3+1)-dimensional new negative-order-KdV-CBS model is investigated in this study. The suggested model combines the Korteweg-de Vries (KdV) and Calogero-Bogoyavlenskii-Schiff (CBS) equations. This research provides multiple soliton solutions and traveling wave solutions for the KdV-CBS model. Multiple exp-function methods have been used for extracting soliton solutions. For this aim, the extended sinh-Gordon equation expansion approach was selected to get traveling wave solutions. The findings are graphically examined by selecting appropriate values for arbitrary parameters.

The influence of fractionality and unconstrained parameters on mathematical and graphical analysis of the time fractional phi-four model

In the framework of several nonlinear physical phenomena arising from water wave mechanics, this work recovers some new precise solutions to the time-fractional phi-four equation. For this, the phi-four equation's space and time fractional transformation is converted into an ordinary differential equation (ODE). The improved modified extended tanh function (imETF) approach is then employed by the ODE as a powerful method based on the conformable derivative. Thus, lump solutions, periodic solutions, solitary and multiple soliton solutions, dark-bright soliton solutions, and Jacobi elliptic doubly periodic type solutions are studied. The imETF appro solitons applications in many scientific and engineering fields develops mathematical methodologies, helps research solitons, and advances our knowledge of nonlinear processes. The differences between the results of this investigation and those obtained earlier using alternative methods are analyzed. In terms of fractionality, unconstrained parameters, and applied method sense, all generated wave solutions are found to be new. Physical explanations and a visual representation of the effects of unconstrained parameters and fractionality on the derived solutions are provided. We find that the wave portents change as the number of unconstrained parameters and fractionality increases. In conclusion, we dynamically show that the proper transformation and the applicable imETF method are more effective in analyzing water wave dynamics and might be employed in subsequent studies to shed light on various physical phenomena.

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

Extended (3 + 1)-dimensional Kairat-II and Kairat-X equations: Painlevé integrability, multiple soliton solutions, lump solutions, and breather wave solutions

PurposeThis study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.Design/methodology/approachThe Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.FindingsThis study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.Research limitations/implicationsThe Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.Practical implicationsThis study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.Social implicationsThe work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.Originality/valueThis paper presents an original work that presents two newly developed Painlev\\'{e} integrable models with insightful findings.

Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV system

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves

Abstract The ZK–mZK–BBM equation plays a crucial role in actually depicting the gravity water waves with the long wave region. In this article, the bilinear forms of the (2 + 1)-dimensional ZK–mZK–BBM equation were derived using variable transformation. Then, the multiple soliton solutions of the ZK–mZK–BBM equation are obtained by bilinear forms and symbolic computation. Under complex conjugate transformations, quasi-soliton solutions and mixed solutions composed of one-soliton and one-quasi-soliton are derived from soliton solutions. These solutions are further studied graphically to observe the propagation characteristics of gravity water waves. The results enrich the research of gravity water wave in fluid mechanics.