Homoclinic Breather Research Articles

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

Solitons, lump and interactional solutions of the (3+1)-dimensional BLMP equation in incompressible fluid

Based on the Hirota bilinear method, we systematically investigate the (3+1)-dimensional Boiti-Leon-Manana-Pempinelli (BLMP) equation in incompressible fluids and main results include: (1) the formulas of N-kink-soliton solutions and the bound states of multi solitons are all presented, (2) the lump solution is derived by the positive quadratic function method, (3) the interactional solutions are given, i.e., one lump interacts with one- and two-kink-soliton, (4) some special periodic solutions are discussed, i.e., lump-periodic solutions and homoclinic breather solutions.

Multiwaves, breathers, lump and other solutions for the Heimburg model in biomembranes and nerves

In this manuscript, a mathematical model known as the Heimburg model is investigated analytically to get the soliton solutions. Both biomembranes and nerves can be studied using this model. The cell membrane’s lipid bilayer is regarded by the model as a substance that experiences phase transitions. It implies that the membrane responds to electrical disruptions in a nonlinear way. The importance of ionic conductance in nerve impulse propagation is shown by Heimburg’s model. The dynamics of the electromechanical pulse in a nerve are analytically investigated using the Hirota Bilinear method. The various types of solitons are investigates, such as homoclinic breather waves, interaction via double exponents, lump waves, multi-wave, mixed type solutions, and periodic cross kink solutions. The electromechanical pulse’s ensuing three-dimensional and contour shapes offer crucial insight into how nerves function and may one day be used in medicine and the biological sciences. Our grasp of soliton dynamics is improved by this research, which also opens up new directions for biomedical investigation and medical developments. A few 3D and contour profiles have also been created for new solutions, and interaction behaviors have also been shown.

Analysis of bifurcation, chaotic structures, lump and $ M-W $-shape soliton solutions to $ (2+1) $ complex modified Korteweg-de-Vries system

<abstract><p>This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.</p></abstract>

Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice

Abstract In this article, the equation showing the cold bosonic atoms in a zig-zag optical lattice model for some breathers, M-shaped solution and lump soliton solution, homoclinic breather pulses, breather lump pulses, periodic-cross kink wave, kink cross-rational propagation, and interaction between lump periodic and kink wave was examined. Some M-shaped solution, M-shaped interaction with periodic and kink, M-shaped interaction with rogue and kink, M-shaped rational solution, M-shaped rational solution with one kink, M-shaped rational solution with two kink, solutions for lump soliton waves, lump one kink waves, lump two kink waves, periodic-cross lump wave propagation, periodic wave propagation, rogue wave propagation, and multiwave propagation were also acquired. Likewise, our solution was also graphically presented, and also their stability was checked.

Exploration of interactional phenomena and multi‐wave solutions of the fractional‐order dual‐mode nonlinear Schrödinger equation

This work concerns the fractional‐order dual‐mode nonlinear Schrödinger equation (FDMNLSE), which portrays the augmentation or absorption of dual waves. This model dissects the concurrent generation of two characteristic waves dealing with dual modes and introduces three physical parameters: nonlinearity, phase velocity, and dispersive factor. In the context of photonics, NLSE models the propagation of soliton pulses over intercontinental distances. Throughout this work, the fractional derivative is given in terms of time and space conformable sense. We analyze the multi‐waves method, homoclinic breather approach, and interactional solution with the double ‐functions procedure, and their applications for this equation are obtained using logarithmic transformation. The multi‐wave method is a well‐known phenomenon in nonlinear science that describes the interaction of three waves that satisfy certain resonance conditions. A breather wave is a localized and oscillatory solution that maintains its shape over time. Finally, we will discuss the dynamics of our newly obtained solutions with the help of graphs by assigning appropriate values to the parameters. The proposed methods are straight and aggressive, so the approved form can be extended for more nonlinear models. The findings are exceptional in comparison to previous findings in the literature. These outcomes may have significance for additional investigation of such frameworks to handle the nonlinear issues in applied sciences. The obtained results help us understand fluid propagation and incompressible fluids.

Multi-Peak and Propagation Behavior of M-Shape Solitons in (2 + 1)-Dimensional Integrable Schwarz-Korteweg-de Vries Problem

This paper examines the propagation of M-shape solitons and their interactions with kink waves to the (2 + 1)-dimensional integrable Schwarz-Korteweg-de Vries (ISKdV) problem by applying the symbolic computation with ansatz functions technique and logarithmic transformation. The governing model usually appears in the nonlinear shallow water waves and fluid mechanics. We discuss various nonlinear waves like multiwave solutions (MSs), homoclinic breather (HB), M-shape solitons, single exponential form (one-kink), and double exponential form (two-kink). These waves have lot of applications in fluid dynamics, nonlinear optics, chemical reaction networks, biological systems, climate science, and material science. We also study interaction among M-shape solitons with kink wave. At the end, we discuss the stability characteristics of all solutions.

Analysis of lump solutions and modulation instability to fractional complex Ginzburg–Landau equation arise in optical fibers

In this paper, the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law in nonlinear optics, which simulates soliton propagation in various waveguides in the presence of a detuning component which comes from the nonlinear Schrödinger equation (NLSE) with the inclusion of the growth and damping terms, is examined. The Hirota bilinear method is exercised to retrieve lump solitons such as the 1-kink wave solution, 2-kink wave solution, double exponential wave solution, and homoclinic breather wave solution to the model. We also scrutinize some M-shaped solutions in the forms of M-shaped rational solutions and the M-shaped interaction with rogue and kink waves. In addition, the instability modulation and gain spectra of the CGLE are examined. The originality of the study lies in the secured outcomes, which were never before produced and effectively balance the nonlinear physical aspects. To illustrate the dynamic of these waves, some of the solutions are sketched in three-dimensional, two-dimensional, contour, and density plots. The produced results are encouraging which can be used to describe the phenomena occurring in nonlinear optical or plasma physics. The computed solutions demonstrate that the suggested approaches are skillful, categorical, consistent, and effective in identifying exact solutions to a variety of complicated nonlinear problems that have recently arisen in nonlinear optics, applied sciences, and engineering.

Multi-wave, breather and interaction solutions to (3+1) dimensional sine-Gordon equation arising in nonlinear physical sciences

Based on the Hirota bilinear forms, the multi-wave solutions of the (3 + 1) dimensional sine-Gordon equation with various physical properties were constructed with using symbolic computation. This equation is characterized by constant coefficients that can be integrated in the Painleve sense. Double exponential and homoclinic breather approaches are used to obtain solutions. In addition, the interaction phenomenon between the exponential function, sine function and the hyperbolic sine function are constructed. For a better understanding the dynamics of the underlying equation three dimensional, density and contour graphs are demonstrated for the appropriately selected parameters. Thus, it has become more possible to observe the physical behaviors of the solutions. The results obtained in this study will contribute to the expansion and enrichment of the solution forms available in the literature of the model under consideration. The results obtained could have important applications in nonlinear optics, solid state physics, optical solitons and quantum field theory.

Description of soliton and lump solutions to [formula omitted]-truncated stochastic Biswas–Arshed model in optical communication

This article negotiates the investigation of optical stochastic solitons and other exact stochastic solutions with the fractional stochastic Biswas–Arshed equation (FSBAE) describing the multiplicative white noise of optical signal propagation in birefringent fibers using the modified F-expansion method and Hirota Bilinear method. In order to manage this model, we used Itô calculus. After utilizing the aforementioned techniques and computational software, different stochastic solitary wave solutions are retrieved, including dark, bright, periodic, singular, hyperbolic, rational, combo, and trigonometric function solutions. Additionally, we also investigate several wave solutions, such as the cross-kink rational wave solution, the homoclinic breather wave solution, and M-shaped rational solution. To examine the different kinds of solitons and their dynamical behaviors, the solutions have been simulated by graphs. The discovered solutions are essential for illuminating the wave dynamics in diverse models. The obtained stochastic solitary wave solutions may be crucial in nonlinear science and engineering fields. It is impressive to see that the chosen methodologies are simple, appropriate, and effective scientific tools for determining stochastic solitary wave solutions for nonlinear engineering models (NLEMs). This is the first investigation into the lump solutions of the FSBA equation with multiplicative white noise, and the findings show how optical solitons propagate in nonlinear optics.

A study of breather lump wave, rogue wave, periodic cross kink wave, multiwave, M-shaped rational and their interactions for generalized nonlinear Schrödinger equation

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave, and multiwave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present 3D, 2D, contour, and stream plot illustrations for our graphs.

A study of breather lump wave, rogue wave, periodic cross kink wave, multi-wave, M-shaped rational and their interactions for generalized nonlinear Schrödinger equation

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two-kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave and multi-wave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two-kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present three-dimensional, two-dimensional, contour, and stream plot illustrations for our graphs.

Study of Stochastic–Fractional Drinfel’d–Sokolov–Wilson Equation for M-Shaped Rational, Homoclinic Breather, Periodic and Kink-Cross Rational Solutions

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves. We also derive the M-shaped interaction with rogue and kink waves and the M-shaped interaction with periodic and kink waves. This model is used in mathematical physics, surface physics, plasma physics, population dynamics and applied sciences. Moreover, we also show our results graphically in different dimensions. We obtain these solutions under some constraint conditions.

Novel rational solitons and generalized breathers for (1+1)-dimensional longitudinal wave equation

This paper deals with the evaluation of multiwave, homoclinic breather, M-shaped rational solitons and interaction of M-shaped rational solitons with single and double kinks for [Formula: see text]-dimensional longitudinal wave equation via the logarithmic transformation and symbolic computation with ansatz functions method. We will obtain two types of M-shaped rational solitons and their dynamics are revealed in figures by choosing the distinct values of involved parameters. In addition, two forms of interaction of M-shaped rational soliton with kink wave will also be evaluated. Furthermore, we will compute kink cross-rational solutions, periodic cross-rational solutions, generalized breathers and Akhmediev breathers for governing model. The solitons behaviors via different values of parameters in the produced solutions will be analyzed.

Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

Multiwave, rogue wave, periodic wave, periodic cross-lump wave, periodic cross-kink wave, lump soliton, breather lump, homoclinic breather, periodic cross-kink, M-shaped rational solutions and their interactions for the Degasperis–Procesi equation

We examine multiwave (MW), rogue wave (RW), periodic wave (PW), homoclinic breather (HB), breather lump wave (BLW), [Formula: see text]-shaped rational solutions, lump soliton, kink cross-rational (KCR), periodic cross-kink rational (PCKR), periodic cross-rational (PCR) solutions for the Degasperis–Procesi (DP) equation in shallow water waves through appropriate polynomial function scheme. We also compute some interactions for stated model including interaction of [Formula: see text]-shaped soliton with one kink, interactional solution with two kinks also lump soliton with kink one and kink two solutions. Furthermore, we made suitable 3D, 2D and contour profiles via appropriate parameters.

Optical solitons in Fiber Bragg Gratings for the coupled form of the nonlinear [formula omitted]-dimensional Kundu-Mukherjee-Naskar equation via four powerful techniques

In this article, four powerful techniques namely, the three wave method, double exponential, homoclinic breather, M-Shaped rational, M-Shaped with one kink, and M-Shaped with two kink are putting into practice to bolster the new optical solutions of the coupled form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation in Fiber Bragg Gratings (FBGs). Implementation of the appropriate functions of the solutions results in a different form of multi-wave solutions with their interaction phenomena such as dark soliton solutions, singular soliton solutions, bright soliton solutions, kink soliton solutions, Multi-Peak soliton solutions, multi dark-bright soliton solutions, multi bright soliton solutions, multi dark soliton solutions with different structure, multi M-Shaped soliton solutions, multi M-Shaped soliton solutions, M-Shaped soliton solutions with different structure, kink soliton solutions with different structure, and periodical M-Shaped soliton solutions. Three-dimensional graphics demonstrating the dependability and productivity of the suggested methodologies are used to describe the dynamics of the produced solutions.

Applications for mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation in water wave flumes and optical fibers

In this work, by applying the symbolic computation with ansatz functions approach and logarithmic transformation, we investigate the analytical rational solitons of the mixed Chen–Lee–Liu derivative nonlinear Schrödinger-equation. We acquire two categories of M-shaped rational solutions and their dynamics are revealed in figures by choosing the proper vales of concerned parameters. Moreover two types of interaction of M-shaped rational solutions with kink wave are computed. Furthermore we evaluate multiwaves solutions, Homoclinic breather, kink cross-rational solutions and periodic cross-rational solutions for governing model. We also investigate the stability property of attained solutions and present in the form of table.

Homoclinic breather, M-shaped rational, multiwaves and their interactional solutions for fractional quadratic-cubic nonlinear Schrödinger equation

Homoclinic breather, M-shaped rational, multiwaves and their interactional solutions for fractional quadratic-cubic nonlinear Schrödinger equation

Optical solitons, multiwave, breather and M-shaped solitons for an nonlinear Schrödinger equation in (2+1)-dimensions with cubic–quintic–septic law

In this paper, we address a newly nonlinear Schrödinger equation (NLSE) in [Formula: see text]-dimensions influenced by cubic–quintic–septic law nonlinearity and spatial dissipations effects. We properly construct the dark and bright optical soliton solutions for our governing model. Furthermore, periodic, other singular solutions, multiwave, homoclinic breather, M-shaped rational and their interaction with kink waves of various structures will be derived. In addition to this, some 3D, 2D and contour profiles will be added to anticipate the wave dynamics.