Hirota Bilinear Method Research Articles

Breather and soliton solutions of a generalized (3 + 1)-dimensional Yu–Toda–Sasa–Fukuyama equation

Fluid mechanics is a branch of physics that focuses on the study of the behavior and laws of motion of fluids, including gases, liquids, and plasmas. The Yu–Toda–Sasa–Fukuyama equation, a class of Kadomtsev–Petviashvili type equations, is a significant integrable model with applications in fluids and other fields. In this paper, we study breather and soliton solutions of a generalized (3 + 1)-dimensional YTSF equation. By utilizing the Hirota bilinear method and Painlevé analysis, we construct solutions in the form of trigonometric and hyperbolic functions and analyze the interaction between waves graphically. We consider the characteristics of wave distribution along characteristic lines to obtain the distance between each wave and the angle generated, which is beneficial for understanding the ocean wave superposition effect. Additionally, we examine the dynamic characteristics of the wave, such as amplitude, velocity, period, shape, position, width, and phase. Furthermore, we investigate the effects of the system parameters on solitons and breathers.

Dark Localized Waves in Shallow Waters: Analysis within an Extended Boussinesq System

We study dark localized waves within a nonlinear system based on the Boussinesq approximation, describing the dynamics of shallow water waves. Employing symbolic calculus, we apply the Hirota bilinear method to transform an extended Boussinesq system into a bilinear form, and then use the multiple rogue wave method to obtain its dark rational solutions. Exploring the first- and second-order dark solutions, we examine the conditions under which these localized solutions exist and their spatiotemporal distributions. Through the selection of various parameters and by utilizing different visualization techniques (intensity distributions and contour plots), we explore the dynamical properties of dark solutions found: in particular, the first- and second-order dark rogue waves. We also explore the methods of their control. The findings presented here not only deepen the understanding of physical phenomena described by the (1+1)-dimensional Boussinesq equation, but also expand avenues for further research. Our method can be extended to other nonlinear systems, to conceivably obtain higher-order dark rogue waves.

Localized waves and interaction solutions to an integrable variable coefficient Date-Jimbo-Kashiwara-Miwa equation

This paper initiates an exploration into the exact solutions of the variable coefficient Date-Jimbo-Kashiwara-Miwa equation, first utilizing the Painlevé analysis method to discuss the integrability of the equation. Subsequently, By employing the Hirota bilinear method, N-soliton solutions for the equation are constructed. The application of the Long wave limit method to these N-soliton solutions yields rational and semirational solutions. Various types of localized waves, encompassing solitons, lumps, breather waves, and others, emerge through the careful selection of specific parameters. By analyzing the image of the solutions, the evolution process and its dynamical behavior are studied.

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the N-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.

Abundant interaction solutions of the integrable (1 + 1)‐dimensional coupled KdV system

Based on the Hirota bilinear method and the consistent tanh expansion method, we study different interaction excitations of the integrable ‐dimensional coupled KdV system, which is a typical and essential water wave model in the two‐layer fluid. Abundant new interaction solutions including lumps, lump–kink solution, breathers, two‐kink solution, resonant soliton, and soliton–cnoidal solution are obtained. All these exact solutions are shown both analytically and graphically with proper constant selection.

New lump solutions and several interaction solutions and their dynamics of a generalized (3+1)-dimensional nonlinear differential equation

Abstract In this paper, we mainly focus on proving the existence of lump solutions to a generalized (3+1)-dimensional nonlinear differential equation. Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a (3+1)-dimensional nonlinear differential equation. Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions. Moreover, the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves. In addition, the breath-wave solutions and several interaction solutions of the (3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

Solitons and lump waves to the elliptic cylindrical Kadomtsev–Petviashvili equation

We construct various kinds of solutions to the elliptic cylindrical Kadomtsev–Petviashvili equation, which can be derived in the context of surface gravity waves. This investigation yields a diverse set of analytical outcomes, encompassing the dynamics of horseshoe-like solitons, lump chains, and lump waves. Based on the Hirota bilinear method, the lump chain is derived under specified conjugate constraint conditions. Additionally, we uncover three distinct types of novel interactions among lump waves through the utilization of the long wave limit method. A detailed analysis is conducted to elucidate the features and patterns inherent in these lump waves. The analytic findings presented may providing a descriptive framework for generic weakly nonlinear and weakly dispersive waves.

Scalar and vector electromagnetic solitary waves in nonlinear hyperbolic media

In this paper, we investigate the problem of electromagnetic wave propagation in hyperbolic nonlinear media. To address this problem, we consider the scalar hyperbolic nonlinear Schrödinger system and its coupled version, namely hyperbolic Manakov type equations. These hyperbolic systems are shown as non-integrable. Then, we examine the propagation properties of both the scalar and vector electromagnetic solitary waves by deriving their exact analytical forms through the Hirota bilinear method. A detailed analysis shows that the presence of hyperbolic transverse dispersion provides an additional degree of freedom to prevent the formation of singularity in both the scalar and vector solitary wave structures in this hyperbolic nonlinear media. Besides this, we realize that the solitary waves in this media possess fascinating propagation properties which cannot be observed in conventional nonlinear media. We believe that the present study will be very useful in analyzing electromagnetic wave propagation in hyperbolic nonlinear metamaterials.

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.

Study on the interaction solution of Zakharov-Kuznetsov equation in quantum plasma

The fundamental difference between quantum and traditional plasmas is the electron and ion composition, the former has a much higher density and extremely lower temperature, and it can be modelled by Zakharov-Kuznetsov (ZK) equation. In this paper, the Hirota bilinear method is used to study its solution properties.

Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method

Construction of M-shaped solitons for a modified regularized long-wave equation <i>via</i> Hirota's bilinear method

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Soliton molecules in Fermi–Pasta–Ulam–Tsingou lattice: Gardner equation approach

We revisit the Fermi–Pasta–Ulam–Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gardner equation. Through the Hirota bilinear method, the fundamental as well as multi-soliton, particularly two-, three- and four-soliton, solutions are obtained for the Gardner equation. Based on these multi-soliton solutions, we show the excitation of an interesting class of table-top soliton molecules in the FPUT lattice through the velocity resonance mechanism. Depending on the condition on the free parameters, which determine the bond length between the soliton atoms, we classify them as dissociated and synthetic type molecules. The main feature of the table-top soliton molecules is that they do not exhibit oscillations in the coalescence region. This property ensures that they are distinct from the soliton molecules, having retrieval force, of the nonlinear Schrödinger family of systems. Further, to study the stability of the soliton molecule we allow it to interact with a single (or multi) soliton(s). The asymptotic analysis shows that their structures remain constant, though the bond length varies, throughout the collision process. Then, from the relative separation distance calculation we also observe that the bond length of each of the soliton molecules is not maintained during soliton molecule–molecule interaction while their structures are preserved. In addition, we consider the FPUT lattice with quadratic nonlinear interaction and FPUT lattice with cubic nonlinearity as sub-cases and point out the nature of the soliton molecules for these cases also systematically. We achieve this based on the interconnections between the solutions of the Gardner, modified Korteweg–de Vries and Korteweg–de Vries equations. Finally, in order to validate the existence of all the soliton structures, especially reported for the Gardner equation in the continuous limit, we simulate the corresponding FPUT chain and verified these various structures numerically and analytically as well. We believe that the present study can be extended to other integrable and non-integrable systems with applications in fluid dynamics, Bose–Einstein condensates, nonlinear optics, and plasma physics.

Rational solutions and some interactions phenomena of a (3+1)-dimensional BLMP equation in incompressible fluids: A Hirota bilinear method and dimensionally reduction approach

This manuscript addresses the rational solutions of a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation, emphasizing key aspects in response to specific questions. The study begins by elucidating the problem’s significance, highlighting the pursuit of rational solutions for the mentioned equation. The methodology involves obtaining the Hirota bilinear (HB) form through a transformative process. Furthermore, the analysis extends to the dimensionally reduced BLMP equation, specifically exploring lump and rogue wave-like (Rwl) solutions, along with the interaction between kink and lump solutions. The HB forms play a crucial role in this examination. Importantly, the solutions presented in this study are characterized as non-singular and localized. Visualizations of these solutions are instrumental in understanding the equation’s behavior, particularly how the solutions evolve under varying physical conditions. The outcomes are visually conveyed through 3D and surface graphs, offering a comprehensive representation for specific parameter values. In conclusion, this work contributes novel insights beyond previous efforts in the literature, showcasing the importance and advancements in the study of the (3+1)-dimensional BLMP equation.

Collision dynamics of high-order localized waves in a novel [formula omitted]-dimensional integrable Boussinesq model

The pursuit of finding precise solutions for complex nonlinear systems in high dimensions has long been a significant goal in the fields of mathematics and physics. This paper focuses on investigating a novel equation, referred to as the (3+1)-dimensional Boussinesq equation, which accurately describes the behavior of gravity waves on the surface of water. Using the Hirota bilinear method, the study successfully derives multiple-soliton and multi-breather solutions for the Boussinesq equation. Moreover, it presents mixed solutions that combine solitons and breathers under specific parameter conditions. By considering the long wave limit for these solitons, rational and semi-rational solutions for the equation are obtained. These rational solutions encompass line rogue waves and lump waves, while the semi-rational solutions emerge from the interaction between line rogue waves and solitons. Additionally, employing symbolic computation, the paper introduces a class of multiple lumps for the (3+1)-d Boussinesq equation, showcasing intriguing triangular formations and circular patterns. Overall, the findings of this research enhance our understanding of unique nonlinear phenomena and offer valuable theoretical support for future experimental investigations.

Novel soliton molecule solutions for the second extend (3+1)-dimensional Jimbo-Miwa equation in fluid mechanics

The main aim of this paper is to investigate the different types of soliton molecule solutions of the second extend (3+1)-dimensional Jimbo-Miwa equation in a fluid. Four different localized waves: line solitons, breather waves, lump solutions and resonance Y-type solutions are obtained by the Hirota bilinear method directly. Furthermore, the molecule solutions consisting of only line waves, breathers or lump waves are generated by combining velocity resonance condition and long wave limit method. Also, the molecule solutions such as line-breather molecule, lump-line molecule, lump-breather molecule, etc. consisting of different waves are derived. Meanwhile, higher-order molecule solutions composed of only line waves are acquired.

Lumps, solitons and stability analysis for the (3 + 1)-dimensional fractional potential-YTSF-like model in weakly dispersive medium

In this study, the fractional (3 + 1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF)-like model is being investigated analytically and semi-analytically. The given model is converted into a bilinear form using the Hirota bilinear method. Which refers to the development of lump waves, collisions between lump waves and periodic waves, collisions between lump waves and single- and double-kink soliton solutions, and collisions between lump, periodic, and single- and double-kink soliton solutions. Furthermore, the Adomian technique and the modified extended Tanh function technique are applied to obtain several forms of innovative combinations for the governing dynamical fractional model. In addition, it has been confirmed that the established results are stable, and it has been helpful to validate the calculations. Moreover, multiple intriguing exact solutions are utilized to illustrate the physical nature of 3D, contour, and 2D graphs. A collection of bright, dark, periodic, rational, and elliptic function solutions are obtained for the solution.

Rogue wave solutions for the [formula omitted]-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, the 3+1-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation is investigated by Hirota bilinear method, from which the first, second and third order rogue wave solutions are derived. The relationships between the parameters and the amplitude, width and center of rogue wave are discussed. At the same time, several evolution and density plots are present to exhibit the dynamical structure of the rogue wave solutions. The results may be useful in nonlinear optics and other physically relevant integrable systems.

Bäcklund transformation and interaction solutions of a generalized Kadomtsev–Petviashvili equation with variable coefficients

Kadomtsev–Petviashvili (KP) equation is a classical nonlinear evolution equation applied in many fields such as plasma physics, fluid dynamics and engineering. In general, variable coefficients are of value in reflecting the inhomogeneities of media, nonuniformities of boundaries, and external forces. This paper mainly studies some exact solutions of a generalized KP equation with variable coefficients by using Hirota bilinear method. Based on the bilinear form, one-, two-, N-soliton solution, Bäcklund transformation and Lax pair are obtained. With the use of the test function method, lump solution and interaction solutions of lump-stripe and lump-soliton types are constructed, and the structure of the solutions are shown in three-dimensional figures. The analysis on the interaction solutions reveals new dynamic behaviors and periodic evolutions.

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.