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In this paper, we investigate a variable-coefficient nonlocal nonlinear Schrödinger equation in an optical fiber, which describes some optical pulses in a self-focusing medium. With the help of the Hirota bilinear method and symbolic computation, we obtain a set of the bilinear forms, the first-, second- and Nth-order breather solutions of the forementioned equation under a constraint. In particular, we graphically describe the effects of the variable coefficients on the first- and second-order breathers and we find that those variable coefficients change the spatial structures of the breathers. With the changes of the variable coefficients, the first-order breathers show periodicity, pulse train, split, breather wall and jagged shape and the second-order breathers show that one of the breathers becomes wider or the distance between the two breathers becomes closer. The results can provide a theoretical basis for the control of optical waves in nonlocal nonlinear media. [ABSTRACT FROM AUTHOR]

In this paper, a two-dimensional extended version of the nonlinear Schrödinger equation is investigated, which is called Maccari system. This system can compete with Davey–Stewartson equation and (2 + 1) -dimensional nonlinear Schrödinger equation in describing nonlinear waves. We obtained N-soliton and breather solutions of Maccari system by using the perturbation expansion method. The elastic collision of semi-rational solutions composed of lump, soliton and breather are also derived by taking a long wave limit of the obtained solitons. Furthermore, the inelastic collisions of first-order semi-rational solutions, namely (a) a hybrid of line rogue wave and dark soliton, (b) a hybrid of fusion lump and dark soliton, and (c) a hybrid of fission lump and dark soliton, are presented in terms of the KP hierarchy. Additionally, the semi-rational solutions composed of fusion lump, soliton and fission lump are derived. Simultaneously, the semi-rational solutions, which consist of fusion lump, fission lump, line rogue wave and dark soliton are also derived. The inelastic collisions of high-order semi-rational solutions of Maccari system have been rarely reported in other nonlinear systems. [ABSTRACT FROM AUTHOR]

The dynamic behavior of the mixed solution and the multiple solitary wave solutions in a (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli type equation is discussed. Based on the Hirota bilinear method, the lump–kink solution is constructed by introducing a new test function, and the existence conditions for the lump–kink solution are provided. The lump–kink solutions describe two types of interaction phenomena: the lump and the kink move head-on and collide elastically, and the lump chases the kink and collides elastically. In addition, the lump can pass through the kink wave, and the waveform and velocity of these localized waves remain unchanged after the collision. What is more, the innovation of this paper is that the multiple periodic-lump solutions are derived by setting some new test functions. Compared with the single period-lump solution in previous research, the multiple periodic-lump solutions exhibit richer dynamic behavior. Finally, the motion of two types of solitary waves on a periodic wave background is studied. Therefore, studying the elastic collision of lump-kink waves and the interaction of multiple periodic lump waves can help people understand nonlinear systems and solve nonlinear problems in nature. [ABSTRACT FROM AUTHOR]

This paper constructs a new (3+1)-dimensional Hirota–Satsuma–Ito equation and employs the Hirota N-soliton condition to determine the conditions for the existence of N-soliton solutions. Under these conditions, the Hirota bilinear method is utilized to derive N-soliton solutions of the equation. Additionally, breather solutions of the equation are obtained through constraints on complex conjugate parameters, while lump solutions are derived using the long-wave limit method. Building upon these results, interaction solutions are also explored. Furthermore, through the multiple-rogue wave method, multi-rogue wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito equation are investigated. A comparison between lump solutions obtained via the Hirota bilinear method and the multi-rogue wave solutions from multiple-rogue wave method reveals significant discrepancies. [ABSTRACT FROM AUTHOR]

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exact solutions of the (3+1)-dimensional pKP-BKP equation for describing certain nonlinear fluctuation phenomena in nonlinear optics, plasma physics and hydrodynamics. By imposing novel constraints on the N-soliton solution of this equation, we derive Y-type soliton solutions. Subsequently, through the application of complex conjugation techniques, these N-soliton solutions are converted into P-order breathing wave solutions. Furthermore, the interaction solutions are obtained by combining the Y-type soliton solutions with the breathing wave solutions during the partial degeneracy process of the N-soliton solutions. Finally, we select specific parameters to demonstrate the dynamic behaviors of the obtained solutions. [ABSTRACT FROM AUTHOR]

The study of nonlinear waves is a significant topic in the field of fluid Physics. In this paper, we study the nonlinear transformed waves, interactions and transformed molecular waves of the (3+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in plasma or ocean dynamics. Based on the condition of state transition, the transformed wave solutions derived from breath waves are obtained. The dynamics of the transformed wave solutions change with time, including peaks and shapes. In addition, we report two kinds of interactions, namely long- and short-lived collisions. The long-lived collision mode reveals the two constituent elements are always intertwined with each other. However, there is only disposable collision in short-lived collision mode, which is also considered as a catch-up case. As special interactions, the transformed molecular waves considered as a collision free mode whose atoms are transformed waves are investigated. The breather-breather molecule, breather-M-shaped molecule and double M-shaped molecule via velocity resonance condition are given. In summary, graphical investigations indicate that the novel transformed wave structures are shown for application in fluid mechanics. [ABSTRACT FROM AUTHOR]

The study delves into the impact of spatiotemporally modulated external potentials on both bright and dark solitons within spin-1 Bose-Einstein condensates. With the Hirota bilinear method, the one-soliton and two-soliton solutions of the (2+1)-dimensional three-coupled Gross-Pitaevskii equation under a spatiotemporal modulated capture potential are obtained. The analysis uncovers the formation of ferromagnetic bright and dark solitons, as evidenced by the computation of the total spin. The findings indicate that a multitude of soliton configurations arise contingent upon the selected external potentials, with their velocities and amplitudes being influenced by spatial and temporal factors. Asymptotic analysis of the bright and dark two-soliton systems reveals the nature of their elastic interactions. Additionally, leveraging a trained Fourier Neural Operator as an alternative predictive model, the temporal evolution of solitons as governed by the Gross-Pitaevskii equation are projected. This approach yields forecasted solutions that are notably in close agreement with the exact solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a generalized $(3+1)$-Kadomtsev-Petviashvili equation which is used to describe waves in a ferromagnetic medium. The equation's bilinear form is created and the new homoclinic test approach based on the Hirota bilinear form is used to find numerous novel precise solutions. These accurate solutions, which are depicted in the contour, two-dimensional and three-dimensional graphs, show the evolution of periodic characteristics. The modulation instability is used to investigate the stability of the obtained solutions. Additionally, the development of the fusion soliton is examined, as well as the fusion phenomenon in the traveling wave solution is described in the physical discussion. For this evolution equation, the study indicates new mechanical structures and various characteristics. The derived results back up the model that was proposed. These discoveries open up a new avenue for us to investigate the concept further.

In this paper, we take the (2+1)-dimensional generalized fifth-order KdV (fKdV) equation as an example. We obtain many wave solutions, including hybrid soliton, breather waves and solution molecules. In order to solve hybrid soliton, we give two transformation forms. For Transform one, real and complex parameters lead to different wave solutions. Soliton solutions are obtained when the parameter values are real, but some breather waves appear when the parameter values are complex. When the parameter value changes, the number of solitons and the position of the waves will also change. For Transform two, we assign the parameter values to real values and obtain the corresponding two-, three- and four-soliton solutions. The most important thing is that, on the basis of Transform one, we add different constraint conditions according to different coordinate systems, and finally get a very interesting phenomenon: soliton molecules. In addition, we give not only the three-dimensional plots but also the density plots. In general, these results will enrich the existing literature on the (2+1)-dimensional generalized fKdV equation and help to understand this kind of KdV equation. [ABSTRACT FROM AUTHOR]

Studying the transformation mechanisms of soliton solutions of some high-dimensional equations can aid in comprehending the physical phenomena of the relevant nonlinear wave interactions. Therefore, the transitions and mechanisms of nonlinear waves in the (2+1)-dimensional sine-Gordon and (2+1)-dimensional modified Boussinesq equations are investigated for the first time by means of characteristic line and phase shift analysis, and the dynamic behavior of various nonlinear transformed waves is analyzed. Firstly, we derive the N-soliton solution by applying the Hirota bilinear method, from which the breather solution is constructed by changing the parameters into a complex form in pairs. In addition, via the characteristic-line analysis, we present the mechanism for the transformation of the breather solution, which is composed of the nonlinear superposition of a solitary wave and periodic wave. When the characteristic lines of the two parts are parallel, we find that various transformed nonlinear wave structures can be obtained, such as M-type solitons, oscillating M-type solitons, multi-peak solitons, quasi-sine waves and so on. Finally, we demonstrate that the geometric properties of the characteristic lines vary with time essentially resulting in the time-varying properties of nonlinear waves, which have never been found in (1+1)-dimensional systems. Overall, the study of breath-wave transitions in the (2+1)-dimensional sine-Gordon equation and the modified Boussinesq equation provides valuable insights into the bahavior of nonlinear systems and wave propagation. [ABSTRACT FROM AUTHOR]

Nonlocal nonlinear Schrödinger equations are among the important models of nonlocal integrable systems. This paper aims to present a general formula for arbitrary-order breather solutions to multi-component nonlocal nonlinear Schrödinger equations by using the Hirota bilinear method. In particular, abundant wave solutions of two- and three-component nonlocal nonlinear Schrödinger equations, including periodic and mixed-wave solutions, are obtained by taking appropriate values for the involved parameters in the general solution formula. Moreover, diverse wave structures of the resulting breather and periodic wave solutions with different parameters are discussed in detail. [ABSTRACT FROM AUTHOR]

In the research paper, the propagation of nonlinear dust ion acoustic solitary waves in a collisionless dusty plasma consisting of negatively charged dust grains, ions, and isothermal electrons in the background of neutral particles is investigated. The reductive perturbation method has been employed to derive the Korteweg–de Vries (KdV) equation for small but finite-amplitude dust ion acoustic waves. The Agrawal technique is used to convert the regular KdV equation to the space-time fractional KdV equation. The Riemann–Liouville definition of the fractional derivative is used to describe the time fractional operator. The Hirota bilinear method is used to derive the multi-soliton solution of the time fractional KdV equation. The effects of the plasma parameters and time fractional parameters on the multisoliton of dust ion acoustic waves for the system are studied. We have also reported the conversion of two-soliton structures to one soliton, depending on the fractional parameter α . This observation in the plasma literature has never been reported before. The result of the present investigation may be applicable to some plasma environments, such as the Earth's mesosphere, cometary tails, Jupiter's magnetosphere, etc. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation is one of the most important models of contemporary physics which is popular throughout the broad fields of fluid movement as well as in the study of thick-water crests, liquid science, refractive optical components and so on. In this paper, we utilize the Hirota bilinear technique and the unified technique to attain various soliton solutions of the governing model analytically. These approaches are robust, powerful and unique also have many applications in different fields of mathematical physics. The solutions attained from these techniques are highly valuable and useful in various fields of sciences specially in the transmissions of optical fibers, also they give different behaviors including V-shaped and periodic soliton solution behavior. Further, the approaches applied here are not applied in this model previously. Therefore, ours is a new work, which summarizes its novelty. The 3D, 2D and contour plots are included to grasp the understanding of solutions' behavior. These findings are valuable in electronic communications such as elliptical circuits and in investigation of solitude controlling. [ABSTRACT FROM AUTHOR]

The Hirota bilinear (HB) is a powerful and widely used technique to find various types of solitons of integrable systems. In this manuscript, we implement HB technique to find bilinear form of a dimensionally reduced (3+1)‐dimensional KdV–Calogero–Bogoyavlenskii–Schiff (KdV‐CBS) equation at z = x, z = y, and z = t. We present various results for distinct auxiliary function to study kink solitons and its fission and fusion dynamics. The MATLAB‐2020 is used to display all the results via 3D and line 2D graphs for appropriate values of parameters. These findings provide a strong new insight into the nonlinear features of the model and lay the foundation for future studies in soliton dynamics and nonlinear events in related systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a generalized (3 + 1)- Kadomtsev-Petviashvili equation which is used to describe waves in a ferromagnetic medium. The equation's bilinear form is created and the new homoclinic test approach based on the Hirota bilinear form is used to find numerous novel precise solutions. These accurate solutions, which are depicted in the contour, two-dimensional and three-dimensional graphs, show the evolution of periodic characteristics. The modulation instability is used to investigate the stability of the obtained solutions. Additionally, the development of the fusion soliton is examined, as well as the fusion phenomenon in the traveling wave solution is described in the physical discussion. For this evolution equation, the study indicates new mechanical structures and various characteristics. The derived results back up the model that was proposed. These discoveries open up a new avenue for us to investigate the concept further. [ABSTRACT FROM AUTHOR]

In this paper, a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is systematically investigated based on the Hirota bilinear method. The explicit N-soliton solution and the bright and dark multi-soliton solutions of it are first derived. Next, various bright and dark higher-order breather solutions, including the periodic line wave solutions, as well as the hybrid solutions composed of solitons, breathers, and periodic line waves, are proposed by virtue of the complex conjugate constraints on the parameters. Then, applying the long wave limit to the N-soliton solution, the bright and dark lump solutions and line rogue wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation are constructed. The semi-rational solutions composed of breathers, lumps, solitons, and line rogue waves are further discussed. These new exact solutions all appear in pairs of bright and dark, which can be interpreted by the uplifts and collapses of energy. In addition, the dynamic behaviors of these exact nonlinear wave solutions are vividly demonstrated by their corresponding three-dimensional diagrams, sectional drawings, and density plots with contours. [ABSTRACT FROM AUTHOR]

In the current research, the new extended direct algebraic method (NEDAM) and the symbolic computational method, along with different test functions, the Hirota bilinear method, are capitalized to secure soliton and lump solutions to the (2+1)-dimensional fractional telecommunication system. Consequently, we derive soliton solutions with sophisticated structures, such as mixed trigonometric, rational, hyperbolic, unique, periodic, dark-bright, bright-dark, and hyperbolic. We also developed a lump-type solution that includes rogue waves and breathers for curiosity’s intellect. These features are important for controlling extreme occurrences in optical communications. Additionally, we investigate modulation instability (MI) in the context of nonlinear optical fibres. Understanding MI is essential for developing systems that may either capitalize on its positive features or mitigate its adverse effects. Also, a comprehensive sensitivity analysis of the observed model is carried out to evaluate the influence of different factors. 3D surfaces and 2D visuals, contours, and density plots of the outcomes are represented with the help of a computer application. Our findings demonstrate the potential of using soliton theory and advanced nonlinear analysis methods to enhance the performance of telecommunication systems.

Abstract 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.

Starting from the bilinear form of the positive Gardner–Kadomtsev–Petviashvili equation, breather-type and N-breather solutions are obtained with the help of the extended homoclinic test approach. Breather-type, one-breather, two-breather and three-breather solutions are illustrated through 3D plots, density plots and contour plots and some appropriate parameter values have been chosen to better demonstrate their time evolution. [ABSTRACT FROM AUTHOR]

In this work, a more accurate analytical solution of nonlinear partial differential equation is sought by setting the generalized activation function in the model of multiple bilinear neural network method. As an example, the 3-2-2-1, 3-2-3-1, 3-3-2-1 and 3-3-3-1 models are selected to study the new (2+1) dimensional nonlinear wave equation equations. Exact analytical solutions with arbitrary activation functions are obtained by selecting different activation functions and the dynamical properties are demonstrated through three-dimensional, two-dimensional and density plots. [ABSTRACT FROM AUTHOR]

Based on the Hirota bilinear method and the consistent tanh expansion method, we study different interaction excitations of the integrable (1+1)$$ \left(1+1\right) $$‐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. [ABSTRACT FROM AUTHOR]

The objective of this article is to investigate soliton, breather and lump solutions for the fractional (3+1)D–Yu–Toda–Sasa–Fukuyama (YTSF) equation with time-dependent variable coefficients, which is concerned with interfacial waves in a dual-layer liquid or elastic quasi-plane waves within a lattice. In particular case, YTSF equation reduces to the Kadomtsev–Petviashvili equation, which describes the long wave with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid or surface wave and the internal wave in the strait or channel of varying depth and width. Also, the YTSF equation reduces to Calogero–Bogoyavlenskii–Schiff equation, which describes the (2 + 1) -dimensional interaction between a Riemann wave propagating along the z-axis and a long wave propagating along the x-axis. For the first time, the Hirota bilinear method is utilized in the context of a fractional differential equation with varying coefficients. To provide a detailed insight into the dynamic behavior of the wave profile, we explore various types of complex solutions such as multi-solitons, lump, and breather solutions. Using the multi-soliton solutions as a foundation, we derive breather wave, lump and hybrid solutions. Graphical discussion is conducted to explore the impact of time-dependent coefficients and fractional derivative on the solutions. [ABSTRACT FROM AUTHOR]

This paper focuses on the long-distance telecommunication issues in real fiber optics transmission lines and explores the inhomogeneous variable-coefficient Hirota equation, which is from the modified nonlinear Schrödinger equation. The multi-soliton solutions can be given via the Hirota bilinear method. The dynamical structures of multi-soliton are investigated based on the obtained solutions. During soliton transmission, a step-like soliton is observed, capable of carrying more information and increasing communication capacity. The distinctive structures of these solitons enhance their resistance to interference compared to conventional solitons. The results in this paper have theoretical guiding significance in reducing information distortion and enhancing the quality of communication in fiber optics transmission. [ABSTRACT FROM AUTHOR]

In this paper, using the Hirota bilinear method, we investigate the perturbed Gerdjikov–Ivanov equation (PGIE) for lump soliton, rogue wave, periodic and kink cross rational solutions, homoclinic breathers, M-shaped and many other interactions. All these solutions have so many real life applications for example lump wave help to study the behavior of vortices and eddies in the ocean, which are used for climate modeling and fisheries management. Rogue waves are used in maritime safety, M-shaped solitons are used in optical fiber communications to transmit data over long distances without distortion, improving the efficiency and reliability of high-speed data transmission. Breathers can be used in fiber optic systems to generate ultrafast optical pulses, which have applications in telecommunications and laser technology. We also discuss the dynamical behaviour of our solutions in various dimensions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the new generalized stochastic fractional potential-KdV model is investigated analytically which describes the nonlinear optical solitons and photons propagating in electric circuits and multicomponent plasmas. The governing model is transformed into a bilinear form using the Hirota bilinear method. A variety of novel waves structures are developed in the form of lump waves, collisions between lump waves and periodic waves, collisions between lump waves or single and double-kink soliton solutions, collisions between lump, periodic, single and double-kink soliton solutions. Furthermore, the ( G ′ / G 2 ) expansion method and the Adomian decomposition method are employed to construct a number of new configurations for the governing dynamical model. To validate the scientific computations, a comparison study and stability analysis are carried out efficiently. Finally, to understand the dynamical behaviour of the the nonlinear fractional model, the 3D, 2D, and contour plots have also been demonstrated with the appropriate set of parameters. Depending upon the efficiency and the validity of the applied techniques, it is established that these strategies can also be applicable to many contemporary models. [ABSTRACT FROM AUTHOR]

Learned from wrinkle wave motions, we concentrated on bifurcation phenomena in substrate-supported graphene sheets by obtaining the bifurcation solitons of thermophoretic motion equation. We find new soliton solutions by applying the bilinear technique and correctly choosing the auxiliary function involved in the bilinear form. Use of the Hirota bilinear technique combined with symbolic computation yields lump solutions as well as lump-type solutions. The governing model is subject to computation of the lump one strip, lump two strip, Y-type, lump periodic solution, generalised breather, and lump periodic solution. When simulating waves in different physical systems, such as water waves, fibre optics, and plasma physics, lump and lump type solutions are crucial. Rogue waves are studied in relation to large and unforeseen ocean wave phenomena that might affect coastal buildings and ship safety. In communication systems, including optical fibres, generalised breathers are used to comprehend and regulate pulse propagation.

In this paper, we obtain a variety of analytical wave solutions of a (2 + 1)-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the exp (− Υ (g)) -expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, by using of the Hirota bilinear method and the long wave limit method, we consider elastic interactions and resonant interactions of a lump and two parallel line solitary waves of the (4+1)-dimensional Fokas equation. We find that the phase shifts of the obtained lumps play an important role in the interactions and three different types of elastic interactions are obtained when the phase shifts are larger than zero. When the phase shifts tend to infinity, we find partially resonant interactions and completely resonant interactions of a lump and two parallel line solitary waves. The asymptotic interaction behaviors of the obtained solutions are analyzed in detail and illustrated graphically. [ABSTRACT FROM AUTHOR]

In this paper, the goal is to focus on the localized characteristics and dynamic behavior of lump solutions and interaction solutions for the (2 + 1)-dimensional complex systems. The (2 + 1)-dimensional Fokas system was considered as the research model in this paper, based on the Hirota bilinear method, and a series of new exact solutions were constructed by selecting some new test functions. Meanwhile, the conditions for ensuring the positivity of lump solutions are discussed. In addition, the localized characteristics and dynamic behavior of some interaction solutions are described through analysis and graphics, for example, interaction solution between a lump and stripe soliton pairs contains a rogue wave excited from stripe soliton pairs. The main novelty of this paper is that a series of new analytical solutions obtained can explain the interaction mechanism of nonlinear waves in nonlinear physics. It is hoped that our research results can play a certain promoting role in the study of nonlinear theory. The method adopted in this paper can also study high-dimensional and variable-coefficient nonlinear complex systems. [ABSTRACT FROM AUTHOR]

The progression of nonlinear electron-acoustic waves (EAWs) in a magnetized and collision-free plasma made up of cold inertial electrons, inertialess superthermal electrons, and stationary background ions with special reference to Saturn's magnetosphere (SMS) is explored. The method of reductive perturbation (MRP) is employed to obtain the evolution equation (i.e., Zakharov– Kuznetsov equation (ZKE)) that governs the propagation of electron acoustic solitons (EASs). Using the elegant and efficient Hirota bilinear method (HBM), multi-soliton solutions (MSSs) of the ZKE are determined. The impact of the effects of hot-to-cold electron density ratio, magnetic field (MF) strength, and superthermality on single as well as the interaction of EASs is examined. Estimates of the values of the electric field at several radii of SMS (i.e., 12 R s − 17.8 R s , where R s is the radius of Saturn) are presented, which are found in μV / m to mV / m range and are in perfect agreement with the data from Cassini radio and plasma wave science wideband receiver. Moreover, the influence of the relevant plasma parameters on the interaction time and spatial extent of the interacting EASs is also explored. [ABSTRACT FROM AUTHOR]

For the sake of intellectual curiosity, in this manuscript we analyse the soliton solutions of a dynamical model namely, the (2 + 1) -dimensional generalised Bogoyavlensky–Konopelchenko equation arising in mathematical physics. We apply the Hirota bilinear method and the unified method to produce a variety of novel solutions. By using the aforementioned techniques, we establish fresh sorts of solutions for the concern problem, including breather wave, lump-periodic, two wave solutions, dark, periodic, and rational soliton solutions. In addition, we describe the stability analysis of our chosen model. For certain values of the required free parameters, the propagation of the derived well-furnished outcomes is visualised in different profiles using Mathematica 13.0. All of the findings in this study are crucial to comprehending the physical significance and behaviour of the investigated equation, which highlights the significance of examining various nonlinear wave phenomena in mathematical physics. The acquired results demonstrate that the computational strategies used are efficient, competent, and concise and may be used in conjunction with representative computations to apply to more complicated phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the (2+1)- dimensional coupled nonlinear Schrödinger equations with variable coefficients which are used to describe the propagation of beams in inhomogeneous and nonlinear birefringent fibers, taking into account the components in the two polarized directions. To tackle this problem, we employ the Hirota bilinear method. Multiple solitons, rogue waves, breather waves and their interaction solutions relating to the suitable choice of time-dependent coefficients are obtained. Through manipulating the relevant parameters, the propagation control and evolution are investigated for them. Several interesting transition phenomena are revealed, such as, the transitions from the bright soliton to breather, from bright rogue to periodic humps of rogue wave structures overlay on the one- and two-peak bright soliton and from the plane wave background to the periodic ones. [ABSTRACT FROM AUTHOR]

Three types of solutions (namely, localized wave solutions and their two distinct types of interaction solutions) to a (3+1)-dimensional nonlinear evolution equation were acquired based on the Hirota bilinear method. The basic steps commenced with obtaining the Hirota bilinear form of the original equation, then solving it by using mathematical software, and ultimately the solutions to the original equation were derived by the bilinear transformation. The solutions of the Hirota bilinear equation were assumed as the superposition of arbitrary number of functions. For the first type of localized wave solutions, the solutions of the corresponding bilinear equation were the superposition of n quadratic positive functions. The second type of mixed solutions which reflected the interaction between localized waves and multi-kink waves, consisted of n positive quadratic function, n 1 exponential function and n 2 hyperbolic cosine functions. The last type solutions were the interaction solutions between the localized wave solutions and the k-kink ( k = 1 , 2 ) wave solutions. All three types of solutions contained localized wave solutions which are not homogeneous. Finally, an analysis of the dynamic properties of each type of solutions was conducted by introducing specific parameter values and generating plots. [ABSTRACT FROM AUTHOR]

The nonlinear Hirota–Satsuma–Ito (HSI) equation is an important (2 + 1) -dimensional model arising in fluid dynamics, plasmas, and other domains. In this study, the Hirota bilinear method is employed to derive a set of innovative soliton interaction solutions within the framework of the HSI system. Second-order solitons are the focus of our investigation, leading to the identification of four distinct solution types through a systematic analysis of their respective phases. These solutions are characterized by a rich spectrum of collision dynamics, encompassing phenomena, such as fusion, fission, and the intersection of trajectories. A comprehensive characterization of critical properties, including asymptotic forms, velocities, amplitudes, and interaction region lengths, is carried out. For third-order cases, seven hybrid soliton solutions are constructed, and their properties are elucidated. The results offer new analytical insights into the intricate resonant soliton behaviors permitted by the HSI system. The understanding of multidimensional soliton interactions carries significant implications spanning from optics to Bose–Einstein condensates. The Hirota bilinear technique demonstrates strong capabilities in uncovering novel soliton features in complex nonlinear wave models. Extending these methods promises additional discoveries at the intersection of nonlinear fluid dynamics, integrable systems, and soliton theory. [ABSTRACT FROM AUTHOR]

In present paper, multiple rogue wave solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation were studied by applying the traveling wave transformation and the Hirota bilinear method. We obtained its 1-rogue wave, 3-rogue wave, 6-rogue wave and 8-rogue wave solutions through symbolic computation. Three-dimensional, contour and density plots of corresponding solutions vividly shown the physical structure and properties. Furthermore, these solutions have greatly enriched the exact solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation on the existing literature. [ABSTRACT FROM AUTHOR]

Waves advantageous properties have recently been exploited to enable a wide range of applications, including impact reduction, switching, focussing, and asymmetric transmission. We find that the motion of lumps, rogue waves, lump periodic waves, Ma-breather waves, lump one-strip, and generalised breather waves may also be used to build flexible constructions crawl. Symbolic computation and the Hirota bilinear method are used to create these nonlinear waves. Combining the aforementioned approaches, we demonstrate that such pulse-driven locomotion achieves optimal effectiveness when the pulses are solitons. Our research shows how nonlinear waves can be used in a variety of new ways, including as a platform for developing adaptable robots that can move. [ABSTRACT FROM AUTHOR]

A variable coefficient Schrödinger equation which is derived by using the multi-scale expansion and coordinate expansion transformation method from nonlinear inviscid barotropic nondivergent vorticity equation in a β -plane is discussed in this paper. It is different from the previous model that the background basic flow is assumed as a function of time t. Then, the bilinear forms are obtained based on a transformation. By using the Hirota bilinear method, exact analytical solutions to the Schrödinger equation are achieved. These solutions include single-soliton, two-soliton and three-soliton solutions, whose interactions have been presented in the form of 3-d solid figure or density graphic to describe the dynamic characteristics, and might help to describe Rossby waves more suitable. Furthermore, the effects on solitons of coefficients except which relate to the dispersion relation of the model are discussed. [ABSTRACT FROM AUTHOR]