Hirota's Bilinear Research Articles

Trajectory equation of a lump before and after collision with other waves for generalized Hirota–Satsuma–Ito equation

Based on the Hirota bilinear and long wave limit methods, the hybrid solutions of m-lump with n-soliton and n-breather wave for generalized Hirota–Satsuma–Ito (GHSI) equation are constructed. Then, by approximating solutions of the GHSI equation along some parallel orbits at infinity, the trajectory equation of a lump wave before and after collisions with n-soliton and n-breather wave are studied, and the expressions of phase shift for lump wave before and after collisions are given. Furthermore, it is revealed that collisions between the lump wave and other waves are elastic, the corresponding collision diagrams are used to further explain.

Resonantly interacting lump chains in the Mel'nikov equation

We constructed resonant lump chain solutions to the Mel'nikov equation by utilizing Hirota bilinear approach, mainly includes the resonance between lump chains, resonance between lump and lump chains. By calculating the asymptotic form of the solution, the resonant interaction of these chains results in infinite phase shifts, which is analogous to the line soliton interactions in the Kadomtsev-Petviashvili-II equation. The interactions are classified into two types, oblique and parallel cases, depending on the velocities of the individual lump chains. Our study highlights several cases of obliquely collided lump chains with a Y-shaped structure. Furthermore, the parallel resonance of lump chains can lead to the transmission, splitting, or absorption of another lump chain. The interactions between lump and lump chains are classified into two types, lump is semi-localized in time or completely localized in time.

Hirota bilinear method and multi-soliton interaction of electrostatic waves driven by cubic nonlinearity in pair-ion–electron plasmas

Multi-soliton interaction of nonlinear ion sound waves in a pair-ion–electron (PIE) plasma having non-Maxwellian electrons including Kappa, Cairns, and generalized two spectral index distribution functions is studied. To this end, a modified Korteweg–de Vries (mKdV) equation is obtained to investigate the ion-acoustic waves in a PIE plasma at a critical plasma composition. The effects of temperature and density ratios and the non-Maxwellian electron velocity distributions on the overtaking interaction of solitons are explored in detail. The results reveal that both hump (positive peak) and dip (negative peak) solitons can propagate for the physical model under consideration. Two and three-soliton interactions are presented, and the novel features of interacting compressive and rarefactive solitons are highlighted. The present investigation may be useful in laboratory plasmas where PIE plasmas have been reported.

Soliton propagation for a coupled Schrödinger equation describing Rossby waves

We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method. For the coupled Schrödinger equation, we obtain the Manakov model of all-focusing, all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions. Especially, we find that the all-defocusing type Manakov model admits bright-bright soliton solutions. Furthermore, we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions. Therefrom, we go over how the free parameters affect the Manakov model’s all-focusing type’s two-soliton and three-soliton solutions’ collision locations, propagation directions, and wave amplitudes. These findings are useful for setting a simulation scene in Rossby waves research. The answers we have found are helpful for studying physical properties of the equation in Rossby waves.

EVOLUTIONARY BEHAVIOR OF THE INTERACTION SOLUTIONS FOR A (3+1)-DIMENSIONAL GENERALIZED BREAKING SOLITON EQUATION

The interaction solutions have attracted the attention of many scholars because of they are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves. In this paper, we investigate the interaction and rational solutions of a (3+1)-dimensional generalized breaking soliton equation by employing the Hirota bilinear and parameter limit methods along with symbolic computations. By studying the Hirota bilinear form of the equation, abundant interaction and rational solutions are derived by choosing appropriate parameters of the test function. The evolutionary behavior of the interaction solutions is also analyzed theoretically and graphically. Compare with the published literatures, we get some completely new results of the equation in this paper.

A new (3+1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws

This study's subject is a (3 + 1) dimensional new Hirota bilinear (NHB) equation that appears in the theory of shallow water waves. We investigate how particular dispersive waves behave in an NHB equation. In this regard, we first test the Painlavé integrability using the WTC-Kruskal method and second perform Lie point symmetry analysis on NHB. The algorithm outputs the Lie algebra, the symmetry reductions, and group invariant solutions. Using Lie point symmetries, NHB transforms into an ordinary differential equation. The integration architectures to solve this equation are the Bernoulli sub-ODE, 1/ G, and modified Kudryashov methods. We plot their graphs and observe how the solutions behaved in understanding the physical phenomenon. Additionally, we discuss the model utilizing nonlinear self-adjointness and Ibragimov's approach to generate conservation laws for each Lie symmetry generator.

Integrability features of a new (3+1)-dimensional nonlinear Hirota bilinear model: multiple soliton solutions and a class of lump solutions

PurposeThis paper aims to propose a new (3+1)-dimensional integrable Hirota bilinear equation characterized by five linear partial derivatives and three nonlinear partial derivatives.Design/methodology/approachThe authors formally use the simplified Hirota's method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.FindingsThe Painlevé analysis shows that the compatibility condition for integrability does not die away at the highest resonance level, but integrability characteristics is justified through the Lax sense.Research limitations/implicationsMultiple-soliton solutions are explored using the Hirota's bilinear method. The authors also furnish a class of lump solutions using distinct values of the parameters via the positive quadratic function method.Practical implicationsThe authors also retrieve a bunch of other solutions of distinct structures such as solitonic, periodic solutions and ratio of trigonometric functions solutions.Social implicationsThis work formally furnishes algorithms for extending integrable equations and for the determination of lump solutions.Originality/valueTo the best of the authors’ knowledge, this paper introduces an original work with newly developed Lax-integrable equation and shows new useful findings.

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

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

Transformation and interactions among solitons in metamaterials with quadratic-cubic nonlinearity and inter-model dispersion

In this paper, we investigate multiple soliton interactions and other solitary wave solutions (SWS) for a perturbed nonlinear Schrödinger equation (NLSE) with negative index material having quadratic-cubic nonlinearity (NLSE-QCN). Due to its high order dispersion term, this model yields sub-picosecond impulses useful in mode-locked ring lasers. Hirota bilinear method (HBM) will be used to study soliton interaction. By controlling the parameters, we will obtain [Formula: see text], [Formula: see text], parabolic and anti-parabolic, butterfly, bright and dark shaped solitons. On the other hand, we will obtain some other solitary wave solutions with the help of Sine-Gordon expansion (SGE) scheme.

Variety interaction between [formula omitted]-lump and [formula omitted]-kink solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we investigate the (3+1)-dimensional Burger system which is employed in soliton theory and generated by considering the Hirota bilinear equation. We conclude some novel analytical solutions, including 2-lump-type, interaction between 2-lump and one kink, two lump and two kink of type I, two lump and two kink of type II, two lump and one periodic, two lump and kink-periodic, and two lump and periodic–periodic wave solutions for the considered system by symbolic estimations. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences. Then we apply explicit numerical methods, most of which are recently introduced by many scholars, to reproduce the analytical solutions. The test results show that the best algorithms, especially the Hirota bilinear, are very efficient and severely outperform the other methods.

A novel method for solving third-order nonlinear Schrödinger equation by deep learning

The nonlinear Schrödinger equation is a classic integrable equation, especially plays an important role in nonlinear optics. Therefore, a large number of scholars have studied the solution of this equation and proposed many effective solutions, such as Hirota Bilinear transformation method, Darboux transformation, neural network, and so on. This paper obtain numerical solutions by improving physics-informed neural network (IPINN) method which embeds the physical law, an adaptive activation function and slope recovery term into a traditional neural network. This method can simulate the soliton solution and the rogue wave solution of the nonlinear Schrödinger equation with very little data. The Adam and L-BFGS optimizer are used to optimize network parameters to minimize the loss function. The dynamic behavior of soliton and rogue wave solution of the third-order nonlinear Schrödinger equation is first revealed by using the IPINN method, and the errors obtained by this method are compared with the PINN method. Numerical experiments indicate that this method has better numerical results than the PINN method, which opens up a new way to simulate other physical problems by using the IPINN method.

Superthermal electron’s effects on lump solitons structures in magnetized auroral plasma

Based on the considerations of the acknowledgements of the numerous space observations by satellites in the auroral plasma, astrophysics plasmas, spacecraft observations, various plasma models have been framed and revealed different interesting features, like electrostatic structures, solitary waves, double layers, supersolitons, etc. Soliton theory is a very efficient and competent way to describe nonlinear features. Using Viking satellite data in the auroral plasma, we have derived lump soliton solutions of the Kadomstev-Petviashvili (KP) equation by employing Hirota bilinear method. Due to its wide range of applications, the study of lump soliton is very attractive and important too. It has been shown that the lump solitons structures as well as in the one-dimensional form of lump soliton are varied with associated parameters in the auroral magnetized plasma. During the analysis of the features of the lump solitons, it is found that the system parameters play a pivotal role on the lump solitons structures.

Trajectory equation of a lump before and after collision with other waves for (2+1)-dimensional Sawada–Kotera equation

In this paper, we employ the Hirota bilinear and long wave limit methods to obtain the hybrid solutions consisting of mth order lumps and nth order breathers, mth order lumps and n line waves for (2+1)-dimensional Sawada–Kotera (SK) equation. Then, according to the characteristics of a lump wave moving along a straight line, the trajectory equation of a lump before and after collision with a line wave, two line waves and breather waves are obtained by approximating the solution of SK equation along some parallel orbits at infinity. Furthermore, we generalize the above cases to the collision of a lump wave with n line waves and the nth order breather waves, and the corresponding trajectory equations are presented. We also give the expression of phase shift for the lump wave, and verify that when ϑi≠0, the height of the lump wave before and after collision does not change, which reveals that collisions between a lump and other waves are elastic.

Study of lump soliton structures in quantum electron‐positron‐ion magnetoplasma

AbstractSoliton theory is a very efficient and competent way to describe nonlinear features. Using Hirota bilinear method, we have obtained a lump soliton solution of the Kadomstev‐Petviashvili equation which is the reduced canonical form of the collisionless quantum electron‐positron‐ion magnetoplasma system. Due to its wide range of applications, the study of lump soliton structures is very much attractive and important. The amplitude of lump solitons is varied for different system parameters. During the analysis of the features of the lump solitons, it is found that the quantum diffraction parameter (He), statistical temperature ratio (σ), positrons number density (μp), magnetic intensity parameter (Ω) have a considerable impact on the lump solitons structures. Compressive as well as rarefactive lump soliton has been found during the investigation.

Lump Collision Phenomena to a Nonlinear Physical Model in Coastal Engineering

In this study, a dimensionally nonlinear evolution equation, which is the integrable shallow water wave-like equation, is investigated utilizing the Hirota bilinear approach. Lump solutions are achieved by its bilinear form and are essential solutions to various kind of nonlinear equations. It has not yet been explored due to its vital physical significant in various field of nonlinear science. In order to establish some more interaction solutions with some novel physical features, we establish collision aspects between lumps and other solutions by using trigonometric, hyperbolic, and exponential functions. The obtained novel types of results for the governing equation includes lump-periodic, two wave, and breather wave solutions. Meanwhile, the figures for these results are graphed. The propagation features of the derived results are depicted. The results reveal that the appropriate physical quantities and attributes of nonlinear waves are related to the parameter values.

A direct method for generating rogue wave solutions to the (3+1)-dimensional Korteweg-de Vries Benjamin-Bona-Mahony equation

In this paper, we provide a generating mechanism to obtain rogue wave solutions from N soliton of Hirota's bilinear method. Based on the long wave limit method, the phase parameters are reconstructed to generating rogue wave solutions of Korteweg–de Vries Benjamin-Bona-Mahony equation. The rogue wave solutions are expressed explicitly in rational forms. Besides fundamental pattern of rogue waves, the triangle or pentagon pattern have been revealed. Both types of triangle and pentagon patterns are resolved upon different choices of free parameters introduced.

Soliton-mediated ionic pulses and coupled ionic excitations in a dissipative electrical network model of microtubules

Many cellular activities are mediated by microtubules (MTs), with most electrophysiological processes depending on ionic flow through MT cylinders. This paper addresses such conductive features by representing the MT as a nonlinear electrical transmission line composed of capacitive and dissipative properties. Thanks to the semi-discrete approximation near the continuum limit, coupled ionic pulses flowing along cellular microtubules are described by a set of coupled cubic complex Ginzburg-Landau (CGL) equations whose one of the solutions is a plane wave. The stability of the latter is checked, under weak modulation, using an explicit analytical expression for the modulational instability (MI) growth rate. The parametric analysis of the instability growth rate allows detecting parameter regions where ionic conductivity, through modulated waves, in the MT network is likely to occur. Dissipative bright-bright pulse soliton solutions for the nonlinearly coupled cubic CGL equations are constructed using a modified Hirota's bilinear method. A generalized coupled mode for the discrete ionic signal is proposed and used as the initial condition to be propagated in the nonlinear electrical transmission lattice of MT under direct numerical simulations. The ionic pulse transfer, mediated by the nonlinear interaction between oscillatory modes, is manifested by the formation of trains of modulated waves whose behaviors depend on the right choice of system parameters. Those results theoretically suggest that coupled ionic signals may facilitate information processing involving MTs.

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

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

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

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

Lump interaction phenomena to the nonlinear ill-posed Boussinesq dynamical wave equation

In this manuscript, we discuss the dynamical behavior of ill-posed Boussinesq (IPB) dynamical wave equation. This equation depicts how long wave made in shallow water propagates due to the influence of gravity. The different wave structures are extracted by the utilization of Hirota's bilinear method (HBM) and different test function approaches. The wave structures in the forms of novel breather waves, lump solutions, two-wave solutions, and rogue wave solutions are obtained. Furthermore, with assistance of appropriate parameters, the acquired solutions' physical behavior is drawn out in 3 dimensional, and contour profiles. As a consequence of the obtained results, we can claim that the used methodology is simple, dynamic, extremely effective, and will serve as a useful tool for solving more severely nonlinear problems in a variety of domains, most notably ocean and coastal engineering. Additionally, our findings provide an important first step in comprehending the structure and physical behavior of complex structures. We hope that our findings will contribute significantly to our understanding of ocean waves. This work, we believe, is timely and will be of interest to a broad range of professionals involved in ocean engineering models. The obtained results are useful in grasping the fundamental scenarios of nonlinear sciences. With the help of Mathematica, the obtained results are verified by inserting them into the governing equation.