Hirota Bilinear Operator Research Articles

The bilinear neural network method for solving Benney–Luke equation

Benney–Luke equation, the estimation of water wave propagation on the water’s surface, is significantly important in studying the tension of water waves in physics. This paper focuses on the Bilinear Neural Network method (BNNM) to find the solutions to the Benney–Luke equation. Using the Hirota bilinear operator, the Benney–Luke equation is transformed into a bilinear expression and establishes models of neurons. The method construction is straightforward, well-organized, and effective for finding various exact analytic solutions of the Benny–Luke equation consisting of one-soliton solutions, two-soliton solutions, one-soliton hyperbolic wave solutions, two-soliton hyperbolic wave solutions, mixed-soliton hyperbolic solutions, and mixed-multi hyperbolic solutions by choosing the appropriate test functions. The solutions are rigorously depicted in three dimensions to observe the mutable behavior.

On soliton solutions, periodic wave solutions, asymptotic analysis and interaction phenomena of the (3+1)-dimensional JM equation

In this paper, the M-lump solutions, the periodic type, and cross-kink wave solutions are acquired. Here, the Hirota bilinear operator is employed. By utilizing the symbolic computation and employing the utilized method, the (3+1)-dimensional Jimbo–Miwa (JM) equation is investigated. Based on the Hirota bilinear form, the soliton solution and periodic wave solution to the mentioned equation, respectively, are obtained. We gained plenty of multiple collisions of lumps. Next, the periodic wave and cross-kink wave have greatly enriched the existing literature on the JM equation. Through the three-dimensional designs, contour design, density design, and two-dimensional design by using Maple, the physical features of these soliton solutions are explained all right. The forms of the attained solutions are one-lump, two-lumps, and three-lumps wave solutions. Then, a class of rogue waves-type solutions to the (3+1)-dimensional JM equation within the frame of the bilinear equation is found. These results can help us better understand interesting physical phenomena and mechanisms.

Lump and lump-kink-type rogue-wave solutions of the homologous (3+1)-dimensional Hirota-bilinear-like equation

In this article, a new dynamical system equation is constructed, named the (3+1)-dimensional Hirota-bilinear-like equation. The new ‘like’ equation has more nonlinear terms than the original equation while they have the same bilinear form. The generalized Hirota bilinear operator and its logarithmic transformation are applied to obtain the new ‘like’ equation and to find its solutions. Some different types of symbolic solutions are obtained, which are the lump solutions and the lump-kink-type rogue wave solutions. The propagation images of the lump waves and the interaction which is between the lump-type waves and two-kink soliton-waves are simulated to analysis the shapes and movements of these solutions.

On soliton solutions, periodic wave solutions and asymptotic analysis to the nonlinear evolution equations in (2+1) and (3+1) dimensions

In this paper, the (2+1)-dimensional generalized fifth-order KdV equation and the extended (3+1)-dimensional Jimbo-Miwa equation were transformed into the Hirota bilinear forms with Hirota direct method. In this process, the Hirota bilinear operator played a significant role. Based on the Hirota bilinear forms, the single soliton solutions and the single periodic wave solutions of these two types of equations were obtained respectively. Meanwhile, the figures of the single soliton solutions and the single periodic wave solutions were plotted. Furthermore, the results shed light on that when the amplitude of water wave approaches 0, the single periodic wave solutions tend to the single soliton solutions. The conclusion has been generalized from (2+1)-dimensional equations to (3+1)-dimensional equations.

Multi-soliton solutions and long-time asymptotic behavior of the modified Korteweg–de Vries equations

In this paper, two modified KdV (MKdV) equations, one is real and the other is complex, are investigated. By applying the Hirota bilinear operator theory and computer algebra, the corresponding bilinear forms of these two MKdV equations are successfully derived, and then their multi-soliton solutions are obtained and expressed in explicit forms. In order to study the dynamic properties of the obtained multi-soliton solutions, we further analyze their long-time asymptotic behaviors. It follows that the collisions between two solitons are elastic which are graphically shown.

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.

Formula omitted]-soliton solutions and dynamic property analysis of a generalized three-component Hirota–Satsuma coupled KdV equation

In this paper, a generalized three-component Hirota–Satsuma coupled KdV equation describing the interactions of two long waves with different dispersion relations, is investigated. Applying Hirota bilinear operator theory, the bilinear form of the proposed model is first obtained, and then its N-soliton solutions are given in explicit forms. Finally, the analysis of the dynamic property shows that the collisions between two solitons are elastic.

Multi-Waves, Breathers, Periodic and Cross-Kink Solutions to the (2+1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The present article deals with multi-waves and breathers solution of the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, solitary wave solutions, periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the multi-waves and breathers, all cases the periodic and cross-kink solutions can be captured from the 1- and 2-soliton. The obtained solutions are extended with numerical simulation to analyze graphically, which results in 1- and 2-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. We have shown that the assigned method is further general, efficient, straightforward, and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

One-, two- and three-soliton, periodic and cross-kink solutions to the (2 + 1)-D variable-coefficient KP equation

This paper deals with [Formula: see text]-soliton solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation by virtue of the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic and cross-kink solutions, which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the [Formula: see text]-soliton solutions, all cases of the periodic and cross-kink solutions can be captured from the one-, two- and three-soliton solutions. The obtained solutions are extended with numerical simulation to analyze graphically, which results into one-, two- and three-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Breather wave, periodic, and cross‐kink solutions to the generalized Bogoyavlensky‐Konopelchenko equation

The present article deals with multi‐waves and breather wave solutions of the generalized Bogoyavlensky‐Konopelchenko equation by virtue of the Hirota bilinear operator method and the semi‐inverse variational principle. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic, and cross‐kink solutions in which have been investigated by the approach of the bilinear method. With certain parameter constraints in the multi‐waves and breather, all cases of the periodic and cross‐kink solutions can be captured from the one and two soliton(s). The obtained solutions are extended with numerical simulation to analyze graphically, which results into 1‐ and 2‐soliton solutions and also periodic and cross‐kink solutions profiles, that will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.

Edge States and Chiral Solitons in Topological Hall and Chern–Simons Fields

The multi-component extension problem of the (2+1) D -gauge topological Jackiw–Pi model describing the nonlinear quantum dynamics of charged particles in multi-layer Hall systems is considered. By applying the dimensional reduction (2 + 1) D → (1 + 1) D to Lagrangians with the Chern–Simons topologic fields , multi-component nonlinear Schrodinger equations for particles are constructed with allowance for their interaction. With Hirota‘s method, an exact two-soliton solution is obtained, which is of interest in quantum information transmission systems due to the stability of their propagation. An asymptotic analysis t →±∞ of soliton-soliton interactions shows that there is no backscattering processes. We identify these solutions with the edge (topological protected) states – chiral solitons – in the multi-layer quantum Hall systems. By applying the Hirota bilinear operator algebra and a current theorem, it is shown that, in contrast to the usual vector solitons, the dynamics of new solutions (chiral vector solitons) has exclusively unidirectional motion. The article is published in the author’s wording.

Kink Multi-soliton Solutions for the (2+1)-Dimensional Bogoyavlenskii-Schiff Equation

In this work, (2+1)-dimensional Bogoyavlenskii-Schiff equation, which cann't be converted to a complete form of Hirota's bilinear operator, is considered. By using modification of extended homoclinic test approach new kink breather soliton and kink multi-solitons solutions are obtained and exhibited, respectively.

Exact Solutions for Domain Walls in Coupled Complex Ginzburg–Landau Equations

The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background. We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.

Exact solitary- and periodic-wave modes in coupled equations with saturable nonlinearity

We demonstrate that the known method, based on the Hirota bilinear operator, generates classes of exact solutions to a system of coupled nonlinear Schrödinger (CNLS) equations with nonpolynomial nonlinearity, either rational or algebraic (the latter involves a square root). The choice of the CNLS equations is suggested by known models for photorefractive media and Bose–Einstein condensation. The solutions are, generally, periodic, and they form families expressed in terms of the Jacobi elliptic functions, the elliptic modulus k being a free parameter of the family. In the limit case corresponding to the infinite period ( k = 1 ), the solutions amount to solitons. In some cases, the exact solutions may feature patterns with two peaks per period. Exact solutions are also found in a single NLS equation with a rational saturable nonlinearity and periodic potential in the form of squared Jacobi sine.

Doubly periodic and multiple pole solutions of the sinh-Poisson equation: Application of reciprocal transformations in subsonic gas dynamics

Vortex patterns associated with the sinh-Poisson equation arise in a remarkable manner as relaxation states of the Navier–Stokes equations. Here, doubly periodic and multiple-pole solutions of the sinh-Poisson equation are generated via the Hirota bilinear operator formalism and exploitation of the phenomenon of coalescence of wave numbers. It is then shown how the multi-parameter reciprocal transformations of gas dynamics may be applied to a seed doubly periodic solution of the sinh-Poisson equation to generate associated periodic vortex structures valid in the subsonic flow of a generalized Kármán–Tsien gas.

Coalescence of Wavenumbers and Exact Solutions for a System of Coupled Nonlinear Schrödinger Equations

An exact 2-soliton expression is obtained for the Manakov system, a special, coupled set of nonlinear Schrödinger equations. The solution permits different asymptotic states for the components in the far field. A `coalescence' of wavenumbers is considered from the perspective of the Hirota bilinear operator. This is roughly equivalent to a double (or in general multiple) pole solution in the language of the inverse scattering transform. Physically counterpropagating waves will occur. With the help of computer algebra software a 3-soliton solution is derived. Coalescence of eigenvalues is investigated. Temporal modulation of the amplitude is observed.

`Solitoff' Solutions of Nonlinear Evolution Equations

Dromions are exact, localized solutions of (2+1) dimensional evolution equations and decay exponentially in all directions. `Solitoffs' of the Davey-Stewartson equations constitute an intermediate state between dromions and plane solitons, since they decay exponentially in all directions except a preferred one. Here solitoffs are rederived by the Hirota bilinear operator and extended to a variety of nonlinear evolution equations.

Exact and Approximate Analytical Solutions of Stationary Vortexlike Modes in thed-Dimensional AnisotropicClassical O(2) (XY) Spin Model

Analytical solutions to nonlinear difference equations describing stationary states of the d -dimensional classical O(2) (XY) spin model are studied. The Hirota bilinear operator formalism is employed to obtain exact and approximate stationary vortexlike-mode solutions having a form somewhat similar to a discrete version of vortex solutions to the two-dimensional sine-Gordon equation. The former exists for a specific case, while the existence condition for the latter is much less restrictive. The obtained analytical result is demonstrated in numerical calculations of vortex profiles for a two-dimensional model O(2) spin system, in which a single vortex and a periodic array of vortex-antivortex pairs are shown.

A reduce program for Hirota's bilinear operator and Wronskian operations

A REDUCE program for Hirota's bilinear operator and the Wronskian form of Freeman and Nimmo is presented. The program is useful for solving nonlinear evolution equations.

Solitons and wavetrains: unified approach

The direct employment of the T-function defined as a limiting transformation of the Riemann theta function leading to a system of algebraic dispersion equations, yields solutions of some nonlinear partial differential equations in the form of solitons on a background of quasi-periodic waves. Pure solitons and pure quasi-periodic solutions appear as particular cases. The method is illustrated by the example of the KdV equation and is also compared with the Hirota bilinear operator technique.