Hirota Satsuma Research Articles

Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Framework of extended BKP equations possessing N‐wave solutions in (2+1)‐dimensions

A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)‐dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)‐dimensional generalized Hirota–Satsuma–Ito (HSI) equation, the special fourth‐order (2+1)‐dimensional nonlinear equation, and the fifth‐order Korteweg–de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions.

Construction of analytical solution for Hirota–Satsuma coupled KdV equation according to time via new approach: Residual power series

In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.

New interaction of high-order breather solutions, lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

New interaction of high-order breather solutions, lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

A (2+1)-dimensional generalized Hirota–Satsuma–Ito equations: Lie symmetry analysis, invariant solutions and dynamics of soliton solutions

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov’s theorem.

Investigation of Hirota equation: Modified double Laplace decomposition method

In this article, the non-linear coupled Hirota equations are considered attributed to localized excitations. By employing a modified double Laplace transform decomposition method, we derived the general approximate solutions for the coupled Hirota and coupled Hirota Satsuma equations. It is observed that the proposed method is a systematic and powerful scheme for solving such nonlinear systems. The main advantage of the method is to calculate solutions in a series form without any linearization, discretization, or perturbation. Using convergence analysis, we observed that the proposed method is converging to the exact solution of the system. Numerical analysis for these solutions shows the confine carrier wave and ensures localized wave packets. A degree enhancement reduction in the dispersion nonlinearly coefficients alters both the spatial width and amplitude of the waves. At vanishingly small dispersion, internal oscillations in the pulse-shaped solitary waves reduced the wave breaking effect. We have noticed that the numerical solutions converge to the exact solutions, confirming the importance of the proposed technique.

Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation

This research paper targets the fractional Hirota’s analytical solutions–Satsuma (HS) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

STUDY OF NONLINEAR HIROTA–SATSUMA COUPLED KdV AND COUPLED mKdV SYSTEM WITH TIME FRACTIONAL DERIVATIVE

This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the solution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are evaluated in an iterative process with the help of homotopy perturbation method manipulating He’s polynomials. The fractional derivatives are considered in the Caputo sense. The obtained results confirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in the form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with the exact solution are presented.

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Formula omitted]-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation

Within the Hirota bilinear formulation, we construct N-soliton solutions and analyze the Hirota N-soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N-soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation, as specific examples.

Localized interaction solution and its dynamics of the extended Hirota–Satsuma–Ito equation

In this research, we will introduce and study the localized interaction solutions and their dynamics of the extended Hirota–Satsuma–Ito equation (HSIe), which plays a key role in studying certain complex physical phenomena. By using the Hirota bilinear method, the lump-type solutions will be firstly constructed, which are almost rationally localized in all spatial directions. Then, three kinds of localized interaction solutions will be obtained, respectively. In order to study the dynamic behaviors, numerical simulations are performed. Two interesting physical phenomena are found: one is the fission and fusion phenomena happening during the procedure of their collisions; the other is the rogue wave phenomena triggered by the interaction between a lump-type wave and a soliton wave.

Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics

Abstract This research paper uses a direct algebraic computational scheme to construct the Jacobi elliptic solutions based on the conformal fractional derivatives for nonlinear partial fractional differential equations (NPFDEs). Three vital models in mathematical physics [the space-time fractional coupled Hirota Satsuma KdV equations, the space-time fractional symmetric regularized long wave (SRLW equation), and the space-time fractional coupled Sakharov–Kuznetsov (S–K) equations] are investigated through the direct algebraic method for more explanation of their novel characterizes. This approach is an easy and powerful way to find elliptical Jacobi solutions to NPFDEs. The hyperbolic function solutions and trigonometric functions where the modulus and, respectively, are degenerated by Jacobi elliptic solutions. In this style, we get many different kinds of traveling wave solutions such as rational wave traveling solutions, periodic, soliton solutions, and Jacobi elliptic solutions to nonlinear evolution equations in mathematical physics. With the suggested method, we were fit to find much explicit wave solutions of nonlinear integral differential equations next converting them into a differential equation. We do the 3D and 2D figures to define the kinds of outcome solutions. This style is moving, reliable, powerful, and easy for solving more difficult nonlinear physics mathematically.

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.

Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Abstract In this article, a generalized Hirota–Satsuma coupled Korteweg–de Vries (KdV) system is investigated from the group standpoint. This system represents an interplay of long waves with distinct dispersion correlations. Using Lie’s theory several symmetry reductions are performed and the system is reduced to systems of non-linear ordinary differential equations (NLODEs). Subsequently, the simplest equation method is invoked to find exact solutions of the NLODE systems, which then provides the solitary wave solutions for the system under discussion. Finally, we construct conservation laws of generalized Hirota–Satsuma coupled KdV system with the aid of general multiplier approach.

Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

Numerical calculation of N-periodic wave solutions to coupled KdV-Toda-type equations.

In this paper, we study the N-periodic wave solutions of coupled Korteweg-de Vries (KdV)-Toda-type equations. We present a numerical process to calculate the N-periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N-periodic wave solutions to the coupled Ramani equation, the Hirota-Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak-Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

Alternative solitons in the Hirota–Satsuma system via the direct method

Multisolitons and multi-hump solitons are constructed for the coupled Korteweg–de Vries (cKdV) system introduced by Hirota and Satsuma (1981) in the context of the propagation of two waves interacting with different dispersion relations. The direct method analytically exhibit a new class of complementary or alternative soliton solutions functionally different to the traditional or classical multisolitons known (Hirota and Satsuma, 1981; Ramani et al., 1983; Parra Prado and Cisneros-Ake, 2020). As in the classical case, our alternative multisolitons are found to obey elastic interactions of the N-KdV and M-cKdV solitons. To show this, we inductively construct tables to describe in detail the mechanism of the different wave interactions and the coefficients involved in them. As a particular scenario, it is found that each of the 1-cKdV and 1-KdV-1cKdV interaction cases may produce two independent criteria for the appearance of one-soliton solutions consisting of two humps in shape. A proper combination of these two conditions gives place to multi-hump solitons. We explicitly show the two-one, two-two, three-one and three-two humps for the two-component wave solutions. • The direct method due to Hirota allows to find complementary two-component multisolitons in a coupled KdV system. • Symmetric one-soliton solutions with two humps in shape are obtained from two independent criteria. • Multi-hump alternative solitons are constructed from two different two-hump conditions.

Extension of optimal homotopy asymptotic method with use of Daftardar–Jeffery polynomials to Hirota–Satsuma coupled system of Korteweg–de Vries equations

Abstract In this study, the Daftardar–Jeffery polynomials are incorporated in the homotopy of the optimal homotopy asymptotic method (OHAM) for solving the generalized Hirota–Satsuma coupled system of Korteweg–de Vries equations. The results are displayed through graphs and tables. The results obtained in this study are also compared with the published work on OHAM, which shows that OHAM-DJ is more explicit, reliable, and an efficient analytical technique. The exactness of the developed method can be improved by performing the higher iterations.

N-soliton solutions and the Hirota conditions in (2+1)-dimensions

We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations.