Multi-soliton Wave Research Articles

Integrability and multiple-rogue and multi-soliton wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito equation

Integrability and multiple-rogue and multi-soliton wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito equation

Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.

Dynamical behaviors of various multi-solutions to the (2+1)-dimensional Ito equation

We explored the multi-solutions of the (2+1)-dimensional Ito equation based on the superposition formula. Firstly, via the superposition of exponential functions, we derived the multi-soliton wave solutions. Secondly, a new type of mixed solutions between multi-arbitrary functions and multi-kink solitons is introduced. Finally, we constructed the multi-localized wave solutions by utilizing the superposition of N-even power functions. Besides, we obtained novel interaction solutions based on the superposition formula of multi-localized wave solutions and multi-arbitrary functions. We illustrated the dynamic analysis and properties of these solutions using 3D, contour and density plots. These findings are essential for showing the propagation behavior of nonlinear waves and for explaining specific physical phenomena. They contribute significantly to the study of nonlinear waves.

Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.

Abundant solitons for the generalized Hirota–Satsuma couple KdV system with an efficient technique

The generalized Hirota–Satsuma Coupled Korteweg–de Varies (GHSC-KdV) system is considered for investigation to study the exact solutions by using the extended generalized G′/G-expansion technique. The GHSC–KdV system describes the interaction of two long waves with different dispersion relations. The dynamical structures of the obtained solutions are solitary-type waves, multi-soliton waves, and singular-kink type waves. It contains information about the hyperbolic, trigonometric, and rational function solutions for the above system. Furthermore, the derived solutions of the system that comprises arbitrary functional parameters and other constant parameters can be used to enhance the advanced behavior of physical situations.

Exact solutions for the Bogoyavlensky-Konopelchenko equation with variable coefficients with an efficient technique

In this investigation, a different form of extended generalized G′G2-expansion technique with variable coefficient is proposed. The advantage of the extended generalized G′G2–expansion technique is that it can be used to solve nonlinear evolution equations with both constant and variable coefficients, whereas the basic G′G2 method can only be used with constant coefficients. The proposed technique is used to solve the Bogoyavlensky–Konopelchenko (BK) equation with variable coefficients, which depicts the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Further, the BK equation with variable coefficients is applied for stratified internal waves, shallow-water waves, ion-acoustic waves, and water propagation in a liquid. The hyperbolic, trigonometric, and rational form solutions of the BK equation are dynamically represented as the annihilation of three-dimensional kink waves, multi-soliton waves, single solitons, and so on. Furthermore, the derived solutions of the considered equation, which comprise arbitrary functional parameters and other constant parameters, can be used to enhance the advanced behaviours of physical situations.

A study of nonlinear extended Zakharov–Kuznetsov dynamical equation in (3+1)-dimensions: Abundant closed-form solutions and various dynamical shapes of solitons

In this work, we execute the generalized exponential rational function (GERF) method to construct numerous and a large number of exact analytical solitary wave solutions of the nonlinear extended Zakharov–Kuznetsov (EZK) dynamical equation in (3+1)-dimensions. The implemented method is one of the best, most reliable, and efficient techniques in the present time for determining numerous closed-form wave analytic solutions to NPDEs. We have accomplished a variety of solitary wave solutions related to some arbitrary parameters under various family cases. These solutions take the following forms based on the free parameters chosen: exponential functions form, trigonometric functions form, and hyperbolic functions form. The obtained solutions are dissimilar and entirely new from the previous findings available in the literature. The dynamics of obtained solutions, namely, soltion, singular soliton wave, a periodic wave, bell-shape, anti-bell-shape wave, breather wave, and multisoliton wave solutions by the special-choice of parameters, are shown graphically in 3D, 2D, and corresponding density profiles. The results demonstrate that the employed computational strategy is efficient, direct, concise, and can be executed in various complex phenomena with symbolic computations. Furthermore, it is revealed that the generalized exponential rational function technique can be effectively utilized for several other NPDEs in engineering, sciences, and mathematical physics.

Fast magnetosonic solitonic structures in a quasi-neutral collision-free finite-beta plasma

The equations of the model of two fluids of a two component collision-free, isotropic and fully ionized plasma in thermal equilibrium consisting of singly-ionized ions and electrons in a uniform magnetic field with heated ions and electrons are treated. We focus our attention on determining the global hydromagnetic fast magnetosonic traveling solitonic waves for plane wave equations. Ordinary solitary waves, bifurcating from the zero wave number, exist for the angles of inclination θ of the magnetic field vector to the direction x of the wave propagation, located in some left neighborhood of π/2. A global family of solitary waves containing solitary waves of all possible amplitudes is numerically determined. The dynamic stability of these solitary waves is investigated by numerical methods: the problem of the evolution of solitary waves via the time dependent field equations of the quasi-neutral collision-free finite beta-plasma is analyzed. We also concern the domains in parameter space where traveling generalized and envelope solitary waves exist and compute the possible forms of these waves and of so-called multi-soliton waves with several humps. The evolution of a localized perturbation is also treated numerically.

An efficient technique of [formula omitted]–expansion method for modified KdV and Burgers equations with variable coefficients

The extended generalized G′G–expansion is well defined and an efficient technique, which is used to obtain the exact traveling wave solutions to the governing nonlinear equations with constant coefficients as well as variable coefficients. In this paper, the modified Korteweg–de Vries (mKdV) equation and Burgers equation with variable coefficients are investigated through the extended generalized G′G–expansion method, which are exceptional cases of the nonlinear evolution equations widely used in a two-layer fluid system, in fluid-filled elastic tubes, in an atmospheric and oceanic dynamical system, traffic flow, turbulence in fluid dynamics, dusty plasma, ion-acoustic waves in a plasma system. New families of exact closed-form solutions are obtained in hyperbolic, trigonometric, and rational function solutions with the available free constants. With the help of computerized symbolic computation work, the newly formed closed-form solutions are validated by back substituting them into the equations using the computational mathematical software. Furthermore, the graphical representations of all these obtained solutions are discussed and demonstrated by giving the suitable best values of arbitrary functions and constants via three-dimensional surface and density plots. The dynamics of the solution profiles demonstrate the annihilation of 3D kink-type soliton waves, shock waves, double solitons, and multi-soliton wave structures.

A study of resonance Y-type multi-soliton solutions and soliton molecules for new (2+1)-dimensional nonlinear wave equations

<abstract> <p>In this study, a fourth-order nonlinear wave equation with variable coefficients was investigated. Through appropriate choice of the free parameters and using the simplified linear superposition principle (LSP) and velocity resonance (VR), the examined equation can be considered as Hirota–Satsuma–Ito, Calogero–Bogoyavlenskii–Schiff and Jimbo–Miwa equations. The main objective of this study was to obtain novel resonant multi-soliton solutions and investigate inelastic interactions of traveling waves for the above-mentioned equation. Novel resonant multi-soliton solutions along with their essential conditions were obtained by using simplified LSP, and the conditions guaranteed the existence of resonant solitons. Furthermore, the obtained solutions were used to investigate the dynamic and fission behavior of Y-type multi-soliton waves. For an accurate investigation of physical phenomena, appropriate free parameters were chosen to ascertain the impact on the speed of traveling waves and the initiation time of fission. Three-dimensional and contour plots of the obtained solutions are presented in <xref ref-type="fig" rid="Figure1">Figures 1</xref>–<xref ref-type="fig" rid="Figure6">6</xref>. Additionally, two nonlinear equations were formulated and investigated using VR, and the related soliton molecules were simultaneously extracted. The reported resonant Y-type multi-soliton waves and equations are new and have not been previously investigated. They can be used to explain modeled physical phenomena and can provide information about dynamic behavior of shallow water waves.</p> </abstract>

A study on the resonant multi-soliton waves and the soliton molecule of the (3+1)-dimensional Kudryashov–Sinelshchikov equation

In this paper, a reduced (3+1)-dimensional nonlinear wave equation describing the dynamic of liquid with gas bubble, namely Kudryashov-Sinelshchikov (KS) equation is carefully investigated. After applying the simplified linear superposition principle (LSP) and the velocity resonance (VR) the fact that there has no existence of soliton molecule to KS equation is formally proofed. Simultaneously, two new resonant multi-soliton solutions constructed with distinct physical structures are also generated. Comparing with the existing ones our solutions are much more general. Most of all, their formula forms reveal the influences of the arbitrary coefficients of the bubble-liquid-dispersion, the y-transverse-perturbation and the z-transverse-perturbation to the traveling soliton waves. Finally, the propagations of inelastic interactions of resonant two-soliton waves are shown in Figures 1-6. Particularly, by specifying special values to the resonant multi-soliton solution gives three-bound-soliton waves moving with the same speed and direction which is in the manner similar to the property of soliton molecule, as shown in Figure 6. The extracted results not only show the efficiency of the simplified LSP but also have potential applications to the physical phenomena of soliton molecules and the resonance of multi-soliton waves .

Evolutionary behavior and novel collision of various wave solutions to (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili equation

In the present paper, taking advantage of the bilinear method, we attained N-soliton solutions of the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH-KP) equation. By combining the long wave limit method and the complex conjugate method to N-soliton solutions, M-lump wave solutions including one-lump wave, two-lump wave and three-lump wave and two types of hybrid solutions including hybrid solution between lump wave and solitons and between M-lump wave and soliton were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physically interpreted via symbolic computation. Finally, resonant multi-soliton wave solutions were obtained by carrying out the linear superposition principle and the progresses of wave motions with physical interpretation represented graphically. Furthermore, the received results have immensely augmented the exact solutions of (3+1)-dimensional gCH-KP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. In addition, the proposed methods will be a strong boost to the calculation method of nonlinear equations.

Novel solitary and resonant multi-soliton solutions to the (3 + 1)-dimensional potential-YTSF equation

In this study, the (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation arising from the (3 + 1)-dimensional Kadomtsev–Petviashvili equation is investigated in detail by using two powerful approaches. First, the generalized resonant multi-soliton solution is generated via the simplified linear superposition principle. Second, after applying the simplest equation method, the generalized single solitary solution is extracted. The results show that the obtained solutions are perfect. The physical explanation of the obtained solutions is depicted in various 3D and 2D figures, which are used to illustrate that the interactions of resonant multi-soliton waves are inelastic. Ultimately, the study reveals that the inelastic interactions can be determined by the sign of the wave related number [Formula: see text].

Rational solutions, and the interaction solutions to the (2 + 1)-dimensional time-dependent Date–Jimbo–Kashiwara–Miwa equation

In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in -dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is studied. We construct rational solutions and the interaction physical phenomena between rational solution and multi-soliton wave to the time-dependent DJKM equation. The multi-soliton solutions are revealed by using the Hirota simple method, and then via the long-wave method, the M-lump solutions for two types of time-dependent are derived. Moreover, the interaction phenomena between rational solution and single-, double-soliton solutions are studied.

Novel resonant multi-soliton solutions and inelastic interactions to the (3 + 1)- and (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equations via the simplified linear superposition principle

In this paper, the resonant solitary waves to the (3 + 1)- and (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equations are exposed via the simplified linear superposition principle (LSP). Based on the related Hirota’s bilinear equations, the resonant multi-soliton solutions and the corresponding wave numbers are formally generated. Abundant inelastic interactions, i.e., fission and fusion of traveling N-solitary waves, are presented analytically and graphically. Besides, we show that the velocity resonance (VR) is not applicable to handle the (3 + 1)- and (4 + 1)-dimensional BLMP equations, due to that the dispersion relation is constructed as $$\omega = - k^{3}$$ . The results show that the simplified LSP provides enough freedom to construct resonant multi-soliton waves that has potential applications to a large variety of real physical phenomena.

Homoclinic breather waves, rouge waves and multi-soliton waves for a (2+1)-dimensional Mel’nikov equation

PurposeThe purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets.Design/methodology/approachThe author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method.FindingsThe results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures.Research limitations/implicationsThis paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations.Originality/valueThe outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

Anomalous errors of direct scattering transform.

Theory of direct scattering transform for nonlinear wave fields containing solitons is revisited to overcome fundamental difficulties hindering its stable numerical implementation. With the focusing one-dimensional nonlinear Schrödinger equation serving as a model, we study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases phase and space position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense "uncatchable." Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analysed using high-precision arithmetics even in the presence of noise opening broad perspectives in nonlinear physics.

Resonant multi-soliton solutions to two fifth-order KdV equations via the simplified linear superposition principle

In this paper, the simplified linear superposition principle is presented and employed to handle two versions of the fifth-order KdV equations, called the (2[Formula: see text]+[Formula: see text]1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation and the (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation, respectively. Two general forms of resonant multi-soliton solutions are formally obtained. The paper proceeds step-by-step with increasing detail about the derivation process. Firstly, illustrate the algorithms of the linear superposition principle which paves the way for solving the wave related numbers. Then, demonstrate its application that exposes the proposed approach provides enough freedom to construct resonant multi-soliton wave solutions. Finally, some graphical representations of obtained solutions are portrayed by taking some definite values to free parameters, which describe various versions of inelastic interactions of resonant multi-soliton waves. The associated propagations may be related to large variety of real physical phenomena.

Resonant multi-soliton solutions to new (3+1)-dimensional Jimbo–Miwa equations by applying the linear superposition principle

In this paper, the linear superposition principle and Hirota bilinear equations are simultaneously employed to handle two new (3+1)-dimensional Jimbo–Miwa equations. The corresponding resonant multi-soliton solutions and the related wave numbers are formally established, which are totally different from the previously reported ones. Moreover, the extracted N-soliton waves and dispersion relations have distinct physical structures compared to solutions obtained by Wazwaz. Finally, five graphical representations are portrayed by taking definite values to free parameters which demonstrates various versions of traveling solitary waves. The results show the proposed approach provides enough freedom to construct multi-soliton waves that may be related to a large variety of real physical phenomena and, moreover, enriches the solution structure.

A novel method for finding new multi-soliton wave solutions of the completely integrable equations

This paper presents a novel simplest equation method (SEM) to construct multi-soliton wave solutions. The paper proceeds step-by-step with increasing detail about derivation process, first illustrating the algorithm of the SEM and then exploiting an even deeper connection between Burgers’ equation and multi-soliton solutions and the simplest equation. Comparing with the Hirota’s method, our method provides the possibility for constructing multi-soliton solutions of the completely integrable equations with variable coefficients concisely. Applying such a method to the Gardner equation and the Whitham-Broer-Kaup (WBK) shallow water model, new general forms of multi-soliton solutions are obtained.