Three-soliton Solutions Research Articles

Wronskian solutions and Pfaffianization for a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma

In this paper, we investigate a (3 + 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (GVCKP) equation in a fluid or plasma. The Nth-order Wronskian solutions for that equation are derived and proved under certain variable-coefficient constraints, where N is a positive integer. One-, two-, and three-soliton solutions in the Wronskian for that equation are given. By means of the Pfaffianization procedure, a coupled (3 + 1)-dimensional GVCKP system is constructed from that equation. Bilinear form for that coupled system is exported. Under certain variable-coefficient constraints, those Wronski-type and Gramm-type Pfaffian solutions for that coupled system are obtained and proved with the help of the Pfaffian identities.

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.

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

This article investigates the Hirota–Satsuma–Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method. The equation is proved to be Painlevé integrable by Painlevé analysis. On the basis of the bilinear form, the forms of two-soliton solutions, three-soliton solutions, and four-soliton solutions are studied specifically. The appropriate parameter values are chosen and the corresponding figures are presented. The breather waves solutions, lump solutions, periodic solutions and the interaction of breather waves solutions and soliton solutions, etc. are given. In addition, we also analyze the different effects of the parameters on the figures. The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions. These are important for describing water waves in nature.

Complex Kraenkel-Manna-Merle system in a ferrite: N-fold Darboux transformation, generalized Darboux transformation and solitons

Ferromagnetic materials such as the ferrites are used in the electronic and energy industries. Here, we concentrate on a complex Kraenkel-Manna-Merle system in a ferrite, under some coefficient constraints. An N-fold Darboux transformation of that system is presented via an existing Lax pair, where N is a positive integer. An N-fold generalized Darboux transformation, which admits one spectral parameter, is proposed through a limit procedure. One-, two- and three-soliton solutions of that system are determined via that N-fold Darboux transformation. The second-order and third-order degenerate soliton solutions of that system are derived via that N-fold generalized Darboux transformation. Those solitons are graphically represented for the magnetization and external magnetic field related to a ferrite.

All-optical switches based on three-soliton inelastic interaction and its application in optical communication systems

In high-speed optical communication systems, in order to improve the communication rate, the distance between pulses must be compressed, which will cause the problem of the interaction between optical pulses in optical communication systems, which has been widely concerned by researches. In this paper, the bilinear method will be used to analyze the coupled high-order nonlinear Schrödinger equations and obtain their three-soliton solutions. Then, the influence of the relevant parameters in the three-soliton solution on the soliton inelastic interaction is studied. In addition, the constraint conditions of each parameter in the three-soliton solution are analyzed, the inelastic interaction properties of optical solitons under different parameter conditions are obtained, and the relevant laws of the inelastic interaction of solitons are studied. The results will have potential applications in the soliton control, all-optical switching and optical computing.

Soliton solutions and lump-type solutions to the (2+1)-dimensional Kadomtsev-Petviashvili equation with variable coefficient

In this article, the integrability, bilinear form, N-soliton solutions, lump solutions and rational function solutions of (2+1)-dimensional Kadomtsev-Petviashvili equation with variable coefficient (vcKP) are investigated. This has important implications for mathematicians and physicists to solve problems related to long water waves. The equation is proved to be Painlevé integrable by Painlevé analysis. With the help of Hirota bilinear method, the bilinear form of vcKP is obtained. Based on the bilinear form, the forms of four different types of solutions are constructed. In addition, taking the long wave limit method on two-soliton solutions, three-soliton solutions and four-soliton solutions to construct the lump solutions and the rational function solutions. As a result, some new solutions of vcKP are obtained, such as soliton solutions, lump solutions and interaction of different kinds of solutions. To describe better the change in solution with time, the dynamical behaviours of the exact solutions to the (2+1)-dimensional vcKP are analysed. The influence of parameters on the soliton solutions is analyzed. Moreover the evolution of several types of solutions is investigated depending on the change in time.

Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrödinger equations.

In this paper, using the algorithm due to Ablowitz et al. [Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)], we explore the anomalous dispersive relations, inverse scattering transform, and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrödinger (fHONLS) equations, containing the fractional third-order NLS (fTONLS), fractional complex mKdV (fcmKdV), and fractional fourth-order nonlinear Schrödinger (fFONLS) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann-Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N-soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two-, and three-soliton solutions with anomalous dispersions of fTONLS and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional one-soliton solutions are related to the power laws of their amplitudes. Moreover, we also deduce the formula for the fractional N-soliton solutions of all fHONLS equations and analyze some velocities of the one-soliton solution. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.

On multi-soliton solutions to a generalized inhomogeneous nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain

A generalized inhomogeneous higher-order nonlinear Schrödinger (GIHNLS) equation for the Heisenberg ferromagnetic spin chain system in (1+1)-dimensions under zero boundary condition at infinity is taken into account. The spectral analysis is first performed to generate a related matrix Riemann–Hilbert problem on the real axis. Then, through solving the resulting matrix Riemann–Hilbert problem by taking the jump matrix to be the identity matrix, the general bright multi-soliton solutions to the GIHNLS equation are attained. Furthermore, the one-, two-, and three-soliton solutions are written out and analyzed by figures.

Observation of resonant solitons and associated integrable properties for nonlinear waves

This paper is concerned with the integrability of a (2+1)-dimensional nonlinear evolution equation. The Painlevé analysis proves that this equation possesses the Painlevé property. Other integrable properties, including the bilinear Bäcklund transformation, Bell-polynomial-typed Bäcklund transformation, Lax pairs and infinite conservation laws, are derived directly by virtue of the Hirota bilinear method and Bell polynomials. The general form of the resonant soliton solutions are constructed based on the linear superposition principle. The resonant two-soliton solutions consist of three waves, each of which is one-soliton profile. For the resonant three-soliton solutions, the resonance of waves may cause some waves to disappear or appear. We hope that the various resonant phenomena discussed here will be helpful to understand the propagation of nonlinear waves.

Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber

In this Letter, we investigate an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Via the truncated Laurent expansions, auto-Bäcklund transformations are obtained. According to the Ablowitz-Kaup-Newell-Segur procedure, we derive a Lax pair under certain optical-fiber coefficient constraints. Via the Hirota method, we obtain some bilinear forms, bright two-soliton and bright three-soliton solutions under certain optical-fiber coefficient constraints.

The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution

It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.

Numerical Experiments on Extreme Waves Through Oblique–Soliton Interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

Kraenkel-Manna-Merle saturated ferromagnetic system: Darboux transformation and loop-like soliton excitations

In this article, we reinvestigate the Kraenkel-Manna-Merle (KMM) system, which can describe the nonlinear ultra-short wave pulse motions in certain saturated ferromagnetic materials. Starting from the Lax pair of the system, we construct the Darboux transformation of this system. Then, new one-, two- and three-soliton solutions are obtained where free functions with respect to the time variable is involved. A series of novel loop-like soliton structures are excited by choosing the free functions as certain particular functions. The soliton excitation dynamics is also discussed in detailed.

New multi-wave solutions of the conformable LPD model with nonlinear wave phenomena arise in mathematical physics

In this paper, the conformable Lakshmanan–Porsezian–Daniel equation with parabolic law nonlinearity is solved and the newly acquired optical solutions are analysed. To achieve this, two robust analytical techniques are implemented namely, the three-wave method and the interaction phenomena form. The newly obtained solutions are in the form of dark, w-shaped, periodic, peakon, bright and three-soliton solutions and their interaction solution in a kinky base. These characteristics are analysed by the graphical representation which demonstrates the reliability and virtue of the proposed methods. The most fascinating part of the reported solutions is their use in science and engineering. The novel aspect of this study is the obtained solutions, which were not before achieved and demonstrate a good balance between the nonlinear physical components. Furthermore, the solutions can facilitate estimating the performance of wave patterns in fibers when needed in a design.

Novel multiple soliton and front wave solutions for the 3D-Vakhnenko–Parkes equation

The multiple exp-function technique, as an extension to Hirota’s perturbation algorithm, is employed to find N-soliton solutions of the 3D Vakhnenko–Parkes (3D-VP) equation. This is a straightforward and effective alternative method. Consequently, one- two- and three-soliton solutions besides the front wave solutions and their wave numbers are obtained, which are completely distinct from the other published literature. 3D and density plots were investigated via the help of the Maple package program by suitably choosing the free parameters in the solutions. Further, the physical meaning for these solutions is clarified. We mention that the propagation of high-frequency waves in a relaxing medium can be investigated by discussing the dynamical behavior of the obtained solutions for the 3D-VP equation.

Integrability and multisoliton solutions of the reverse space and/or time nonlocal Fokas–Lenells equation

This paper studies reverse space and/or time nonlocal Fokas–Lenells (FL) equation, which describes the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are considered, by Hirota bilinear method. Firstly, we construct variable transformations from reverse space nonlocal FL equation to reverse time and reverse space-time nonlocal FL equations. Secondly, the multisoliton and quasi-periodic solutions of the reverse space nonlocal FL equation are derived through Hirota bilinear method, and the soliton solutions of reverse time and reverse space-time nonlocal FL equations are given through variable transformations. Also, dynamical behaviors of the multisoliton solutions are discussed in detail by analyzing their wave structures. Thirdly, asymptotic analysis of two- and three-soliton solutions of reverse space nonlocal FL equation is used to investigate the elastic interaction and inelastic interaction. Finally, the infinite conservation laws of three types of nonlocal FL equations are found by using their lax pairs. The results obtained in this paper possess new properties that different from the ones for FL equation, which are useful in exploring novel physical phenomena of nonlocal systems in nonlinear media.

Generalized fifth-order nonlinear evolution equation for the Sawada-Kotera, Lax, and Caudrey-Dodd-Gibbon equations in plasma physics: Painlevé analysis and multi-soliton solutions

This research aims to investigate a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and other nonlinear sciences. The Painlevé analysis is used to determine the integrability of the equation, and the simplified Hirota technique is applied to construct multiple soliton solutions with an investigation of the dispersion relation and phase shift of the equation. We utilize a linear combination approach to construct a system of equations to obtain a general logarithmic transformation for the dependent variable. We generate one-soliton, two-soliton, and three-soliton wave solutions using the simplified Hirota method and showcase the dynamics of these solutions graphically through interaction between one, two, and three solitons. We investigate the impact of the system’s parameters on the solitons and periodic waves. The SK, Lax, and CDG equations have a wide range of applications in nonlinear dynamics, plasma physics, oceanography, soliton theory, fluid dynamics, and other sciences.

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term.

Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

In this paper, we use the Riemann–Hilbert (RH) approach to examine the integrable three-coupled Lakshmanan–Porsezian–Daniel (LPD) model, which describes the dynamics of alpha helical protein with the interspine coupling at the fourth-order dispersion term. Through the spectral analysis of Lax pair, we construct the higher-order matrix RH problem for the three-coupled LPD model, when the jump matrix of this particular RH problem is a $$4\times 4$$ unit matrix, the exact N-soliton solutions of the three-coupled LPD model can be exhibited. As special examples, we also investigate the nonlinear dynamical behaviors of the single-soliton, two-soliton, three-soliton and breather soliton solutions. Finally, an integrable generalized N-component LPD model with its linear spectral problem is discussed.

Riemann–Hilbert problem and dynamics of soliton solutions of the fifth-order nonlinear Schrödinger equation

The theory of inverse scattering is developed to investigate the initial-value problem for the fifth-order nonlinear Schrödinger (foNLS) equation under the zero boundary conditions at infinity. The spectral analysis is performed in the direct scattering process, including the establishment of the analytical, asymptotic and symmetric properties of the scattering matrix and the Jost functions. In the inverse scattering process, a suitable Riemann–Hilbert (RH) problem is successfully established by using the modified eigenfunctions and scattering data, and the relationship between the potential function and the solution of the RH problem is successfully established. In order to further analyze the propagation behavior of the solutions of the foNLS equation, we present some new phenomena of studying three-soliton solutions corresponding to simple zeros and two-soliton solutions corresponding to double zeros in scattered data.