Multi-soliton Solutions Research Articles

The Painlevé integrability and abundant analytical solutions for the potential Kadomtsev–Petviashvili (pKP) type coupled system with variable coefficients arising in nonlinear physics

Acknowledging the inhomogeneities of media and non-uniformity of boundaries, the nonlinear evolution equations with variable coefficients may show more realistic situations dealing with time-varying environment, inhomogeneous medium, etc. In this order, this article deals with the variable-coefficients (2+1)-dimensional potential Kadomtsev–Petviashvili (pKP) type coupled equations. The Painlevé analysis approach is being used to assess the integrability of the governing model. The Laurent series, which appears when using the Painlevé analysis technique, has been truncated in order to build an auto-Bäcklund transformation (ABT). Three different analytical solutions in the form of hyperbolic, exponential, periodic and rational have been evaluated using the ABT approach. The multi-soliton solutions are also derived by using Hirota bilinear method for the aforesaid equation. All the solutions are displayed as 3D plots in which the variable coefficients and parameters can be varied to see the effects. In these plots, one can see the many different behaviors of the examined coupled system, such as periodic waves, kink-soliton, anti-kink soliton, kink anti-kink wave, complex periodic waves, complex kink anti-kink wave, and multi-soliton surfaces.

Multi-soliton solutions for the two-component modified short pulse equation and its nonlocal reductions via Bäcklund transformations

Multi-soliton solutions for the two-component modified short pulse equation and its nonlocal reductions via Bäcklund transformations

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.

Multisoliton and general Wronskian solutions of a variable‐coefficient Korteweg–de Vries equation

With the help of a suitable transformation of the potential function, a variable‐coefficient Korteweg–de Vries (vcKdV) equation is transformed into a quadrilinear form. This form is further transformed into a bilinear form by the introduction of an auxiliary variable. The multisoliton solution is obtained with the aid of the perturbation technique. Furthermore, using the Laplace expansion of the determinant and a set of conditions, we verify that the potential function expressed in the form of the Wronskian determinant satisfies the given bilinear equation. Lastly, some rational solutions of the vcKdV equation are also obtained.

Deep learning data-driven multi-soliton dynamics and parameters discovery for the fifth-order Kaup–Kuperschmidt equation

In this paper, we extend the multi-layer physics-informed neural networks (PINNs) to successfully learn the data-driven multi-soliton solutions and discover the coefficients of fifth-order Kaup–Kuperschmidt (KK) equation with the aid of multi-soliton data. In the forward problems, the corresponding accuracies for one-, two- and three-soliton solutions are O(10−6), O(10−4) and O(10−3), respectively. Moreover, some basic elements affecting the performance of PINNs are analyzed in detail, such as activation functions, collocation points, optimizers, neural network structures, and so on. For the inverse problems, the coefficients of the equation are discovered by the data of one-, two- and three-soliton solutions, respectively. Meanwhile, the robustness of the PINNs algorithm is explored under different noises. The accuracy of the identified coefficients can reach O(10−3) when 1% initial noise is added to the training data. And the prediction accuracy can still reach O(10−2) even if 3% initial noise is added. These numerical experiments not only show the effectiveness of PINNs to solve and discover the KK equation with some known information, but also embody the applicability to reveal the multi-soliton dynamic behaviors in higher-order nonlinear wave equations.

Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method

The aim of this article is to investigate the time fractional coupled nonlinear Schrödinger equation (TFCNLSE) which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. The TFCNLS equation plays a crucial role in soliton wavelength division multiplexing and pulse propagation through a two-mode optical fiber. For the sake of getting different analytical solutions, we capitalize modified version of the extended tanh-expansion method. Fractional nonlinear partial differential equations (FNLPDEs) have become popular because they can explain complex physical phenomena and have dynamic structures of localized wave solutions. The study obtained various soliton solutions including bright, dark, periodic, and singular solitons. The results are presented in graphical form with appropriate parameter values to aid visualization. Such solutions are critical for explaining some mesmerizing and complex physical phenomena. In addition, some results from earlier studies are generalized which justify the novelty of the current study. This study proves that the computational method used is efficient, brief, and widely applicable in finding analytical solutions to diverse complex nonlinear problems arising in the recent era of nonlinear optics, applied sciences and engineering.

Overtaking interaction of electron-acoustic solitons in Saturn’s magnetosphere

The progression of nonlinear electron-acoustic waves (EAWs) in a magnetized and collision-free plasma made up of cold inertial electrons, inertialess superthermal electrons, and stationary background ions with special reference to Saturn’s magnetosphere (SMS) is explored. The method of reductive perturbation (MRP) is employed to obtain the evolution equation (i.e., Zakharov– Kuznetsov equation (ZKE)) that governs the propagation of electron acoustic solitons (EASs). Using the elegant and efficient Hirota bilinear method (HBM), multi-soliton solutions (MSSs) of the ZKE are determined. The impact of the effects of hot-to-cold electron density ratio, magnetic field (MF) strength, and superthermality on single as well as the interaction of EASs is examined. Estimates of the values of the electric field at several radii of SMS (i.e., 12 R s − 17.8 R s, where R s is the radius of Saturn) are presented, which are found in μV/ m to mV/ m range and are in perfect agreement with the data from Cassini radio and plasma wave science wideband receiver. Moreover, the influence of the relevant plasma parameters on the interaction time and spatial extent of the interacting EASs is also explored.

The deformed modified Korteweg–de Vries equation: Multi-soliton solutions and their interactions

The deformed modified Korteweg–de Vries equation: Multi-soliton solutions and their interactions

Reverse-time type nonlocal Sasa–Satsuma equation and its soliton solutions

In this work, we study the Riemann–Hilbert problem and the soliton solutions for a nonlocal Sasa–Satsuma equation with reverse-time type, which is deduced from a reduction of the coupled Sasa–Satsuma system. Since the coupled Sasa–Satsuma system can describe the dynamic behaviors of two ultrashort pulse envelopes in birefringent fiber, our equation presented here has great physical applications. The classification of soliton solutions is studied in this nonlocal model by considering an inverse scattering transform to the Riemann–Hilbert problem. Simultaneously, we find that the symmetry relations of discrete data in the special nonlocal model are very complicated. Especially, the eigenvectors in the scattering data are determined by the number and location of eigenvalues. Furthermore, multi-soliton solutions are not a simple nonlinear superposition of multiple single-solitons. They exhibit some novel dynamics of solitons, including meandering and sudden position shifts. Also, they have the bound state of multi-soliton entanglement and its interaction with solitons.

Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

Multisolitons in the surface gravity waves and internal waves

In this paper, a five-order Korteweg–de Vries (KdV) equation is studied, which is used to describe the nonlinear phenomena in the fluids, especially those of the surface gravity waves and internal waves in the stratified fluids. (a) Via the symbolic calculation, this KdV equation cannot pass the Painlevé test without any constraint conditions. By virtue of the ansatz method, bell-shape and kink soliton solutions of this KdV equation are attained. (b) Via the bilinear method, multisoliton solutions of this KdV equation are obtained under some constraint conditions. Propagation and interaction of the multisoliton are discussed. Soliton interaction is elastic, that is to say, they have no effect on each other’s amplitude and speed except for phase shift. We hope that our results will be useful for experimental studies of surface gravity waves and internal waves since the coefficients of this KdV equation are all expressed in terms of physical constants, depths, and densities of the fluid.

An extended Hirota bilinear method and new wave structures of (2+1)-dimensional Sawada–Kotera equation

An extended Hirota bilinear method is proposed via changing the summation rules of τ function in the multiple soliton solution expressions while investigating the (2+1)-dimensional Sawada–Kotera equation using the classical Hirota bilinear method. Multi-soliton solutions possessing n arbitrary molecule parameters σi(i=1,…,n),n(n−1)/2 interaction parameters bij(i<j=2,3,…,n), and showing a series of complicated dynamic behaviors are obtained. Asymptotic analysis unveils the dynamical evolution characteristics of the new two-soliton solutions. Abundant wave structures involving single peak soliton, M-shaped soliton, molecular soliton, web-like soliton and various interaction structures are presented and discussed simultaneously.

Loop dynamics of a fully discrete short pulse equation

In this paper, a fully discrete short pulse (SP) equation is presented as an integrability condition of a linear system of difference equations (also known as discrete Lax pair). Additionally, two semi-discrete versions of the SP equation have also been obtained from fully discrete SP equation under continuum limits. Darboux transformation is employed to compute multi-soliton solutions of fully discrete and semi-discrete SP equations. Explicit expressions of first and second nontrivial soliton solutions are computed. We also derived explicit expression of breather solution for fully discrete SP equation. The dynamics of single loop soliton and interaction mechanism of loop–loop and loop–antiloop solutions have been explored and illustrated.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin–Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

Propagation of coupled quartic and dipole multi-solitons in optical fibers medium with higher-order dispersions

We present the discovery of two types of multiple-hump soliton modes in a highly dispersive optical fiber with a Kerr nonlinearity. We show that multi-hump optical solitons of quartic or dipole types are possible in the fiber system in the presence of higher-order dispersion. Such nonlinear wave packets are very well described by an extended nonlinear Schrödinger equation involving both cubic and quartic dispersion terms. It is found that the third- and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole solitons into double-, triple-, and multi-humped solitons. We provide the initial conditions for the formation of coupled multi-hump quartic and dipole solitons in the fiber. Numerical results illustrate that propagating multi-quartic and multi-dipole solitons in highly dispersive optical fibers coincide with a high accuracy to our analytical multi-soliton solutions. It is important for applications that described multiple-hump soliton modes are stable to small noise perturbation that was confirmed by numerical simulations. These numerical results confirm that the newly found multi-soliton pulses can be potentially utilized for transmission in optical fibers medium with higher-order dispersions.

Integrable aspects, analytic solutions and their asymptotic analysis for a discrete relativistic Toda lattice system

In this paper, we investigate a discrete relativistic Toda lattice (mathrm{dRTL}_{+}(alpha )) system, which may describe particle vibrations in lattices with an exponential interaction force. First, we construct its discrete generalized (m,2N-m)-fold Darboux transformation, from which we can explicitly give its analytic solutions, such as discrete multi-soliton solutions, position controllable rational and semi-rational solutions and their hyperbolic-and-rational mixed solutions, whose properties and dynamics are analyzed and shown graphically. Second, the asymptotic behaviors of diverse exact solutions are analyzed, which shows that the interactions among different solutions are always elastic. In particular, the position of controllable rational solutions and asymptotic state analysis of discrete hyperbolic-and-rational mixed solutions are obtained and discussed for the first time. Finally, we study some integrable properties of this system, such as the integrable hierarchy and relevant Hamiltonian structures and conservation laws from a discrete spectral problem. These results may be helpful for understanding nonlinear lattice dynamics.

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schr ödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detail and it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

On the propagation and interaction of ion-acoustic solitary, periodic, shock, and breather waves in a non-Maxwellian electron–positron–ion magnetoplasma

This investigation analyzes the propagation of nonlinear ion-acoustic waves (IAWs) in an unmagnetized, collisionless plasma composed of inertial positive ions and inertialess Maxwellian positrons as well as the inertialess non-Maxwellian electrons that obey (r, q)-distribution. To observe the impact of particle trapping on the nonlinear IAWs in an electron–positron–ion plasma, the Korteweg–De Vries (KdV) and modified KdV (mKdV) equations are derived using a reductive perturbation method. In the distribution function, the spectral parameters (r, q) put up their contribution to the flatness and high-energy tails, respectively. An important aspect of this investigation is the determination of well-known quasi-periodic solutions, multi-soliton solutions, breathers, and shocks under the variation of different physical parameters, especially spectral indices (r, q). Finally, the interaction of solitons is also presented for discussion of the complete profile. In addition, a detailed comparison, especially in a periodic wave, is made between the generalized (r, q)-distribution and the limiting cases of Kappa and Maxwellian distributions. The results presented in this study contribute to a better understanding of the characteristics of both high- and low-energy parts of the electron distribution function as well as the formation of periodic, soliton, multi-soliton, breathers, and shocks in space and astrophysical plasmas.

The Sasa-Satsuma Equation on a Non-Zero Background: The Inverse Scattering Transform and Multi-Soliton Solutions

The Sasa-Satsuma Equation on a Non-Zero Background: The Inverse Scattering Transform and Multi-Soliton Solutions