Multi-soliton Solutions Research Articles

Nonlocal M-component nonlinear Schrödinger equations: Bright solitons, energy-sharing collisions, and positons.

The general set of nonlocal M-component nonlinear Schrödinger (nonlocal M-NLS) equations obeying the PT-symmetry and featuring focusing, defocusing, and mixed (focusing-defocusing) nonlinearities that has applications in nonlinear optics settings, is considered. First, the multisoliton solutions of this set of nonlocal M-NLS equations in the presence and in the absence of a background, particularly a periodic line wave background, are constructed. Then, we study the intriguing soliton collision dynamics as well as the interesting positon solutions on zero background and on a periodic line wave background. In particular, we reveal the fascinating shape-changing collision behavior similar to that of in the Manakov system but with fewer soliton parameters in the present setting. The standard elastic soliton collision also occurs for particular parameter choices. More interestingly, we show the possibility of such elastic soliton collisions even for defocusing nonlinearities. Furthermore, for the nonlocal M-NLS equations, the dependence of the collision characteristics on the speed of the solitons is analyzed. In the presence of a periodic line wave background, we notice that the soliton amplitude can be enhanced significantly, even for infinitesimal amplitude of the periodic line waves. In addition to these solutions, by considering the long-wavelength limit of the obtained soliton solutions with proper parameter constraints, higher-order positon solutions of the nonlocal M-NLS equations are derived. The background of periodic line waves also influences the wave profiles and amplitudes of the positons. Specifically, the positon amplitude can not only be enhanced but also be suppressed on the periodic line wave background of infinitesimal amplitude.

The dynamic behaviors between multi-soliton of the generalized $$\pmb {(3+1)}$$-dimensional variable coefficients Kadomtsev–Petviashvili equation

In this work, the generalized $$(3+1)$$ -dimensional variable coefficients Kadomtsev–Petviashvili equation, widely used in fluids or plasmas, is analyzed via the unified method and its general form. The multi-soliton rational solutions are obtained including single- and double-soliton rational solutions. Single-soliton shaping and the interactions of double-soliton are graphically discussed in different choices of coefficients. Single-soliton wave keeps its shape, velocity and amplitude unchanged and propagates periodically in a certain direction. The double-soliton waves do not change in shapes, velocities and amplitudes before and after the collisions. We conclude that collisions between the double-soliton waves are elastic and they are not affected by the coefficients of the equation.

Soliton solutions of Kundu-Eckhaus equation in birefringent optical fiber with inter-modal dispersion

We study the dynamics of optical soliton in birefringent optical fiber with the effect of inter-modal dispersion. We consider the model of the Kundu-Eckhaus (KE) equation for birefringent optical fiber with the inclusion of inter-modal dispersion. Multisoliton solutions of KE equation with the presence of inter-modal dispersion are found analytically by using mapping method and a symbolic computation system. Initially, the solutions are expressed in terms of the Jacobi elliptic function which is also recovered by the use of limiting case of modulus of ellipticity. With the help of graphical illustrations of our solutions, we understand the effect of inter-modal dispersion on the soliton evolution in optical birefringent optical fiber.

Overtaking collisions of oblique isothermal ion‐acoustic multisolitons in ultra‐relativistic degenerate dense magnetoplasmas

AbstractOvertaking collisions of oblique isothermal ion‐acoustic multisolitons are studied in an ultra‐relativistic degenerate dense magnetoplasma, containing non‐degenerate inertial warm ions and ultra‐relativistic degenerate inertialess electrons and positrons. A non‐linear Korteweg‐de Vries (KdV) equation describing oblique isothermal ion‐acoustic solitons (OIIASs) in such a plasma model is derived. By applying Hirota's bilinear method (HBM), the overtaking collisions of oblique isothermal ion‐acoustic multisoliton solutions are investigated. An in‐depth discussion shows that the amplitude, the width, and the phase shift of isothermal ion‐acoustic multisolitons increase as the obliqueness and the chemical potential of electrons increase. The deviation of the trajectories decreases with increasing concentration of fermions and the ion cyclotron frequency. The present finding of this study is applicable in compact objects, such as white dwarfs and neutron stars, having degenerate ultra‐relativistic dense electrons and positrons.

Three-wave interactions in a more general (2+1)-dimensional Boussinesq equation

A more general (2+1)-dimensional Boussinesq equation with five arbitrary constants is studied. Multi-soliton solution, multi-breather solution, higher-order lump solution are obtained by employing the Hirota bilinear method and the long-wave limit approach. By selecting suitable value of parameters, the interaction mixed solutions are constructed, which are composed of three waves for lumps, breathers and solitons. Some novel dynamical processes are shown by figures.

Positive and negative integrable lattice hierarchies: Conservation laws and [formula omitted]fold Darboux transformations

Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and N−fold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual 1−fold DT, this kind of N−fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive N−fold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.

An effective approach for constructing novel KP-like equations

In this paper, an effective algorithm for constructing nonlinear evolution equations (NLEEs) has been proposed. Particularly, the existence of resonant multi-soliton solutions in the newly generated NLEEs is verified and demonstrated, and the accuracy of the extracted resonant multi-soliton solutions has been proved at the same time. Firstly, via the linear superposition principle along with reverse engineering two new NLEEs arising from the B-type Kadomtsev–Petviashvili (BKP) equation are established and investigated as well. The first new NLEE is constructed with three time derivative terms, and the second one is constructed with three space dissipative terms, respectively. Besides, the infinite resonant multi-soliton solutions are extracted which enjoy a variety of inelastic interactions due to the fact that they are constructed with variable parameters. Then, the reliable judgments to the multi-soliton solutions are carried out. Finally, the Painlevé test is applied to examine the new equations and none of them passed the test. It is important to highlight that the presented method and NLEEs could be extended to diversify the problem of physical nature.

Riemann–Hilbert approach and N-soliton solutions for a new two-component Sasa–Satsuma equation

A new two-component Sasa–Satsuma equation associated with a $$4\times 4$$ matrix spectral problem is proposed by resorting to the zero-curvature equation. Riemann–Hilbert problems are formulated on the basis of spectral analysis of the $$4\times 4$$ matrix Lax pair for the two-component Sasa–Satsuma equation, from which zero structures of the Riemann–Hilbert problems are investigated. As applications, N-soliton formulas of the two-component Sasa–Satsuma equation are obtained by solving a particular Riemann–Hilbert problem corresponding to the reflectionless case. Further, the obtained N-soliton formulas are expressed by the ratios of determinants, which are more compact and convenient for symbolic computations. Moreover, the interaction dynamics of the multi-soliton solutions are analyzed and graphically illustrated.

N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation

In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with $$N=1$$ , from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena.

Multiple breathers and high-order rational solutions of the new generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

In this paper, we consider a new generalized KP equation which is obtained by adding the extra term $$u_{tz}$$ in the previous equation. Based on these detailed discussions in the previous reference documentations, we know that solitons, breathers, lump waves, and rogue waves are four typical local waves. Therefore, we mainly focus on investigating the multi-soliton solutions, high-order breather solutions, and high-order rational solutions. The high-order breather solutions can be derived by taking complex conjugate parameters in the multi-soliton solutions. Applying the long wave limit method to the multi-soliton solutions, we conclude Theorem 3.1 which can be used directly to obtain high-order rational solutions. Meanwhile, for the case of three-soliton and five-soliton, the elastic interaction solutions among two parallel breathers and one soliton as well as between one breather and one soliton also can be derived, respectively. For all these types of exact solutions, we provide corresponding graphics to illustrate their dynamical characteristics in the end.

Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation

The nonlocal defocusing nonlinear Schrodinger (ND-NLS) equation is comparatively studied via the Riemann–Hilbert approach. Firstly, via spectral analysis, the spectral structure of the ND-NLS equation is investigated, which is different to those of the other three NLS-type equations, i.e., the local focusing nonlinear Schrodinger (LF-NLS) equation, the local defocusing nonlinear Schrodinger (LD-NLS) equation and the nonlocal focusing nonlinear Schrodinger (NF-NLS) equation. Secondly, by solving the Riemann–Hilbert problem corresponding to the reflectionless cases, multi-soliton solutions are obtained for the ND-NLS equation. Thirdly, we prove that, if parameters are suitably chosen, the multi-soliton solutions of the ND-NLS equation can be reduced to those of the LF-NLS equation and the LD-NLS equation, respectively. Fourthly, the multi-soliton solutions of the ND-NLS equation are demonstrated to possess repeated singularities generally, but they can also remain analytic for appropriate soliton parameters. Moreover, the multi-soliton dynamics are graphically illustrated using Mathematica symbolic computations. These results show that the solution structure and the nonlinear dynamics in the ND-NLS equation are rather different from those of the LF-NLS equation, the LD-NLS equation and the NF-NLS equation.

Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

AbstractWe study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

Exact multisolitons of noncommutative and commutative Lakshmanan–Porsezian–Daniel equation

We propose and study a noncommutative generalization of Lakshmanan–Porsezian–Daniel equation. For this generalization, we demonstrate its integrability by constructing its zero curvature representation, Darboux transformation and multisoliton solutions. The soliton solutions up to fifth order are presented explicitly within framework of quasideterminants.

Inverse Scattering Transform and Soliton Classification of Higher-Order Nonlinear Schrödinger-Maxwell-Bloch Equations

We investigate higher-order nonlinear Schrodinger-Maxwell-Bloch equations using the Riemann-Hilbert method. We perform a spectral analysis of the Lax pair and construct a Riemann-Hilbert problem according to the spectral analysis. As a result, we obtain three types of multisoliton solutions. Based on the analytic solution and with a choice of corresponding parameter values, we obtain solutions of the breather type and a bell-shaped solution and find an interesting phenomenon of the collision of two soliton solutions. We hope that these results can be useful in modeling the wave propagation of a nonlinear optical field in an erbium-doped fiber medium.

Nonlinear waves behaviors for a coupled generalized nonlinear Schrödinger–Boussinesq system in a homogeneous magnetized plasma

Under investigation in this paper is a coupled generalized nonlinear Schrodinger–Boussinesq system, which describes the coupled upper-hybrid and magnetoacoustic modes in a homogeneous magnetized plasma for the bidirectional propagation near the magnetoacoustic speed. Based on the Hirota method, the expressions for the multi-soliton solutions are given. Effects of the group velocity, group dispersion coefficient for the upper-hybrid, and the properties of the magnetic field on the soliton are discussed. Based on the asymptotic analysis, interaction between two solitons is proved to be elastic through the asymptotic analysis. Position at which the maximal distortion occurs is obtained. Multi-soliton interaction is illustrated and investigated. Two prerequisites of the formation and features of the bound state are discussed. For the cases of three solitons, inelastic interaction occurs with phase shifts. Characteristics of the breather and its relation with the bound state and the breather are investigated. Interaction between the bound state (even the breather) and a single soliton is discussed for both cases that they are parallel or not.

Soliton interaction for Maxwell-Bloch system

In this paper, scalar Maxwell-Bloch equations are studied by virtue of the Darboux transformation, which can be used to describe the optical pulse propagation in the two-level optical medium. Multi-soliton solutions of the integrable Maxwell-Bloch equations are obtained. Propagation and interaction of the solitons are analyzed: Spectral parameter has effect on the propagation direction of the one solitons; varies of kinds interaction pattern for the two- and three-soliton are attained by virtue of chooseing the different spectral parameters.

A two-component modified Korteweg–de Vries equation: Riemann–Hilbert problem and multi-soliton solutions

In this paper, the inverse scattering transform of a two-component modified Korteweg–de Vries equation is under investigation, which is one of the important nonlinear models in mathematics and physics. The spectral analysis with a Lax pair is performed for the two-component modified Korteweg–de Vries equation, from which a Riemann Hilbert problem is strictly formulated. Moreover, multi-soliton solutions to the two-component modified Korteweg–de Vries equation are obtained through a particular Riemann–Hilbert formulation. Finally, some figures are presented to reveal the soliton features of the two-component modified Korteweg–de Vries equation.

Soliton molecules, nonlocal symmetry and CRE method of the KdV equation with higher-order corrections

The soliton molecules of the Korteweg–de Vries (KdV) equation with higher-order corrections are studied by using the velocity resonance mechanism and the multi-soliton solution. The interaction between a soliton molecule and one-soliton of the KdV equation with higher-order corrections is elastic by means of analytical and graphical ways. The nonlocal symmetry of the KdV equation with higher-order corrections is derived by the truncate Painlevé analysis. An nonauto-Bäcklund theorem is established by solving the initial value problem of the Lie’s first principle of the nonlocal symmetry. In the meanwhile, the KdV equation with higher-order corrections is proved to be a consistent Riccati expansion (CRE) solvable system by exploiting the CRE method.

Multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit: Darboux transformation approach

We propose a new approach for calculating multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit by combining a reciprocal transformation with the Darboux transformation of the negative flow of the Kaup–Kupershmidt hierarchy. In particular, different specifications of the soliton parameters lead to two different types of soliton solutions of the Degasperis–Procesi equation.

Interactions among solitons for a fifth-order variable coefficient nonlinear Schrödinger equation

A fifth-order variable coefficient nonlinear Schrodinger equation based on original inhomogeneous model is proposed and studied. Bright multi-soliton analytic solutions of the equation are calculated through the Hirota method and auxiliary function. The effect of different constraints between fifth-order dispersion with third-order dispersion on soliton transmission is researched. Besides, their propagation and interaction dynamics are analyzed. Moreover, based on the obtained three-soliton solutions, we change the value of $$\beta (x)$$ to discuss the soliton propagation. Three different propagation and interaction structures are derived via adjusting the nonlinear term. The results can enrich the inhomogeneous model and apply in nonlinear optical fiber.