Multi-soliton Solutions In Terms Research Articles

Bright soliton solutions to the nonlocal Manakov equations of reverse-space type

Utilizing the bilinear method combined with the KP hierarchy reduction technique, bright multi-soliton solutions in terms of determinant to the nonlocal Manakov equations of reverse-space type are derived in the focusing case. Our results show that these solutions only allow an even number of solitons and each paired soliton exhibits the head-on collision with the same velocity. In the dynamic behaviors illustrated, the fundamental paired soliton supports valley and hump at the center of collision, as well as one degenerate soliton with position shifts. The higher-order soliton solutions exhibit the interaction of paired solitons, where the particular bound states include the mixed line-breathing patterns and the single breather with position shifts.

Three-wave resonant interactions: dark–bright–bright mixed N- and high-order solitons, breathers, and their structures

In this paper, a three-wave resonant interaction system, which describes the mixture of waves with different frequencies in a weakly nonlinear and dispersive medium, is investigated with the help of the Darboux transformation and its extension. Firstly, we construct the dark–bright–bright mixed multi-soliton solutions in terms of the determinant. Dark–bright–bright solitons are found to arise in the focusing, defocusing and mixed cases. Bound state of the two dark–bright–bright solitons is also analyzed. Secondly, we derive the dark–bright–bright mixed high-order semi-rational solitons and find that their structures are different from the multi-soliton ones. Finally, we present the Nth-order breather solutions in terms of the determinant, where N is a positive integer. Line-shaped and three wing-shaped structures of the breathers are obtained.

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.

Broken and unbroken $$\varvec{\mathcal {PT}}$$-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

In this letter, we obtain multi-soliton solutions in terms of ratio of ordinary determinants for semi-discrete nonlocal nonlinear Schrodinger (sd-NNLS) equation by employing the Darboux transformation. We construct explicit expressions of single and double soliton solutions in zero background. We obtain symmetry-broken and symmetry-unbroken soliton solutions of sd-NNLS equation by using appropriate eigenfunctions. We notice that for symmetry non-preserving case, the potential term exhibits stable structure whereas individual fields display instability. We also obtain blowup or oscillating singular-type soliton solutions for symmetry-preserving case.

Dynamics of dark multisoliton and rational solutions for three nonlinear differential-difference equations

In this paper, three nonlinear differential-difference equations (NDDEs) from the same hierarchy are investigated using the generalised perturbation $$(n,N-n)$$ -fold Darboux transformation (DT) technique. The dark multisoliton solutions in terms of determinants for three equations are obtained by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied through numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such dark multisolitons without a noise. We find that the solutions of lower-order NDDEs in the same hierarchy are more robust against a small noise than their corresponding higher-order NDDEs. The discrete generalised perturbation $$(1,N-1)$$ -fold DT is used to derive some discrete rational and semirational solutions of the first equation, and a few mathematical features are also discussed. Results in this paper might be helpful for understanding some physical phenomena.

Fission and fusion interaction phenomena of the discrete kink multi-soliton solutions for the Chen–Lee–Liu lattice equation

Chen–Lee–Liu (CLL) lattice equation is an integrable discretization of the CLL equation which can be used to model the evolution of the self-steepening optical pulses without self-phase modulation. In this paper, the discrete N-fold Darboux transformation (DT) is used to derive the discrete kink multi-soliton solutions in terms of determinant for CLL lattice equation. Soliton fission and fusion interaction structures of such solutions are shown graphically. The details of their evolution are investigated by using numerical simulations, showing that a small noise with amplitude less than or equal to 0.01 produces a strong oscillation and instability of these kink soliton solutions. The discrete generalized perturbation [Formula: see text]-fold DT is constructed to express some rational solutions in terms of the determinants of CLL lattice equation by modifying the discrete N-fold DT. Infinitely many conservation laws for CLL lattice equation are constructed based on its Lax representation. Results in this paper might be helpful for understanding the propagation of optical pulses.

Darboux transformation and multi-soliton solutions of local/nonlocal N-wave interactions

We study matrix Darboux transformation (MDT) for the generalized N-wave interactions (local/nonlocal) and express multi-soliton solutions in terms of ratios of determinants. In order to explain our construction, we obtain explicit expressions of single and double soliton solutions for local/nonlocal 3-wave and 4-wave interactions.

Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

Complex and coupled complex negative order AKNS equation

In this paper, we propose a complex negative order AKNS equation and a coupled complex negative order AKNS equation. They are integrable due to the existence of Lax pairs. Furthermore, we find their multi-soliton solutions in terms of Pfaffians by virtue of Hirota’s bilinear method. One- and two-soliton solutions are investigated in detail exhibiting favorable evolution properties. Especially, the same as the coupled nonlinear Schrödinger equation, inelastic interactions and energy re-distribution happen between two solitons, that has potential applications and deserve further study.

Complex short pulse and coupled complex short pulse equations

In the present paper, we propose a complex short pulse equation and a coupled complex short equation to describe ultra-short pulse propagation in optical fibers. They are integrable due to the existence of Lax pairs and infinite number of conservation laws. Furthermore, we find their multi-soliton solutions in terms of pfaffians by virtue of Hirota’s bilinear method. One- and two-soliton solutions are investigated in details, showing favorable properties in modeling ultra-short pulses with a few optical cycles. Especially, same as the coupled nonlinear Schrödinger equation, there is an interesting phenomenon of energy redistribution in soliton interactions. It is expected that, for the ultra-short pulses, the complex and coupled complex short pulses equation will play the same roles as the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equation.

Darboux Transformation and Multisoliton Solutions of the Short Pulse Equation

We study Darboux transformation for the short pulse (SP) equation and obtain its soliton solutions. We express the multisoliton solutions in terms of Wronski determinants. We also obtain an explicit expression of single soliton solution of the SP equation using the Darboux transformation.

DARBOUX TRANSFORMATION AND MULTI-SOLITON SOLUTIONS OF PRINCIPAL CHIRAL AND WZW MODELS

In this paper we present Darboux transformation for the principal chiral and WZW models in two dimensions and construct multi-soliton solutions in terms of quasideterminants. We also establish the Darboux transformation on the holomorphic conserved currents of the WZW model and expressed them in terms of the quasideterminant. We discuss the model based on the Lie group SU (n) and obtain explicit soliton solutions for the SU(2) model.

Bäcklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for (2+1)-Dimensional Breaking Soliton Equations with Symbolic Computation

In this paper, two types of the (2+1)-dimensional breaking soliton equations are investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinear forms and Bäcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilinear equations. Via the Wronskian technique, it is proved that the Bäcklund transformations obtained are the ones between the (N − 1)- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonic properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.

Multisoliton solutions to the lattice Boussinesq equation

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the two-dimensional lattice, having three-dimensional consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discretization of the exponential as [(a+k)/(a−k)]n, we need two different terms [(a−ωk)/(a−k)]n and [(a−ω2k)/(a−k)]n, where ω is a cubic root of unity ≠1.

Quasideterminant solutions of the generalized Heisenberg magnet model

In this paper we present the Darboux transformation for the generalized Heisenberg magnet (GHM) model based on the general linear Lie group GL(n) and construct multi-soliton solutions in terms of quasideterminants. Further we relate the quasideterminant multi-soliton solutions obtained by means of Darboux transformation with those obtained by the dressing method. We also discuss the model based on the Lie group SU(n) and obtain explicit soliton solutions of the model based on SU(2).

Inelastic interactions and double Wronskian solutions for the Whitham–Broer–Kaup model in shallow water

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) model for the dispersive long wave in shallow water. Connection between the WBK model and a second-order Ablowitz–Kaup–Newell–Segur (AKNS) system is revealed. By means of the Darboux transformation for the second-order AKNS system, the multisoliton solutions in terms of the double Wronskian determinant for the WBK model are derived. Inelastic interactions are graphically discussed. Our results could be helpful for interpreting certain nonlinear wave phenomena in shallow water.

Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications

In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schrödinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Bäcklund transformation transforms between (N−1)- and N-soliton solutions.