Mixed Soliton Research Articles

Numerical study of bright-bright-dark soliton dynamics in the mixed coupled nonlinear Schrödinger system

An efficient, accurate numerical method is developed to study the mixed (bright-bright-dark) solitons in a three component mixed coupled nonlinear Schrödinger system, arising in nonlinear optics. For this purpose, a compact finite difference method involving different boundary conditions for the bright and dark solitons is employed to derive an implicit nonlinear numerical scheme of fourth order in space and second order in time. Particularly, the scheme is shown to be unconditionally stable by von Neumann stability method. Also, the exact soliton solutions and the conserved quantities are used to assess the proposed method. Propagation dynamics as well as the standard elastic and the fascinating energy-sharing collisions of multicomponent mixed solitons are studied in this framework. Significantly, the energy-sharing collision scenario of solitons is found to be robust against small perturbations.

Exact solutions of nonlocal Fokas–Lenells equation

In this paper we propose a nonlocal Fokas–Lenells (FL) equation which can be derived from the Kaup–Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several types of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.

Interactions between soliton and rogue wave for a (2+1)-dimensional generalized breaking soliton system: Hidden rogue wave and hidden soliton

A (2+1)-dimensional generalized breaking soliton system is under investigation, which can be used to describe the interaction between Riemann wave and long wave along with two space directions in nonlinear media. Rogue wave solutions, two types of mixed soliton and rogue wave solutions are constructed through appropriately choosing the functions in the bilinear forms. The rogue wave structures and the interaction characteristics between the soliton and rogue wave are studied in details. Novel hidden rogue wave and hidden soliton, which are invisible but work, are observed during the interactions.

Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation

By using the Darboux transformation, we obtain two new types of exponential-and-rational mixed soliton solutions for the defocusing nonlocal nonlinear Schrödinger equation. We reveal that the first type of solution can display a large variety of interactions among two exponential solitons and two rational solitons, in which the standard elastic interaction properties are preserved and each soliton could be either the dark or antidark type. By developing the asymptotic analysis method, we also find that the second type of solution can exhibit the elastic interactions among four mixed asymptotic solitons. But in sharp contrast to the common solitons, the mixed asymptotic solitons have the t-dependent velocities and their phase shifts before and after interaction also grow with |t| in the logarithmical manner. In addition, we discuss the degenerate cases for such two types of mixed soliton solutions when the four-soliton interaction reduces to a three-soliton or two-soliton interaction.

Then-component nonlinear Schrödinger equations: dark–bright mixedN- and high-order solitons and breathers, and dynamics

The generaln-component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their generalNth-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

Soliton solutions for coupled nonlinear Schrödinger equations with linear self and cross coupling terms

In this paper, coupled nonlinear Schrödinger equations with linear self and cross coupling terms are investigated analytically. We obtain the exact soliton solutions which exist a periodic-wave background for this system, including one-, two- and N-soliton solutions. We find the period of oscillation for periodic wave background is related to the coupling terms. In addition, by adjusting other parameters we can get many types of mixed soliton solutions on the plane-wave background. Moreover, the interactions of the two and three solitons have been discussed through a detailed analysis for the asymptotic behavior and it displays that they are all elastic.

Bright-Dark Mixed N-Soliton Solution of the Two-Dimensional Maccari System**Supported by the Global Change Research Program of China under Grant No 2015CB953904, the National Natural Science Foundation of China under Grant Nos 11675054 and 11435005, and the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No ZF1213.

The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.

Bright-Dark MixedN-Soliton Solution of Two-Dimensional Multicomponent Maccari System

AbstractBased on the KP hierarchy reduction method, we construct the general bright-dark mixedN-soliton solution of the two-dimensional (2D) (M+1)-component Maccari system comprised ofM-component short waves (SWs) and one-component long wave (LW) with all possible combinations of nonlinearities. We firstly consider two types of mixedN-soliton solutions (two-bright-one-dark and one-bright-two-dark solitons in SW components) to the (3+1)-component Maccari system in detail. Then by extending our analysis to the (M+1)-component Maccari system, its generalm-bright-(M–m)-dark mixedN-soliton solution is obtained. The formula obtained also contains the general all-bright and all-darkN-soliton solutions as special cases. For the two-bright-one-dark mixed soliton solution of the (3+1)-component Maccari system, it can be shown that solioff excitation and solioff interaction take place in the two SW components supporting bright solitons, whereas the SW component supporting dark solitons and the LW component possess V-type solitary and interaction.

Three-component Gross-Pitaevskii equations in the spin-1 Bose-Einstein condensate: Spin-rotation symmetry, matter-wave solutions, and dynamics

We report new matter-wave solutions of the one-dimensional spin-1 Bose-Einstein condensate system by combining global spin-rotation states and similarity transformation. Dynamical behaviors of non-stationary global spin-rotation states derived from the SU(2) spin-rotation symmetry are discussed, which exhibit temporal periodicity. We derive generalized bright-dark mixed solitons and new rogue wave solutions and reveal the relations between Euler angles in spin-rotation symmetry and parameters in ferromagnetic and polar solitons. In the modulated spin-1 Bose-Einstein condensate system, new solutions are derived and graphically illustrated for different types of modulations. Moreover, numerical simulations are performed to investigate the stability of some obtained solutions for chosen parameters.

General bright–dark soliton solution to (2 + 1)-dimensional multi-component long-wave–short-wave resonance interaction system

In this paper, we derive a general mixed (bright–dark) multi-soliton solution to a two-dimensional (2D) multi-component long-wave–short-wave resonance interaction (LSRI) system, which include multi-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients including positive, negative and mixed types. With the help of the KP hierarchy reduction method, we firstly construct two types of general mixed N-soliton solution (two-bright–one-dark soliton and one-bright–two-dark one for SW components) to the 2D three-component LSRI system in detail. Then by extending the corresponding analysis to the 2D multi-component LSRI system, a general mixed N-soliton solution in Gram determinant form is obtained. The expression of the mixed soliton solution also contains the general all bright and all dark N-soliton solution as special cases. In particular, for the soliton solution which include two-bright–one-dark soliton for SW components in three-component LSRI system, the dynamics analysis shows that solioff excitation and solioff interaction appear in two SW components which possess bright soliton, while V-type solitary and interaction take place in the other SW component and LW one.

The superposition solitons for 3-coupled nonlinear Schrödinger equations

In this paper, a Hirota bilinear method is developed for applying to the 3-coupled nonlinear Schrödinger equations. With a reasonable assumption the exact two-superposition-one-dark (TSD) and one-bright-two-superposition (BTS) soliton solutions are constructed analytically. It shows that they can transform into general mixed (dark–bright) soliton solutions in the special conditions. Moreover, the asymptotic behavior analysis shows that the collision of TSD and BTS two solitons are all elastic.

General Mixed Multi-Soliton Solutions to One-Dimensional Multicomponent Yajima–Oikawa System

In this paper, we derive a general mixed (bright-dark) multi-soliton solution to a one-dimensional multicomponent Yajima-Oikawa (YO) system, i.e., the (M+1)-component YO system comprised of M-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients including positive, negative and mixed types. With the help of the KP-hierarchy reduction method, we firstly construct two types of general mixed N-soliton solution (two-bright-one-dark soliton and one-bright-two-dark one for SW components) to the (3+1)-component YO system in detail. Then by extending the corresponding analysis to the (M+1)-component YO system, a general mixed N-soliton solution in Gram determinant form is obtained. The expression of the mixed soliton solution also contains the general all bright and all dark N-soliton solution as special cases. Besides, the dynamical analysis shows that the inelastic collision can only take place among SW components when at least two SW components have bright solitons in mixed type soliton solution. Whereas, the dark solitons in SW components and the bright soliton in LW component always undergo usual elastic collision.

Mixed solitons in a (2+1)-dimensional multicomponent long-wave-short-wave system.

We derive a (2+1)-dimensional multicomponent long-wave-short-wave resonance interaction (LSRI) system as the evolution equation for propagation of N-dispersive waves in weak Kerr-type nonlinear medium in the small-amplitude limit. The mixed- (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher-dimensional LSRI system, are obtained by applying the Hirota's bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for an N-component LSRI system with N>3, if the bright solitons appear in at least two components, interesting collision behavior takes place, resulting in energy exchange among the bright solitons. However, the dark solitons undergo standard elastic collision accompanied by a position shift and a phase shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher-dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.

Multi-loop soliton solutions and their interaction in the Degasperis–Procesi equation

In this paper, we construct loop soliton solutions and mixed soliton–loop soliton solutions for the Degasperis–Procesi equation. To explore these solutions, we adopt the procedure given by Matsuno (2005 Inverse Problems21 1553). Appropriately modifying the τ-function given in the above paper we derive these solutions. We present the explicit form of one- and two-loop soliton solutions and mixed soliton–loop soliton solutions and investigate the interaction between (i) two-loop soliton solutions in different parametric regimes and (ii) a loop soliton with a conventional soliton in detail.

Soliton complexes in dissipative systems: Vibrating, shaking, and mixed soliton pairs

We show, numerically, that coupled soliton pairs in nonlinear dissipative systems modeled by the cubic-quintic complex Ginzburg-Landau equation can exist in various forms. They can be stationary, or they can pulsate periodically, quasiperiodically, or chaotically, as is the case for single solitons. In particular, we have found various types of vibrating and shaking soliton pairs. Each type is stable in the sense that a given bound state exists in the same form indefinitely. New solutions appear at special values of the equation parameters, thus bifurcating from stationary pairs. We also report the finding of mixed soliton pairs, formed by two different types of single solitons. We present regions of existence of the pair solutions and corresponding bifurcation diagrams.

Mixed Stack Compounds, Phonons and Solitons

Abstract Phase transitions in organic mixed-stack compounds, known as “neutral-ionic” transitions, are reinterpreted as the onset of bond dimerization. The phase diagram is derived and the effects of temperature, pressure and the difference of donor-acceptor potentials are correlated. Electron-multi phonon couplings lead to renormalization effects both above and below the transition. A soft mode at the transition is predicted. Analysis of the phonon spectra is an efficient test for any quantitative theory of these compounds. Solitons are possible excitations below the transition and may account for conductivity and ESR data. The incommensurate situation is described by a soliton lattice.

Multiple soliton and bisoliton bound state solutions of the sine-Gordon equation and related equations in nonlinear optics

A prescription is given for obtaining mixed multiple soliton and bisoliton bound state solutions of the sine-Gordon equation and of related equations in nonlinear optics. Some applications in nonlinear optics and other branches of physics are sketched.