Short Wave Resonance Interaction Research Articles

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

AbstractGeneral higher order breather and rogue wave (RW) solutions to the two‐component long wave–short wave resonance interaction (2‐LSRI) model are derived via the bilinear Kadomtsev–Petviashvili hierarchy reduction method and are given in terms of determinants. Under particular parametric conditions, the breather solutions can reduce to homoclinic orbits, or a mixture of breathers and homoclinic orbits. There are three families of RW solutions, which correspond to a simple root, two simple roots, and a double root of an algebraic equation related to the dimension reduction procedure. The first family of RW solutions consists of bounded fundamental RWs, the second family is composed of bounded fundamental RWs coexisting with another fundamental RWs of different a bounded state ( being positive integers), while the third one has fundamental bounded RWs ( being nonnegative integers). The second family can be regarded as the superpositions of the first family, while the third family can be the degenerate case of the first family under particular parameter choices. These diverse RW patterns are illustrated graphically.

Bright, dark and breather soliton solutions of the generalized long-wave short-wave resonance interaction system

In this paper, a generalized long-wave short-wave resonance interaction system, which describes the nonlinear interaction between a short-wave and a long-wave in fluid dynamics, plasma physics and nonlinear optics, is considered. Using the Hirota bilinear method, the general N-bright and N-dark soliton solutions are deduced and their Gram determinant forms are obtained. A special feature of the fundamental bright soliton solution is that, in general, it behaves like the Korteweg-deVries soliton. However, under a special condition, it also behaves akin to the nonlinear Schrödinger soliton when it loses the amplitude-dependent velocity property. The fundamental dark-soliton solution admits anti-dark, gray, and completely black soliton profiles, in the short-wave component, depending on the choice of wave parameters. On the other hand, a bright soliton-like profile always occurs in the long-wave component. The asymptotic analysis shows that both the bright and dark solitons undergo an elastic collision with a finite phase shift. In addition to these, by tuning the phase shift regime, we point out the existence of resonance interactions among the bright solitons. Furthermore, under a special velocity resonance condition, we bring out the various types of bright and dark soliton bound states. Also, by fixing the phase factor and the system parameter \(\beta \), corresponding to the interaction between long and short wave components, the different types of profiles associated with the obtained breather solution are demonstrated.

Painlevé integrability and multi-wave pattern for (2+1)-dimensional long wave–short wave resonance interaction system

Painlevé integrability and multi-wave pattern for (2+1)-dimensional long wave–short wave resonance interaction system

Resonant collision of lumps with homoclinic orbits in the two-dimensional multi-component long-wave–short-wave resonance interaction systems

The resonant collisions of lumps and homoclinic orbits are studied for the two-dimensional (2D) multi-component long-wave–short-wave resonance interaction (M-LSRI) system. Compared with a usual lump, which keeps its shape and velocity unchanged in the course of the evolution from t→−∞ to t→+∞, the lumps in the resonant collision with homoclinic orbits have completely different dynamical behaviours. The resonant collisions are classified into partial resonant collisions and complete resonant collisions, depending upon the changes in the dynamical behaviours of the lumps. In the partial resonant collisions, the lumps are partially localized in time, and mainly exhibit two opposite behaviours: (i) they do not exist at t→−∞ but suddenly emerge from the background and remain existing at t→+∞, and (ii) they exist at t→−∞ and unexpectedly merge into the background and disappear at t→+∞. In the complete resonant collisions, the lumps are localized in time as well as in the two-dimensional space; they disappear at t→±∞, and at the intermediate collision process they first emerge from the background and then they merge into the background again after living for a very short period of time.

Nondegenerate Bright Solitons in Coupled Nonlinear Schrödinger Systems: Recent Developments on Optical Vector Solitons

Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton-based promising potential applications. Mathematically, the dynamics of vector solitons can be understood from the framework of the coupled nonlinear Schrödinger (CNLS) family of equations. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable CNLS framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two coupled nonlinear Schrödinger family of equations, namely the Manakov system, mixed 2-CNLS system (or focusing-defocusing CNLS system), coherently coupled nonlinear Schrödinger (CCNLS) system, generalized coupled nonlinear Schrödinger (GCNLS) system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Furthermore, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.

Interaction of the variable-coefficient long-wave–short-wave resonance interaction equations

Long-wave–short-wave resonance interaction (LSRI) equations have been studied in the plasmas, gravity waves, nonlinear electron-plasma and ion-acoustic waves. By virtue of the bilinear method, two soliton solutions of the variable-coefficient LSRI equations are attained. Interaction of the solitons are studied when the coefficients are taken as the generalized Gauss functions. New types of the soliton interaction are exhibited. Position and width of the disturbances can be controlled.

General soliton solutions to a nonlocal long-wave–short-wave resonance interaction equation with nonzero boundary condition

Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric potential. This equation offers PT symmetry analogues of the classical integrable LSRI equation and may be important for the occurence of such equations in nonlinear optics as the nonlocal NLS equation. General soliton solutions to the nonlocal LSRI equation with nonzero boundary condition are derived by using the Hirota’s bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions are expressed in terms of Gramian determinants and include dark–dark solitons, dark–antidark solitons and antidark–antidark solitons. Three typical cases of the two solitons, namely two dark–dark solitons, two dark–antidark solitons and two antidark–antidark solitons, are demonstrated.

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.

Mixed-type solitons and soliton interaction for the (2+1)-dimensional two-component long wave–short wave resonance interaction equations in a two-layer fluid through the Bell polynomials

In this work, we investigate the ( 2 + 1 )-dimensional two-component long wave–short wave resonance interaction equations in a two-layer fluid. Through the Bell polynomials, bilinear forms and mixed-type soliton solutions are derived. Such soliton phenomena as the V-type solitons, two parallel solitons with the periodic interaction, breather-type solitons and semi-breather-type solitons are observed, and the relevant parameter conditions are presented.

Dynamics of solitons in multicomponent long wave–short wave resonance interaction system

In this paper, we study the formation of solitons, their propagation and collision behaviour in an integrable multicomponent (2 + 1)-dimensional long wave–short wave resonance interaction (M-LSRI) system. First, we briefly revisit the earlier results on the dynamics of bright solitons and demonstrate the fascinating energy exchange collision of bright solitons appearing in the short-wave components of the M-LSRI system. Then, we explicitly construct the exact one- and two-multicomponent dark soliton solutions of the M-LSRI system by using the Hirota’s direct method and explore its propagation dynamics. Also, we study the features of dark soliton collisions.

Multicomponent long-wave-short-wave resonance interaction system: Bright solitons, energy-sharing collisions, and resonant solitons.

We consider a general multicomponent (2+1)-dimensional long-wave-short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multisoliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head-on and overtaking collisions of bright solitons and explore two types of energy-sharing collisions as well as standard elastic collision. We have also corroborated the obtained analytical one-soliton solution by direct numerical simulation. Also, we discuss the formation and dynamics of resonant solitons. Interestingly, we demonstrate the formation of resonant solitons admitting breather-like (localized periodic pulse train) structure and also large amplitude localized structures akin to rogue waves coexisting with solitons. For completeness, we have also obtained dark one- and two-soliton solutions and studied their dynamics briefly.

Elliptic waves in two-component long-wave–short-wave resonance interaction system in one and two dimensions

We consider (2+1) and (1+1) dimensional long-wave short-wave resonance interaction systems. We construct an extensive set of exact periodic solutions of these systems in terms of Lam\'e polynomials of order one and two. The periodic solutions are classified into three categories as similar, mixed, superposed elliptic solutions. We also discuss the hyperbolic solutions as limiting cases.

The existence of a global attractor for the [formula omitted]-dimensional long wave–short wave resonance interaction equation

In this paper, we prove the existence of a global attractor and an asymptotic smoothing effect of the solution for the (2+1)-dimensional long wave–short wave resonance interaction equation.

Inverse scattering transform for the Yajima-Oikawa equations with nonvanishing boundary conditions

Application of the Yajima-Oikawa (YO) equations describing the long wave--short wave resonance interaction to study of vector solitons of mixed bright-dark types in quasi-one-dimensional spinor Bose-Einstein condensates is considered. The inverse scattering transform for the YO equations with the mixed vanishing and nonvanishing boundary values is constructed. Examples of management of the high-frequency waves by changing initial and boundary values associated with the low-frequency waves are presented.

NOTE ON THE TWO-COMPONENT ANALOGUE OF TWO-DIMENSIONAL LONG WAVE – SHORT WAVE RESONANCE INTERACTION SYSTEM

AbstractAn integrable two-component analogue of the two-dimensional long wave – short wave resonance interaction (2c-2d-LSRI) system is studied. Wronskian solutions of 2c-2d-LSRI system are presented. A reduced case, which describes resonant interaction between an interfacial wave and two surface wave packets in a two-layer fluid, is also discussed.

The global solution of the (2+1)-dimensional long wave–short wave resonance interaction equation

Abstract

Application of the Weierstrass Elliptic Expansion Method to the Long-Wave and Short-Wave Resonance Interaction System

With the aid of symbolic computation, nine families of new doubly periodic solutions are obtained for the (2+1)-dimensional long-wave and short-wave resonance interaction (LSRI) system in terms of the Weierstrass elliptic function method. Moreover Jacobian elliptic function solutions and solitary wave solutions are also given as simple limits of doubly periodic solutions.

A new method applied to obtain complex Jacobi elliptic function solutions of general nonlinear equations

Based on computerized symbolic computation, a new complex Jacobi elliptic function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equation be the superposition of different powers of the Jacobi elliptic function. By finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex Jacobi elliptic function. The characteristic feature of this method is that, without transformation, we can derive exact solutions to the general nonlinear equations directly. Some illustrative equations, such as the (1 + 1)-dimensional Klein–Gordon–Zakharov equation, (2 + 1)-dimensional long-wave–short-wave resonance interaction equation, are investigated by this means to find the new exact solutions.

Two-component analogue of two-dimensional long wave–short wave resonance interaction equations: a derivation and solutions

The two-component analogue of two-dimensional long wave–short wave resonance interaction equations is derived in a physical setting. Wronskian solutions of the integrable two-component analogue of two-dimensional long wave–short wave resonance interaction equations are presented.

Application of dressing method for long wave–short wave resonance interaction equation

In this paper we investigate the application of the Zakharov-Shabat dressing method [Funct. Anal. Appl. 8, 226 (1974)] to (2+1)-dimensional long wave–short wave resonance interaction equation (LSRI). Using this method, we can construct the exact N-soliton solution of this equation depending on arbitrary constants. It contains both solutions which do not decay along N different directions in space infinity and “dromion” ones or localized solutions that decay exponentially in all directions.