Research paper
Dynamics of interaction between lumps and solitons in the Mel’nikov equation

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Abstract

Two families of semi-rational solutions to the Mel’nikov equation (ME), which is a known model of the interaction between long and short waves in two dimensions, are reported. These semi-rational solutions describe interaction between lumps and solitons. The first family of semi-rational solutions is derived by employing the KP-hierarchy reduction method. The fundamental (first-order) semi-rational solutions, consisting of a lump and a dark soliton, feature three different interactions, depending on speeds of the corresponding lump, Vlump, and soliton, Vsoliton: fusion of the lump into the dark soliton (at Vlump < Vsoliton); splitting of the lump from the soliton (at Vlump > Vsoliton); and lump-soliton bound states (at Vlump=Vsoliton). Three subclasses of non-fundamental semi-rational solutions, namely, higher-order, multi-, and mixed semi-rational solutions, are produced. These non-fundamental semi-rational solutions also represent different interaction: fusion of mutli-lumps into mutli-dark solitons, fission of multi-lumps from multi-dark solitons, multi-lump-soliton bound states, partial merger or partial splitting of lumps into/from lump-soliton bound states, etc. In particular, by selecting specific parameter constraints, the first family of semi-rational solutions reduces to solutions of a newly proposed partially spatial-reversed nonlocal ME. The second family of semi-rational solutions is constructed by using the Hirota method combined with a perturbative expansion and a long-wave limit, which describes a lump permanently propagating on the background of a dark soliton. The second family of the solutions indicates not all interactions between lumps and solitons in the ME give rise to fission or fusion. Besides that, a semi-rational solution to the coupled Schrödinger-Boussinesq equation, consisting of a rogue wave and a dark soliton, is obtained as a reduction of a semi-rational solution belonging to the second family.

Introduction

Nonlinear-wave interactions give rise to rich dynamics in nonlinear physical systems, including the formation of various nonlinear modes, such as breathers, solitons, and rogue waves. In particular, the interactions play an important role in many systems which support copropagation of short- and long-wave excitations. The interaction between them is possible because, while the frequencies and wavenumbers of the short and long waves are widely different, their phase and group velocities may be close, thus providing effective enhancement of the coupling. The fundamental model for the short-long wave coupling is provided by the Zakharov’s system, which, in the general form derived in the Zakharov’s work [1], includes the system composed of the linear Schrödinger equation for the short waves, nonlinearly coupled to the long-wave Boussinesq equation. In the Zakharov’s system, the coupling in the Schrödinger equation for the envelope of the short-wave component is represented by an effective potential term induced by the long-wave component [2], [3], [4]. In particular, for Langmuir waves in plasmas, the potential term represents variation of the plasma density caused by the long ion-sound waves. On the other hand, for short surface waves in stratified fluids the potential is induced by the flow current under the surface [5], [6], [7], [8]. Besides, the Boussinesq equation, which governs the propagation of long waves, includes a quadratic term induced by the short waves, which thus also plays the role of the coupling. Accordingly, the coupled Schrö dinger-Boussinesq (SB) system [1], [9], [10] takes the following explicit form in scaled units:iΦt=uΦ+Φxx,uxx+uxxxx+3(u2)xx3utt+κ(|Φ|2)xx=0,where Φ is a complex short-wave component, and u is a real long-wave one. The SB system is integrable, as it passes the Painlevé test [11] and possesses N-soliton solutions [9]. Hu et al. [12] have presented homoclinic-orbit solutions of Eq. (1) by using the Hirota method. Mu and Qin [13] have obtained fundamental rogue-wave (RW) solutions of the SB equations, also by means of the Hirota method. General higher-order RWs were derived in Ref. [14] by reducing τ functions of the Kadomtsev-Petviashvili (KP) hierarchy. Very recently, Song et al.[15] have proposed a two-component coupled SB equations and investigated their soliton and RW solutions. However, to the best of our knowledge, semi-rational RW and soliton solutions have not been reported before.

The Mel’nikov equations (ME),iΦy=uΦ+Φxx,uxt+uxxxx+3(u2)xx3uyy+κ(|Φ|2)xx=0,where κ is a real constant, u is a real long wave-wave filed, and Φ is a complex short-wave one, provides a (2+1)-dimensional version of the SB equations (1). Indeed, in the particular case ofΦ(x,y,t)=Φ(x,y,x),u(x,y,t)=u(x,y,x),Eq. (2) reduces to Eq. (1), with t replaced by y. ME (2) was first proposed by Mel’nikov [16], [17], [18], [19], also in the context of the interaction of long- and short-wave packets. This model is relevant as a two-dimensional one for the specific dispersion law corresponding to the linearization of Eq. (2). The ME may also be considered either as a generalization of the KP (Kadomtsev-Petviashvili) equation with the addition of the complex field, or as a generalization of the nonlinear Schr ödinger (NLS) equation, coupled to the real field [19]. Hase et al. [21] have derived general bright- and dark-solitons of Eq. (2). Lakshmanan et al. [20] and Mu et al. [22] have investigated, respectively, exponentially and rationally localized solutions of Eq. (2). Besides, a multi-component ME was proposed and investigated by Chen et al. [23], [24]. Several families of semi-rational solution to the ME equation were reported in [25], [26]. Sun and Wazwaz [25] exhibited the inelastic collisions between lumps and dark solitons, which give rise to several examples of excitation phenomena: fusion of lumps and dark line solitons into dark line solitons, and fission of dark line solitons into lumps and dark line solitons. Zhang and Chen investigated the interaction between of different types of localized waves [26], such as interaction between line rogue waves and line solitons, interaction between lumps and breathers.

Recently, Ablowitz and Musslimani [27] have introduced the nonlocal time-reversed NLS equationiqt(x,t)+qxx(x,t)+2q2(x,t)q(x,t)=0.This equation features the parity-time (PT) symmetry, i.e., invariance under the joint transformations, xx,tt. As noted by Bender et al. [28], [29], [30], [31], equations possessing the PT symmetry, including nonlocal equations similar to Eq. (4), find important realizations in physics, especially in optics [32], [33], [34], [35], [36], [37], [38], [39]. Although Eq. (4) is also a special reduction of the Ablowitz-Kaup-Newell-Segur hierarchy, its distinctive feature is nonlocality in t. In this context, Yang [40] investigated general N -soliton solutions of Eq. (4). In this work, we follow the works of Ablowitz and Musslimani [35] and Fokas [41], and propose a partially spatial-reversed nonlocal ME,iΦy=uΦ+Φxx,uxt+uxxxx+3(u2)xx3uyy+κ[ΦΦ(x,y,t)]xx=0.The objective of this paper is to investigate relations between solutions of the local ME (2) and nonlocal one (5). Semi-rational solutions of such equations, consisting of lumps and solitons have not been investigated before.

In Refs. [42], [43], Fokas et al. have derived semi-rational solutions describing the interaction of lumps and solitons for the Davey-Stewartson II (DSII) equation and the KPI equations by analyzing the inverse spectral problem for the associated Lax pair. In particular, these solutions are “generic”, i.e., they emerge from appropriate initial conditions as t → ∞. The dynamical behaviors of these solutions in the DSII and KPI equations are significantly different. In the DSII equation, the lump permanently propagates along the background represented by a line soliton. On the other hand, in the framework of the KPI equation, the lump initially propagates on the background of a line soliton. Later, the lump fuses into a bright soliton. Very recently, families of semi-rational solutions, describing the interaction of lumps and dark solitons in the multi-component long-wave–short-wave resonantly-interaction (LSRI) system, have been obtained in Ref. [45]. These solutions give rise to two different types of inelastic interactions: fusion of dark solitons and lumps into dark solitons (i.e., absorption of lumps by the dark solitons), and fission of dark solitons into lumps and dark solitons (i.e., lumps originating from dark solitons). Before that, similar semi-rational solutions were studied in Ref. [44] for the third-type Davey–Stewartson (DSIII) equation, which only describe the process of fission of dark solitons into lumps and dark solitons. However, in the semi-rational solutions of the multi-component long-wave–short-wave resonance-interaction system and the DSIII equation, only the case of lumps and solitons traveling in opposite directions was considered, and speeds of the lumps and solitons did not affect the interaction of the lumps and solitons. Yuan et.al. studied the full degeneration process of the high-order breathers to high–order lumps in the KPI equation [46]. Furthermore, when the lumps and solitons move in the same direction, different interactions may be expected for different speeds of the lumps and solitons. Motivated by these facts, we consider semi-rational solutions of ME (2), addressing the following aspects:

  • When the lumps and solitons move in the same direction, how is their interaction affected by their speeds?

  • Do the interaction between lumps and dark soliton always lead to outcomes such as fusion of lumps and dark solitons into dark solitons, and fission of dark solitons into lumps and dark solitons?

  • Can solutions of the partially spatial-reversed nonlocal ME (5) be obtained from solutions of the underlying ME (2)?

  • Does the SB equation (1) admit mixed solutions consisting of RWs and solitons? This kind of solutions has not been reported for the scalar (single-component) NLS equation.

The rest of the paper is organized as follows. In Section 2, the first family of semi-rational solutions of ME (2) is derived by employing the KP hierarchy reduction method. These semi-rational solutions describe interactions of lumps and dark solitons leading to remarkable phenomena, such as fission, fusion, and lump-soliton bound states, depending on the speeds of the lumps and dark solitons. Besides that, upon selecting particular constraints for parameters, these solutions also satisfy the spatial-reversed nonlocal ME (5). In Section 3, the second family of semi-rational solutions, built of a lump and a soliton, is derived by means of the Hirota’s bilinear method combined with a perturbative expansion and a long-wave limit. In these solutions, the interaction between the lump and dark soliton does not give rise to fission or fusion. Under particular parameters constraints, these solutions reduce to mixed ones, composed of a RW and a dark soliton of the SB equation (1). Section 4 contains a summary and discussions.

Section snippets

Semi-rational solutions i

To apply the KP hierarchy reduction method for constructing semi-rational solutions of the ME (2), we employ the dependent-variable transformation,Φ=2gf,u=(2lnf)xx,and allow for solutions Φ and u satisfying asymptotic conditionsΦ2,u0atx,y,t,where f(x, y, t) and g(x, y, t) are, respectively, real and complex functions. Obviously, Φ=2,u=0 is a constant solution of ME (2), and under transformation (6), the ME (2) is cast into the following bilinear form :(Dx2iDy)g·f=0,(Dx4+DxDt3Dy2)f·f=2k(f2

Semi-rational solutions II

In Section 2, the interaction between the lump and dark soliton always leads to either of the two outcomes: fusion of the lump into the dark soliton, or splitting of the lump from the dark soliton. In this section, we construct another kind of semi-rational solutions describing interaction of a lump and a dark soliton without energy exchange between them.

We first start from the three-soliton solution, and take functions f and g in Eq. (6) asf=1+ϵf1+ϵ2f2+ϵ3f3,g=1+ϵg1+ϵ2g2+ϵ3g3,withf1=eζ1+eζ2+eζ3,

Conclusion

In this paper, we have investigated the ME (Mel’nikov equation), which models nonlinear resonant interaction of long and short waves in two spatial dimensions, providing a (2 + 1)-dimensional extension of the coupled SB (Schrödinger-Boussinesq) equations. We have also proposed a partially spatial-reversed nonlocal ME with the PT symmetry, which may find realizations in optics. Two families of semi-rational solutions to the ME, consisting of lumps and solitons, have been constructed. The first

CRediT authorship contribution statement

Jiguang Rao: Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft. Boris A. Malomed: Writing - review & editing. Yi Cheng: Formal analysis, Writing - review & editing. Jingsong He: Conceptualization, Formal analysis, Project administration, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors J. Rao and J. He thank Prof. D. Zuo of University of Science and Technology of China for helpful discussions. This work is supported by the NSF of China under Grant Nos. 11671219 and 11871446.

References (55)

  • E.G. Fan

    Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method

    J Phys A: Math Gen

    (2002)
  • P. Janssen

    The interaction of ocean waves and wind

    (2009)
  • R.J. Wahl et al.

    Estimation of brunt-väisälä frequency from temperature profiles

    Oceanography

    (1983)
  • C. Kharif et al.

    The modulational instability in deep water under the action of wind and dissipation

    J Fluid Mech

    (2010)
  • Y. Hase et al.

    An n-soliton solution for the nonlinear schrödinger equation coupled to the boussinesq equation

    J Phys Soc Jpn

    (1988)
  • N.N. Rao

    Near-magnetosonic envelope upper-hybrid waves

    J Plasma Physics

    (1988)
  • X.B. Hu et al.

    Hon-wah tam, homoclinic orbits for the coupled schrödinger–boussinesq equation and coupled higgs equation

    J Phys Soc Jpn

    (2008)
  • G. Mu et al.

    Rogue waves for the coupled Schrödinger–Boussinesq equation and coupled higgs equation

    J Phys Soc Jpn

    (2012)
  • Zhang X.E., Chen J.C., Chen Y.. General high-order rogue wave to NLS-boussinesq equation with the dynamical analysis....
  • C.Q. Song et al.

    Soliton and rogue wave solutions of two-component nonlinear schrödinger equation coupled to the boussinesq equation

    Chin Phys B

    (2017)
  • V.K. Mel’nikov

    On equations for wave interactions

    Lett Math Phys

    (1983)
  • V.K. Mel’nikov

    Reflection of waves in nonlinear integrable systems

    J Math Phys

    (1987)
  • V.K. Mel’nikov

    A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the xy plane

    Commum Math Phys

    (1987)
  • C.S. Kumar et al.

    Exponentially localized solutions of mel’nikov equation

    Chaos, Solitons, Fractals

    (2004)
  • Y. Hase et al.

    Soliton solutions of the me’lnikov equations

    J Phys Soc Jpn

    (1989)
  • Z. Han et al.

    Bright–dark mixed n–soliton solutions of the multi-component mel’nikov system

    J Phys Soc Jpn

    (2017)
  • Z. Han et al.

    General n–dark soliton solutions of the multi–component mel’nikov system

    J Phys Soc Jpn

    (2017)
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