General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system

General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system

Junchao Chen 11footnotemark: 122footnotemark: 2 Yong Chen 33footnotemark: 3 Bao-Feng Feng 44footnotemark: 4 Ken-ichi Maruno 55footnotemark: 5 Yasuhiro Ohta 66footnotemark: 6 Department of Mathematics, Lishui University, Lishui, 323000, People¡¯s Republic of China Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, People’s Republic of China Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
Abstract

General high-order rogue wave solutions for the (1+1)-dimensional Yajima-Oikawa (YO) system are derived by using Hirota’s bilinear method and the KP-hierarchy reduction technique. These rogue wave solutions are presented in terms of determinants in which the elements are algebraic expressions. The dynamics of first and higher-order rogue wave are investigated in details for different values of the free parameters. It is shown that the fundamental (first-order) rogue waves can be classified into three different patterns: bright, intermediate and dark ones. The high-order rogue waves correspond to the superposition of fundamental rogue waves. Especially, compared with the nonlinear Schödinger equation, there exists an essential parameter to control the pattern of rogue wave for both first- and high-order rogue waves since the YO system does not possess the Galilean invariance.

keywords:
Yajima-Oikawa system, high-order rogue wave, bilinear method, KP-hierarchy reduction
\biboptions

numbers,sort&compress

1 Introduction

Rogue waves, which are initially used for the vivid description of the spontaneous and monstrous ocean surface waves kharif2009rogue (), have recently attracted considerable attention on both experimentally and theoretically. Rogue waves have been observed in a variety of different fields, including optical systems solli2007optical (); hohmann2010freak (); montina2009non (), Bose–Einstein condensates bludov2009matter (); bludov2010vector (), superfluids ganshin2008observation (), plasma moslem2011langmuir (); bailung2011observation (), capillary waves shats2010capillary () and even in finance yan2011vector (). Compared with the stable solitons, rogue waves are the localized structures with the instability and unpredictability akhmediev2009waves (); akhmediev2009extreme (). A typical model for characterizing the rogue wave is the celebrated nonlinear Schrödinger (NLS) equation. The most fundamental rogue wave of the NLS equation is described by Peregrine soliton peregrine1983water (), which is first-order rogue wave and expressed in a simple rational form including the polynomials up to second order. This rational solution has localized behavior in both space and time, and its maximum amplitude attains three times the constant background. The Peregrine soliton can be obtained from a breather solution when the period is taken to infinity. More recently, significant progress on higher order rogue waves has been achieved akhmediev2009rogue (); kedziora2012second (); ankiewicz2011rogue (); kedziora2011circular (); dubard2010multi (); dubard2011multi (); gaillard2011families (); guo2012nonlinear (); ohta2012general (); ankiewicz2010rogue (); he2010generating (); mu2016dynamic (); ling2016multisoliton (); wang2017dynamics (); chen2015rational (); bludov2009rogue (); ankiewicz2010discrete (); ohta2014general (); yan2010nonautonomous (); yan2010threedimensional (); yan2010optical (); wen2015generalized (); yan2015twodimensional (); yang2015rogue (); wen2016dynamics (); wen2017higherorder () since a few special higher order rogue waves from first to fourth order were provided theoretically by Akhmediev et al.akhmediev2009rogue () via the Darboux transformation method. The higher order rogue waves were also excited experimentally in a water wave tank chabchoub2013observation (); chabchoub2012observation (), which guarantees that such nonlinear complicated waves are meaningful physically. In fact, higher-order rogue waves can be treated as the nonlinear superposition of fundamental rogue wave and they are usually expressed in terms of complicated higher-order rational polynomials. These higher-order waves were also localized in both coordinates and could exhibit higher peak amplitudes or multiple intensity peaks.

Another major development of importance is the study of rogue waves in multicomponent coupled systems, as a lot of complex physical systems usually contain interacting wave components with different modes and frequenciesling2012highorder (); guo2011roguewave (); baronio2012solutions (); zhao2013roguewave (); baronio2013solutions (); zhao2016high (); ling2016darboux (); mu2015dynamics (); zhai2013multirogue (); wang2015roguewave (); zhang2017solitons (); ohta2012rogue (); ohta2013dynamics (); mu2017solitons (); wen2015modulational (); wen2016higherorder (). As stated in Ref.ling2012highorder (), the cross-phase modulation term in coupled systems leads to the varying instability regime characters. Due to the additional degrees of freedom, there exist more abundant pattern structures and dynamics characters for rogue waves in coupled systems. For instance, in the scalar NLS equation, because the existence of Galilean invariance, the velocity of the background field does not influence the pattern of rogue waves. However, for the coupled NLS system, the relative velocity between different component fields has real physical effects, and cannot be removed by any trivial transformation. This fact bring some novel patterns for rogue waves such as dark rogue waves guo2011roguewave (), the interaction between rogue waves and other nonlinear waves guo2011roguewave (); baronio2012solutions (), a four-petaled flower structure zhao2013roguewave () and so on. In particular, those more various higher order rogue waves in coupled nonlinear models enrich the realization and understanding of the mechanisms underlying the complex dynamics of rogue waves.

Among coupled wave dynamics systems, the long-wave-short-wave resonance interaction (LSRI) is a fascinating physical process in which a resonant interaction takes place between a weakly dispersive long-wave (LW) and a short-wave (SW) packet when the phase velocity of the former exactly or almost matches the group velocity of the latter. The theoretical investigation of this LSRI was first done by Zakharov zakharov1972collapse () on Langmuir waves in plasma. In the case of long wave propagating in one direction, the general Zakharov system was reduced to the one-dimensional (1D) Yajima-Oikawa (YO) systemyajima1976formation (). This phenomenon has been predicted in diverse areas such as plasma physics zakharov1972collapse (); yajima1976formation (), hydrodynamics grimshaw1977modulation (); djordjevic1977two (); ma1979some () and nonlinear optics kivshar1992stable (); chowdhury2008long (). For instance, this resonance interaction can occur between the long gravity wave and the capillary-gravity one djordjevic1977two (), and between long and short internal waves grimshaw1977modulation () in hydrodynamics. In a second-order nonlinear negative refractive index medium, it can be achieved when the short wave lies on the negative index branch while the long wave resides in the positive index branch chowdhury2008long (). The (1+1) dimensional model equation, which is known as 1D YO system or LSRI system, can be written in a dimensionless form

 iSt−Sxx+SL=0, (1) Lt=−4(|S|2)x, (2)

where and represent the short wave and long wave component, respectively. The 1D YO system was shown to be integrable with a Lax pair, and was solved by the inverse scattering transform method yajima1976formation (). It admits both bright and dark soliton solutions ma1979some (); ma1978complete (). In Refs.cheng1992constraints (); loris1997bilinear (), it is shown that the 1D YO system can be derived from the so-called -constrained KP hierarchy with while the NLS equation with . Very recently, the first-order rogue wave solutions to the 1D YO system have been derived by using the Hirota¡¯s bilinear method wing2013rogue () and Darboux transformation chen2014dark (); chen2014darboux (). These vector parametric solutions indicate interesting structures that the long wave always keeps a single hump structure, whereas the short-wave field can be manifested as bright, intermediate and dark rogue wave. Nevertheless, as far as we know, there is no report about high-order rogue wave solutions for the 1D YO system. Therefore, it is the objective of present paper to study high-order rogue wave solutions of the 1D YO system (1)–(2) by using the bilinear method in the framework of KP-hierarchy reduction. As will be shown in the subsequent section, a general rogue wave solutions in the form of Gram determinant is derived based on Hirota’s bilinear method and the KP-hierarchy reduction technique. This determinant solution can generate rogue waves of any order without singularity.

The remainder of this paper is organized as follows. In Section 2, we start with a set of bilinear equations satisfied by the functions in Gram determinant of the KP hierarchy, and reduce them to bilinear equations satisfied by the 1D YO system (1)–(2). The reductions include mainly dimension reduction and complex conjugate reduction. We should emphasize here that the most crucial and difficult issue is to find a general algebraic expression for the element of determinant such that the dimension reduction can be realized. In Section 3, the dynamical behaviors of fundamental and higher-order rogue wave solutions are illustrated for different choices of free parameters. The paper is concluded in Section 4 by a brief summary and discussion.

2 Derivation of general rogue wave solutions

This section is the core of the present paper, in which an explicit expression for general rogue wave solutions of the 1D YO system (1)–(2) will be derived by Hirota’s bilinear method. To this end, let us first introduce dependent variable transformations

 S=ei[αx+(h+α2)t]gf,  L=h−2∂2∂x2logf, (3)

where is a real-valued function, is a complex-valued function and and are real constants. Then the 1D YO system (1)-(2) is converted into the following bilinear equations

 (D2x+2iαDx−iDt)g⋅f=0, (4) (DxDt+4)f⋅f=4gg∗, (5)

where denotes the complex conjugation hereafter and the is Hirota’s bilinear differential operator defined by

 DnxDmt(a⋅b)=(∂∂x−∂∂x′)n(∂∂t−∂∂t′)ma(x,t)b(x′,t′)∣∣∣x=x′,t=t′.

Prior to the tedious process in deriving the polynomial solutions of the functions and , we highlight the main steps of the detailed derivation, as shown in the the subsequent subsections.

Firstly, we start from the following bilinear equations of the KP hierarchy:

which admit a wide class of solutions in terms of Gram or Wronski determinant. Among these determinant solutions, we need to look for algebraic solutions to satisfy the reduction condition:

 (∂x2+2i∂ta)τn=cτn, (8)

such that these algebraic solutions satisfy the (1+1)-dimensional bilinear equations:

Furthermore, by introducing the variable transformations:

 x1=x,  x2=−it, (11)

and taking , , , and , the above bilinear equations (9)-(10) become

 (D2x+2iαDx−iDt)g⋅f=0, (12) (DxDt+4)f⋅f=4gh. (13)

Lastly, by requiring the real and complex conjugation condition:

 f=τ0:real,  g=τ1,  h=τ−1=g∗, (14)

in the algebraic solutions, then the bilinear equations (12)-(13) are reduced to the bilinear equations (4)–(5), hence the general higher-order rogue wave solutions are obtained through the reductions.

2.1 Gram determinant solution for the bilinear equations in KP hierarchy

In this subsection, through the Lemma below, we present and prove the a pair of bilinear equations satisfied by the functions of the KP hierarchy.

Lemma 2.1 Let , depending on and , be function of the variables , and , and satisfy the following differential and difference relations:

 ∂x1m(n)ij=φ(n)iψ(n)j, ∂x2m(n)ij=[∂x1φ(n)i]ψ(n)j−φ(n)i[∂x1ψ(n)j], ∂tam(n)ij=−φ(n−1)iψ(n+1)j, (15) m(n+1)ij=m(n)ij+φ(n)iψ(n+1)j,

where and are functions satisfying

 ∂x2φ(n)i=∂2x1φ(n)i,  φ(n+1)i=(∂x1−a)φ(n)i,  ∂x2ψ(n)j=−∂2x1ψ(n)j,  ψ(n−1)i=−(∂x1+a)ψ(n)j. (16)

Then the functions of the following determinant form

 τn=det1≤i,j≤N(m(n)ij), (17)

satisfy the following bilinear equations (6) and (7) in KP hierarchy:

Proof: By using the differential formula of determinant

 ∂xdet1≤i,j≤N(aij)=N∑i,j=1Δij∂xaij, (20)

and the expansion formula of bordered determinant

 det(aijbicjd)=−N∑i,jΔijbicj+ddet(aij), (21)

with being the -cofactor of the matrix , one can check that the derivatives and shifts of the function are expressed by the bordered determinants as follows:

 ∂x2τn=∣∣ ∣∣m(n)ij∂x1φ(n)i−ψ(n)j0∣∣ ∣∣−∣∣ ∣∣m(n)ijφ(n)i−∂x1ψ(n)j0∣∣ ∣∣,  ∂taτn=∣∣ ∣∣m(n)ijφ(n−1)iψ(n+1)j0∣∣ ∣∣, (∂x1∂ta−1)τn=∣∣ ∣ ∣ ∣∣m(n)ijφ(n−1)iφ(n)iψ(n+1)j0−1−ψ(n)j−10∣∣ ∣ ∣ ∣∣, (∂x1+a)2τn+1=∣∣ ∣∣m(n)ij∂2x1φ(n)i−ψ(n+1)ja2∣∣ ∣∣+∣∣ ∣ ∣ ∣∣m(n)ij∂x1φ(n)iφ(n)i−ψ(n+1)ja1−ψ(n)j00∣∣ ∣ ∣ ∣∣, (∂x2+a2)τn+1=∣∣ ∣∣m(n)ij∂2x1φ(n)i−ψ(n+1)ja2∣∣ ∣∣−∣∣ ∣ ∣ ∣∣m(n)ij∂x1φ(n)iφ(n)i−ψ(n+1)ja1−ψ(n)j00∣∣ ∣ ∣ ∣∣.

With the help of these relations, one has

 (∂x1∂ta−1)τn×τn−∂x1τn×∂taτn+(−τn−1)(−τn+1) (22) 12(∂2x1+2a∂x1−∂x2)τn+1×τn−(∂x1+a)τn+1×∂x1τn+τn+1×12(∂2x1+∂x2)τn =∣∣ ∣ ∣ ∣∣m(n)ij∂x1φ(n)iφ(n)i−ψ(n+1)ja1−ψ(n)j00∣∣ ∣ ∣ ∣∣×∣∣m(n)ij∣∣−∣∣ ∣∣m(n)ijφ(n)i−ψ(n)j0∣∣ ∣∣×∣∣ ∣∣m(n)ij∂x1φ(n)i−ψ(n+1)ja∣∣ ∣∣+∣∣ ∣∣m(n)ij∂x1φ(n)i−ψ(n)j0∣∣ ∣∣×∣∣ ∣∣m(n)ijφ(n)i−ψ(n+1)j1∣∣ ∣∣. (23)

The r.h.s of both (2.1) and (2.1) are identically zero because of the Jacobi identity and hence the functions (17) satisfy the bilinear equations (18) and (19). This completes the proof.

2.2 Algebraic solutions for the (1+1)-dimensional YO system

This subsection is crucial in the KP-hierarchy reductions. We will construct an algebraic expression for the elements of function of preceding subsection so that the dimension reduction condition (8) is satisfied. The main result is given by the following Lemma.

Lemma 2.2 Suppose the entries of the matrix are

 m(νμn)kl=(A(ν)kB(μ)lm(n))∣∣p=ζ,q=ζ∗, (24)

where

 m(n)=1p+q(−p−aq+a)neξ+η,  ξ=px1+p2x2,  η=qx1−q2x2, ζ=ζr+iζi,  ζr=±√312K2−4α2K,  ζi=112K+13α2K+23α,
 K=(8α3+108+12√12α3+81)1/3,   (α>−3221/3),

and and are differential operators with respect to and , respectively, given by

 A(ν)n=n∑k=0a(ν)k[(p−a)∂p]n−k(n−k)!,n≥0, (25) B(ν)n=n∑k=0b(ν)k[(q+a)∂q]n−k(n−k)!,n≥0, (26)

where and are constants satisfying the iterated relations

 a(ν+1)k=k∑j=02j+2(p−a)2+(−1)j2ip−a+2a(p−a)(j+2)!a(ν)k−j,  ν=0,1,2,⋯, (27) b(ν+1)k=k∑j=02j+2(q+a)2−(−1)j2iq+a−2a(q+a)(j+2)!b(ν)k−j,  ν=0,1,2,⋯, (28)

then the determinant

 τn=det1≤i,j≤N(m(N−i,N−j,n)2i−1,2j−1)=∣∣ ∣ ∣ ∣ ∣ ∣∣m(N−1,N−1,n)11m(N−1,N−2,n)13⋯m(N−1,0,n)1,2N−1m(N−2,N−1,n)31m(N−2,N−2,n)33⋯m(N−2,0,n)3,2N−1⋮⋮⋮m(0,N−1,n)2N−1,1m(0,N−2,n)2N−1,3⋯m(0,0,n)2N−1,2N−1∣∣ ∣ ∣ ∣ ∣ ∣∣, (29)

satisfies the bilinear equations

Proof. Firstly, we introduce the functions , and of the form

 ~m(n)=1p+q(−p−aq+a)ne~ξ+~η,  ~φ(n)=(p−a)ne~ξ,  ~ψ(n)=(−1q+a)ne~η,

where

 ~ξ=1p−ata+px1+p2x2,  ~η=1q+ata+qx1−q2x2.

These functions satisfy the differential and difference rules:

 ∂x1~m(n)=~φ(n)~ψ(n), ∂x2~m(n)=[∂x1~φ(n)]~ψ(n)−~φ(n)[∂x1~ψ(n)], ∂ta~m(n)=−~φ(n−1)~ψ(n+1), ~m(n+1)=~m(n)+~φ(n)~ψ(n+1),

and

 ∂x2~φ(n)=∂2x1~φ(n),  ~φ(n+1)=(∂x1−a)~φ(n),  ∂x2~ψ(n)=−∂2x1~ψ(n),  ~ψ(n−1)=−(∂x1+a)~ψ(n).

We then define

 ~m(νμn)ij=A(ν)iB(μ)j~m(n),  ~φ(νn)i=A(ν)i~φ(n),  ~ψ(μn)j=B(μ)j~ψ(n). (32)

Since the operators and commute with differential operators , and , these functions , and obey the differential and difference relations as well (2.1)-(16). From Lemma 2.1, we know that for an arbitrary sequence of indices , the determinant

 ~τn=det1≤k,l≤N(~m(νk,μl,n)ik,jl)

satisfies the bilinear equations (18) and (19), for instance, the bilinear equations (18) and (19) hold for with arbitrary parameters and . Based on the Leibniz rule, one has,

 [(p−a)∂p]m(p2+2ip−a) = m∑l=0(ml)[2l(p−a)2+(−1)l2ip−a+2a(p−a)][(p−a)∂p]m−l (33) +a2[(p−a)∂p]m,

and

 [(q+a)∂q]m(q2−2iq+a) = m∑l=0(ml)[2l(q+a)2−(−1)l2iq+a−2a(q+a)][(q+a)∂q]m−l (34) +a2[(q+a)∂q]m.

Furthermore, one can derive

 [A(ν)n,p2+2ip−a]=n−1∑k=0a(ν)k(n−k)![((p−a)∂p)n−k,p2+2ip−a] =n−1∑k=0a(ν)k(n−k)!n−k∑l=1(n−kl)(2l(p−a)2+(−1)l2ip−a+2a(p−a))((p−a)∂p)n−k−l,

where is the commutator defined by .

Let be the solution of the cubic equation

 2(p−a)2−2ip−a+2a(p−a)=0,

with , then has an explicit expression given previously. Hence we have

 [A(ν)n,p2+2ip−a]∣∣∣p=ζ=0,

for and

 =n−2∑k=0a(ν)k(n−k)!n−k∑l=2(n−kl){2l(p−a)2+(−1)l2ip−a+2a(p−a)}((p−a)∂p)n−k−l∣∣ ∣∣p=ζ =n−2∑k=0n−k−2∑j=0a(ν)k(j+2)!(n−k−j−2)!{2j+2(p−a)2+(−1)j2ip−a+2a(p−a)}((p−a)∂p)n−k−j−2∣∣ ∣∣p=ζ =n−2∑^k=0⎛⎜⎝^k∑^j=02^j+2(p−a)2+(−1)^j2ip−a+2a(p−a)(^j+2)!a(ν)^k−^j⎞⎟⎠((p−a)∂p)n−2−^k(n−2−^k)!∣∣ ∣∣p=ζ =n−2∑^k=0a(ν+1)^k((p−a)∂p)n−2−^k(n−2−^k)!∣∣ ∣∣p=ζ =A(ν+1)n−2∣∣p=ζ,

for . Thus the differential operator satisfies the following relation

 (35)

where we define for .

Similarly, it is shown that the differential operator satisfies

 (36)

where we define for .

Consequently, by referring to above two relations, we have

 (∂x2+2i∂ta)~m(νμn)kl∣∣p=ζ,q=ζ∗ = [A(ν)kB(μ)l(∂x2+2i∂ta)~m(n)]∣∣p=ζ,q=ζ∗ = (A(ν)kB(μ)l(p2−q2+2i(1p−a+1q+a))~m(n))∣∣∣p=ζ,q=ζ∗ = (A(ν)k(p2+2ip−a)B(μ)l~m(n))∣∣∣p=ζ,q=ζ∗−(A(ν)kB(μ)l(q2−2iq+a)~m(n))∣∣∣p=ζ,q=ζ∗ = {((p2+2ip−a)A(ν)k+A(ν+1)k−2)B(μ)l~m(n)}∣∣∣p=ζ,q=ζ∗ −{A(ν)k((q2−2iq+a)B(μ)l+B(μ+1)l−2)~m(n)}∣∣∣p=ζ,q=ζ∗ = (ζ2+2iζ−iα)~m(νμn)kl∣∣p=ζ,q=ζ∗+~m(ν+1,μ,n)k−2,l∣∣p=ζ,q=ζ∗−(ζ∗2−2iζ∗+iα)~m(νμn)kl∣∣p=ζ,q=ζ∗−~m(ν,μ+1,n)k,l−2∣∣p=ζ,q=ζ∗.

By using the formula (20) and the above relation, the differential of the following determinant

 ~~τn=det1≤i,j≤N(~m(N−i,N−j,n)2i−1,2j−1∣∣p=ζ,q=ζ∗)

can be calculated as

 (∂x2+2i∂ta)~~τn = N∑i=1N∑j=1Δij(∂x2+2i∂ta)(~m(N−i,N−j,n)2i−1,2j−1∣∣p=ζ,q=ζ∗) = N∑i=1N∑j=1Δij[(ζ2+2iζ−iα)~m(N−i,N−j,n)2i−1,2j−1∣∣p=ζ,q=ζ∗+m(N−i+1,N−j,n)2i−3,2j−1∣∣p=ζ,q=ζ∗ −(ζ∗2−2iζ∗+iα)~m(N−i,N−j,n)2i−1,2j−1∣∣p=ζ,q=ζ∗−m(N−i,N−j+1,n)2i−1,2j−3∣∣p=ζ,q=ζ∗] = (ζ2+2iζ−iα)N~~τn+N∑i=1N∑j=1Δij~m(N−i+1,N−j,n)2i−3,2j−1∣∣p=ζ,q=ζ∗−(ζ∗2−2iζ∗+iα)N~~τn−N∑i=1N∑j=1Δij~m(N−i,N−j+1,n)2i−1,2j−3∣∣p=ζ,q=ζ∗,

where is the -cofactor of the matrix . For the term