# Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

Authors: Wei Liu; Zhenyun Qin; Kwok Wing Chow; Senyue Lou
Journal: Complexity
 (2020)

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## Abstract

Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) are widely applicable in a variety of intriguing phenomena in physical sciences and engineering, e.g., biophysics, condensed matter, fluids, optics, particle dynamics, and plasma physics [1–8]. Searching for exact solutions of NLEEs is critically important in these applications, as such solutions offer rich knowledge and penetrating insight for the phenomena being modelled by the NLEEs. Many ingenious methods have been devised to solve these NLEEs analytically. Examples include, but are not limited to, the Darboux transformation [9, 10], Hirota bilinear method [11], homogeneous balance method [12, 13], inverse scattering transform [14, 15], Lie group analysis [16, 17], and many other schemes [18–20].Most of the NLEEs studied in the literature are local equations where the evolution depends only on the local value of the dependent variable and its local space and time derivatives. Nonlocal evolution equations arise in many fields of applications too, where the nonlocal nature frequently comes from a dependence on the global properties, e.g., an integral of the dependent variable [21–23]. Physical scenarios include modulation of nonlinear waves (Whitham’s equation) [21] and reaction diffusion systems [22, 23]. Aside from the global operator, the linear portion of the differential equation may be of the forms of the Duffing, heat, or inviscid Burgers equations. Here, we consider another class of nonlocal evolution equations where the intensity of the wave motion depends on values of the dependent variable at points in the far field. Physically, the significance is that this potential displays a parity time symmetry property studied also intensively in quantum mechanics. Recently, Ablowitz and Musslimani [24] have introduced a nonlocal nonlinear Schrödinger (NLS) equation:(1)qtx,t−iqxxx,t±2iVx,tqx,t=0,Vx,t=qx,tq∗−x,t,which incorporates a parity time (PT) symmetric potential V. Fokas extended the nonlocal NLS equation into multidimensional versions and proposed a new integrable nonlocal Davey–Stewartson (DS) equation [25]:(2)iAt=Axx+Ayy+ϵV−2QA,Qxx−Qyy=ϵVxx,where(3)V=Ax,y,tA−x,−y,t∗,ε=±1,orV=Ax,y,tA−x,y,t∗,ε=±1.Subsequently, several new nonlocal integrable equations were also investigated [26–43].Here, the focus is on a set of nonlinear equations which exhibits partial reverse space time in multidimensional setting. Inspired by the works of Ablowitz, Mussliman, and Fokas, we propose a partial reverse space-time nonlocal Mel′nikov equation:(4)3uyy−uxt−3u2+uxx+κϕϕ∗−x,y,−txx=0,iϕy=uϕ+ϕxx,where u and ϕ are functions of x,y,andt. Obviously, by replacing ϕ∗−x,y,−t with ϕ∗x,y,t, the nonlocal Mel′nikov equation reduces to the usual Mel′nikov equation:(5)3uyy−uxt−3u2+uxx+κϕ2xx=0,iϕy=uϕ+ϕxx.Here, realu is the long wave component and ϕ is the complex valued short wave envelope. As noted by Mel′nikov [44–47], this equation may describe an interaction of long waves with short wave packets, and it could also be considered either as a generalization of the Kadomtsev–Petviashvili (KP) equation with the addition of a complex scalar field or as a generalization of the nonlinear Schrödinger (NLS) equation with a real scalar field [48]. High-order soliton solutions for this equation were calculated by Hase et al. [49]. Rogue wave solutions were derived by Mu and Qin [50], and general N-dark soliton solutions of the multicomponent Mel′nikov system were investigated by Han et al. [51]. Under the variable transformations,(6)ξ=x+t,η=y,τ=t,κ=−1,on neglecting the τ-dependence and rewriting ξ⟶x,η⟶t, the nonlocal Mel′nikov equation reduces to the nonlocal Schrödinger–Boussinesq equation:(7)3utt−uxx−3u2+uxx−ϕϕ∗−x,txx=0,iϕt=uϕ+ϕxx.Rogue wave solutions of the conventional local Schrödinger–Boussinesq equation have been studied by Mu and Qin [52].In recent works [37, 38], 2+1-dimensional breather, rational, and semirational solutions of the partially and fully parity-time (PT) symmetric nonlocal DS equations have been reported. Thus, it is natural to seek exact solutions for other partial reverse space-time nonlocal equations too. In this work, we derive families of rational and semirational solutions to the partial reverse space-time nonlocal Mel′nikov equation (4) by using the Hirota bilinear method.This paper is organized as follows. In Section2, soliton and breather solutions are derived by employing Hirota’s bilinear method. The unusual feature is that both dependent variables are allowed to be complex valued, leading to novel structures in the auxiliary scalar field. In Section 3, the main theorem on the rational solutions is established, and typical features of these rational solutions are demonstrated. In Section 4, semirational solutions consisting of lumps, breathers, and periodic line waves are generated, and their novel dynamics is also elucidated. In Section 5, doubly periodic solutions are computed in terms of theta and elliptic functions. Our results are summarized in Section 6.


## 2. Soliton and Breather Solutions of the Nonlocal Mel′nikov Equation

To activate the Hirota bilinear method, we formulate the following transformation and allow for nonzero asymptotic conditionϕ,u⟶1,0 as x,y,t⟶∞:(8)ϕ=gf,u=2logfxx,where fandg are functions with respect to three variables x,y, and t, which satisfy the condition(9)f∗−x,y,−t=fx,y,t.In other words,f can be complex. This is in strong contrast with the local Mel′nikov equation where the auxiliary scalar field must be real. Equation (4) is written in the following bilinear form:(10)Dx2−iDyg⋅f=0,Dx4+DxDt−3Dy2f⋅f=κf2−gg∗−x,y,−t.Here, the operatorD is the Hirota bilinear differential operator [11] defined by(11)PDx,Dy,Dt,Fx,y,t,…⋅Gx,y,t,…=P∂x−∂x′,∂y−∂y′,∂t−∂t′,…Fx,y,t,…Gx′,y′,t′,…x′=x,y′=y,t′=t,where P is a polynomial of Dx,Dy,Dt,….Following a well-established procedure [11], to obtain N-soliton solutions of the nonlocal Mel′nikov equation, we expand g and f as power series of a small parameter ϵ:(12)g=1+εg1+ε2g2+ε3g3+⋯+εNgN,f=1+εf1+ε2f2+ε3f3+⋯+εNfN,with truncations leading to exact solutions:(13)f=∑μ=0,1exp∑k<jNμkμjAkj+∑k=1Nμkηk,g=∑μ=0,1exp∑k<jNμkμjAkj+∑k=1Nμkηk+iϕk,where(14)ηj=iPjx+Qjy+iΩjt+ηj0,expiϕj=−Pj4−Qj2Pj4+Qj2+i2Pj2QjPj4+Qj2,expAjk=−κ−κcosϕj−ϕk+Pj−PkΩj−Ωk−Pj−Pk4+3Qj−Qk2κ−κcosϕj+ϕk+Pj+PkΩj+Ωk−Pj+Pk4+3Qj+Qk2,and the dispersion relation is(15)ΩjPjPj4+Qj2+Pj4Pj4−2Qj2−2κ−3Qj4=0.Here,PjandQj are real parameters and ηk0 is an arbitrary complex constant. The notation ∑μ=0,1 indicates summation over all possible combinations of μ1=0,1,μ2=0,1,…,μN=0,1; the ∑j<kN summation is over all possible combinations of the N elements with the specific condition j<k.Soliton solutions of the nonlocal DS equations may have singularities [37, 38]. Here, by suitable constraints on the parameters Pj,Qj,andηj0 in equation (13),(16)N=2n,Pn+j=−Pj,Qn+j=Qj,ηn+j0=ηj0∗.General nonsingularn-breather solutions can be generated. For instance, with N=2 and parameter choices,(17)P1=−P2=P,Q1=Q2=Q,η10=η20=η0,where P,Q,andη0 are real, the one breather can be expressed in terms of hyperbolic and trigonometric functions as(18)ϕ=g0f0,u=2logf0xx,where(19)f0=McoshΘ+cosPx+Ωt,g0=Mcos2ψcoshΘ+sin2ψsinhΘ+icosψsinψcoshΘ−sinhΘ+cosPx+Ωtcosψ+isin.M,Θ,ψ, and η0 are defined by(20)M=1+P4Q2,expη0=Mexpη0,expiψ=Q2−P4P4+Q2+i2P2QP4+Q2,Θ=−Qy+η0,Ω=P4−2P4Q2−2κP4−3Q4PP4+Q2.The period ofϕ or u is 2π/P along the x direction in the x,y-plane (Figure 1). In particular, under the parameter constraints,(21)Ω=P,and the variable transformations defined in equation (6), the one-breather solutions defined in equation (18) reduce to one-breather solutions of the nonlocal Schrödinger–Boussinesq equation. Moreover, the higher-order breather solutions can also be generated from equation (13) under the parameter constraints equation (16), which still maintain the features of being periodic in the x direction and localized in the y direction. For example, taking parameters in equation (13) as(22)P1=−P2,P3=−P4,Q1=Q2,Q3=Q4,η10=η2∗0,η30=η4∗0,the second-order breather solutions ϕ,u can be generated (Figure 2).Figure 1
The one-breather solutionsϕ and u of the nonlocal Mel′nikov equation given by equation (18) with parameters κ=1/2,P=2/3,Q=1,η0=0,andt=0.
(a)(b)Figure 2
The two-breather solutionsϕ and u of the nonlocal Mel′nikov equation with parameters N=4,P1=1,P2=−1,P3=1,P4=−1,κ=1/2,Q2=2/3,Q3=1,Q4=1,ηj0=0j=1,2,3,4,andt=0.
(a)(b)In addition to general breather solutions, mixed solutions consisting of breathers and periodic line waves can also be generated by taking parameters in equation (13) as(23)N=2n+1,Pn+j=−Pj,Qn+j=Qj,Q2n+1=0,ηn+j0=ηj0∗.This family of hybrid solutions is also nonsingular and describesn-breather on a background of periodic line waves. The period of the periodic line waves is 2π/P2n+1. For example, with N=3 and parameter choices in equation (13) as P0,Q0,real,(24)P1=−P2=P0,Q1=Q2=Q0,Q3=0,η10=η20.A mixed solution consisting of one breather and periodic line waves is obtained (Figure3). The period of the one breather is 2π/P0, while that of the line waves is 2π/P3. In this case, u is a complex valued function, which is different from the solutions of the local Mel′nikov equation, where the scalar field must be real. Again, when one takes parameter constraints,(25)Ω1=P0,Ω2=−P0,Ω3=P3,and the variable transformations defined in equation (6), the corresponding mixed solutions reduce to hybrid solutions consisting of one-breather solutions and periodic line waves to the nonlocal Schrödinger–Boussinesq equation.Figure 3
The mixed solutionsϕ and u consisting of one breather and periodic line waves in the nonlocal Mel′nikov equation given by equation (18) with parameters κ=1,P0=1,Q0=2,P3=1,Q3=0,η10=0,η20=0,η30=−π,andt=0.
(a)(b)Another popular method to derive exact solutions to soliton equation theoretically is the Darboux transformation [53–60], but we have demonstrated that the bilinear method is a feasible scheme too. Examples of these solutions include solitons, breathers, rogue waves, and many other types of rational solutions. This alternative is especially valuable as most soliton systems possess bilinear forms.


## 3. Rational Solutions of the Nonlocal Mel′nikov Equation

To generate rational solutions to the nonlocal Mel′nikov equation, a long wave limit is now taken with the provision(26)expηj0=−1,1≤j≤N.Indeed, under parameter constraints,(27)Qj=λjPj,ηj0=iπ,1≤j≤N,and the limit Pj⟶0 in equation (13), rational solutions can be obtained.Theorem 1.
The partial reverse space-time nonlocal Mel′nikov equation haveNth−order rational solutions:(28)ϕ=gNfN,u=2logfNxx,where(29)fN=∏j=1Nθj+12∑j,kNαjk∏l≠j,kNθl+⋯+1M!2M∑j,k,…,m,nNαjkαsl⋯αmn︷M∏p≠j,k,…m,nNθp+⋯,gN=∏j=1Nθj+bj+12∑j,kNαjk∏l≠j,kNθl+bl+⋯+1M!2M∑j,k,…,m,nNαjkαsl⋯αmn︷M∏p≠j,k,…m,nNθp+bp+⋯,with(30)θj=ix+λjy−i3λj2+2κλj2t,bj=2iλj,ajk=−4λj−λk2,where the two positive integers j and k are not larger than N and λjandλk are arbitrary real constants. As example, the first four terms of equation (29) can be written explicitly as(31)f1=θ1,f2=θ1θ2+a12,f3=θ1θ2θ3+a12θ3+a13θ2+a23θ1,f4=θ1θ2θ3θ4+a12θ3θ4+a13θ2θ4+a14θ2θ3+a23θ1θ4+a24θ1θ3+a34θ1θ2+a12a34+a13a24+a14a23,g1=θ1+b1,g2=θ1+b1θ2+b2+a12,g3=θ1+b1θ2+b2θ3+b3a12θ3+b3+a13θ2+b2+a23θ1+b1,g4=θ1+b1θ2+b2θ3+b3θ4+b4+a12θ3+b3θ4+b4+a13θ2+b2θ4+b4+a14θ2+b2θ3+b3+a23θ1+b1θ4+b4+a24θ1+b1θ3+b3+a34θ1+b1θ2+b2+a12a34+a13a24+a14a23.
The abovementioned formulae forf and g will be used to calculate explicit forms of rational solutions.Remark 1.
These rational solutions can be classified into two patterns:(1)
By restricting the parameters,(32)N=2n,λn+j=−λj,j=1,2,3,…,n,in Theorem1, the corresponding rational solutions are nonsingular, which are nth-order lumps.(2)
When the parameters do not satisfy constraints defined in equation (32), the corresponding solutions have singularity at the point x,y,t=0,0,0. Thus, hereafter, we just focus on the nonsingular rational solutions.
To demonstrate their typical dynamics, we first consider the first-order lump solutions obtained by taking in equation (29)(33)N=2,λ2=−λ1,λ=λ.
In this case,u and ϕ can be expressed:(34)ϕ=1+g20f2=1+4iλ2y−4λx−2κ/λ+3λ3t2+λ4y2+1,u=2logf2xx=−4λ2x−2k+3λ4t2−λ6y2−λ2λx−2k/λ+3λ3t2+λ4y2+12.
As discussed in Ref. [38], solutions ϕandu are constant along the xt,yt trajectory, where(35)x−2κ+λ42λ2,t=0,y=0.
Moreover, at any fixed time,ϕ,u⟶1,0 when x,y becomes unbounded. Hence, these rational solutions are permanent lumps moving on the constant backgrounds.
Hence, we can discuss the patterns of the lump solutions att=0 without loss of generality. In this case, the two solutions ϕandu have critical points:(36)A1x,y=0,0,A2x,y=3λ,0,A3x,y=−3λ,0,on insisting the first derivatives to vanish ∂ϕ/∂x=∂ϕ/∂y=0∂u/∂x=∂u/∂y=0. The second derivatives at these three critical points are given by(37)ψxxA1=−48λ2,ψxxψyy−ψxy2A2=ψxxψyy−ψxy2A3=768λ6,uxxA1=−24λ4,uxxuyy−uxy2A2=uxxuyy−uxy2A3=412λ10.
Hence,A1 is the maximum point of solutions ϕ and u and A2andA3 are the minimum points. Thus, there are only bright lumps in the nonlocal Mel′nikov equation, a feature different from the local Mel′nikov equation [45] as the latter possesses three patterns of lumps. Furthermore, by comparison with the rational solutions of the local Mel′nikov equation, the expression of g20 in equation (34) only contains the variable y, but the corresponding solutions of the local Mel′nikov equation contain all the variables x,y, and t (Figure 4). In particular, on taking(38)λ=63,κ=−1,in equations (6) and (34), the two-dimensional lump solutions (34) reduce to the rogue waves of the nonlocal Schrödinger–Boussinesq equation, which can be expressed as(39)ψ=1−24it+366x2+4t2+9,u=244t2+9−6x26x2+4t2+92.
In terms of displacements,ψ reaches the maximum amplitude of three (i.e., three times the background amplitude) at the point 0,0, and the minimum amplitude of zero at points ±32/2,0. u reaches the maximum amplitude of 8/3 at the point 0,0, and the minimum amplitude of −1/3 at points ±32/2,0. The function u tends to a zero background in the far field.
Higher-order lumps can be derived for largerN=2nn≥2 and other parameters which meet the constraints defined in equation (32). The solution then describes the interaction of n fundamental lumps. For example, on taking(40)N=4,λ1=−λ3,λ2=−λ4,the second-order lump solutions can be obtained from equation (28), which is(41)ϕ=2g4/f4,u=2logf4xx,where g4 satisfies equation (31) and(42)f4=36x4−1260tx3+15525t2+36y2+365x2+−78750t3−6590t−2340ty2x+140625t4+25850t2+9225t2y2+144y4−359y2+53299.
Peculiar features occur during the interaction of fundamental lumps (Figure5). When the two lumps get close to each other, their shapes are deformed. The amplitudes of both entities decrease and remain less than three (the amplitude of the fundamental lump solution). This feature contrasts sharply with that of the nonlocal DSI equation [38], which displays a higher intermediate amplitude during interactions.
For still largerN and parameters satisfying parameters constraints equation (32), higher-order lumps would be obtained, e.g., for(43)N=6,λ1=−λ2=2,λ3=−λ4=32,λ5=−λ6=1,κ=1,a three-lump solution can be derived:(44)ϕ=1+g60f6,u=2logf6xx,where f6 and g60 are given by equations (A.1) and (A.2) in Appendix A.
These third-order solutions consist of three lumps (Figure6). The maximum value of ϕ stays below three at all times.Figure 4
The one-lump solutionsϕ and u of the nonlocal Mel′nikov equation given by equation (34) with parameters κ=1,λ=1,andt=0.
(a)(b)Figure 5
The interaction of two fundamental lumps at different values in time: (a)t = −2. (b) t = −1. (c) t = 0. (d) t = 1/2. (e) t = 1. (f) t = 2.
(a)(b)(c)(d)(e)(f)Figure 6
The third-order lump solutionsϕandu defined in equation (44).
(a)(b)


## 4. Semirational Solutions of the Nonlocal Mel′nikov Equation

To understand resonant behaviours for the nonlocal Mel′nikov equation, we consider several types of semirational solutions. Similar to the derivation of the rational solutions in the last section, semirational solutions are also computed by taking suitable long wave limits. More precisely, a coalescence of wavenumbers at a finite value is taken, instead of the zero wavenumber regime, which would generate purely rational solutions previously. These semirational solutions describe interactions among lumps, breathers, and periodic line waves.Case 1.
A combination of lumps and periodic line waves. The simplest semirational solutions consisting of a lump and periodic line waves are generated from the third-order soliton solutions. Indeed, taking parameters in (13) as(45)N=3,Q1=λ1P1,Q2=λ2P2,expη10=expη20=−1,Q3=0,and taking a long wave limit,(46)P1,P2⟶0,the functions f and g are rewritten as(47)f=θ1θ2+a12+θ1θ2+a12+a13θ2+a23θ1+a12a23eη3,g=θ1+b1θ2+b2+a12+θ1+b1θ2+b2+a12+a13θ2+b2+a23θ1+b1+a12a23eη3+iϕ3,where(48)as3=−4P32P32+λs2,s=1,2,and a12,bs,ϕs,η3 are given by (30) and (14). Furthermore, by taking parameters as(49)λ1=−λ2=λ0,a semirational solution consisting of a lump and periodic line waves is obtained (Figures 7 and 8). The periodic line waves coexisting with a fundamental lump are periodic in x direction and localized in y direction and the period is 2π/P3. By comparing this type of semirational solutions with that of the nonlocal DSI equation in Ref. [38], the interaction between the lump and periodic line waves can generate either much higher peaks or lower peaks. In Figure 7, the maximum amplitude of the lump can reach 4 (four times the constant background), and thus it is higher than the fundamental lumps. However, in Figure 8, the maximum amplitudes of the lump do not exceed 2 (two times the constant background). In this case, the interaction between the lump and periodic line waves will result in lower peaks.
In particular, when one takes(50)λ=63,κ=−1,Ω3=P3,in equations (6) and (47), we obtain one-dimensional mixed solutions consisting of a fundamental rogue wave and periodic line waves for the nonlocal Schrödinger–Boussinesq equation.
Semirational solutions consisting of more lumps and periodic line waves can also be derived in a similar way for largerN. As illustrative examples, we consider a subclass of semirational solutions consisting of two lumps and periodic line waves, which can be derived from 5-soliton solutions. Indeed, setting the parameters in equation (13) as(51)N=5,Qj=λjPj,expηj0=−1,Q5=0j=1,2,3,4,and then taking a long wave limit(52)Pj⟶0j=1,2,3,4,the functions f and g are changed from exponential functions to a combination of rational and exponential functions, which are explicitly expressed by equation (A.4) in Appendix A. Under the parameter constraints,(53)λ1=−λ2,λ3=−λ4,a family of semirational solutions describing two lumps on a background of periodic line waves is generated. For typical parameters,(54)λ1=−λ2=1,λ3=−λ4=4,P5=1,ϕ and u are shown in Figure 9. The period of these line waves is 2π. The maximum value of the solution ϕ can exceed 4.5 (4.5 times the constant background) at t=0, while u can reach 12. In other words, this interaction between these two lumps and periodic line waves can create very high spikes under proper parameter choices.Figure 7
Semirational solutions constituting of a lump and periodic line waves for the local Mel′nikov equation with parametersλ0=1,P3=1,andη30=−π/6. The panels (b) and (d) are the contour plots of the panels (a) and (c), respectively.
(a)(b)(c)(d)Figure 8
Semirational solutions constituting of a lump and periodic line waves for the nonlocal Mel′nikov equation with parametersλ0=1,P3=1/3,andη30=−π/6. The panels (b) and (d) are the contour plots of the panels (a) and (c), respectively.
(a)(b)(c)(d)Figure 9
Semirational solutions constituting of two lumps and periodic line waves for the nonlocal Mel′nikov equation with parametersκ=1,λ1=−λ2=1,λ3=−λ4=2,P5=1,Q5=0,andη50=−π/6. The panels (b) and (d) are the contour plots of the panels (a) and (c), respectively.
(a)(b)(c)(d)Case 2.
A combination of breathers and periodic line waves
Another type of semirational solutions is a mixed mode consisting of lumps and breathers. Here, we only consider the simplest example, consisting of one lump and one breather. We choose parameters in equation (13) as(55)N=4,Q1=λ1P1,Q2=λ2P2,expη10=expη20=−1,and take a limit of(56)P1,P2⟶0.
Then, functionsf and g of solutions to solve the Mel′nikov equation can be rewritten as equation (A.6) in Appendix A. Furthermore, on taking parameters(57)λ1=−λ2,P3=−P4,Q3=Q4,η30=η40∗,semirational solutions consisting of a fundamental lump and one breather are obtained (Figure 10). The period of this breather is 2π/P3. In this case, u is real. For fixed parameters η30andη40, the distance between the lump and the breather is not altered during their propagation in the x,y-plane. In order to observe the interaction between the lump and breather, we control the location of the breather by adjusting the parameters η30,andη40. For η30⟶0, the distance between the lump and the breather tends to zero (Figure 11). The lump immerses itself in the breather (see the panel of η30=0). The superposition of the lump and breather excites a peak whose height is less than three times the background amplitude. Moreover, the breather possesses a lower amplitude and a smaller period than the previous example and it propagates stably. Apparently, the wave structure of the lump is destroyed due to the interaction, and energy transfer has occurred between the lump and the breather.
Higher-order semirational solutions consisting of higher-order lumps and higher-order breathers can also be calculated in a similar way with largerN and will be reported in the future.Figure 10
Semirational solutionsϕandu consisting of a lump and one breather for the nonlocal Mel′nikov equation with parameters κ=1,λ1=1,λ2=−1,P3=−P4=1,Q3=Q4=1,η30=η40=−2π,andt=0.
(a)(b)Figure 11
The superposition of a lump and one breather with parametersκ=1,λ1=1,λ2=−1,P3=−P4=1,Q3=Q4=1,andt=0 for different values of η30. (a)η30=−2π. (b)η30=0. (c)η30=π. (d)η30=2π.
(a)(b)(c)(d)


## 5. Doubly Periodic Solutions

In addition to breather (singly periodic) and rational (algebraically localized) solutions, the evolution system in one spatial dimension, namely, the Schrödinger–Boussinesq equation (equation (7), also admits doubly periodic solutions. The wave profile will be periodic in x, but also possesses a distinct period in the t direction as well. In terms of the mathematical techniques employed, theta functions can be expressed as Fourier series with exponentially decreasing coefficients, while elliptic functions can be written as ratios of theta functions. The underlying principle is that the Hirota bilinear derivatives of theta functions can be represented in terms of theta functions themselves ([61, 62] and Appendix B). Here, we first generalize equation (7) by introducing the auxiliary function Φ and the real parameter σ:(58a)3utt−uxx−3u2+uxxxx=σΦΦ∗−x,txx,(58b)iΦt+Φxx+uΦ=0,with (functions g complex and f real and Ω being the real angular frequency)(59)u=2logfxx,Φ=exp−iΩtgf.The bilinear form for equations (58a) and (58b) can be obtained (C0 = a constant and D = the Hirota operator):(60)iDt+Dx2+Ωg⋅f=0,3Dt2−Dx2−Dx4−C0f⋅f=σgg∗−x,t.Using combinations of theta functions ofxandt with different moduli τ and τ1 [62], solutions for equations (58a), (58b), and (60) can be expressed as (A0 and Ω being real constants to be determined)(61a)Φ=A0θ2ωt,τ1θ4αx,τ+iθ1ωt,τ1θ3αx,τθ3ωt,τ1θ4αx,τ+θ4ωt,τ1θ3αx,τexp−iΩt,(61b)f=θ3ωt,τ1θ4αx,τ+θ4ωt,τ1θ3αx,τ,g=θ2ωt,τ1θ4αx,τ+iθ1ωt,τ1θ3αx,τ.From the first bilinear form of equation (60), the period parameter ω and the angular frequency of the whole wave packet, Ω, are defined in terms of the theta constants as(62)ω=2α2θ320,τθ420,τθ420,τ1,Ω=−2α2θ2″0,τθ20,τ.One constraint relating the two different moduliτandτ1 is(63)θ320,τ1θ420,τ1=θ340,τ+θ440,τ2θ320,τθ420,τ.The second bilinear form of equation (60) not only defines the amplitude parameter A0 but also imposes additional constraints in terms of the theta constants:(64)α2θ340,τ+θ440,τ+α4θ3″″0,τθ40,τ+θ4″″0,τθ30,τ+6θ3″0,τθ4″0,τθ30,τθ40,τ−α42θ2″″0,τθ20,τ+6θ2″0,τ2θ20,τ2−3ω2θ340,τ1+θ440,τ1=σA02θ320,τ1θ220,τ1,(65)2α2θ320,τθ420,τ+8α4θ1′0,τθ1″″0,τθ220,τ−6ω2θ320,τ1θ420,τ1=σA02θ420,τ1θ220,τ1.The three constraints equations (63)–(65) then determine the parameter A02, τ, and τ1, assuming σ and α are given (and ω through equation (62)). The solution equation (61b) can be expressed in terms of Jacobi elliptic function in a slightly more condensed but less symmetric forms, with again different kandk1:(66)A=A0k1cnst,k1+i1+k1snst,k1dnrx,k1+k1dnrx,k+dnst,k1exp−iΩt,s=2r21+k1,k2=2k11+k1.The advantage is that most computer algebra software can handle Jacobi elliptic functions efficiently.


## 6. Discussion and and Conclusions

In this paper, we have introduced and investigated a nonlocal Mel′nikov equation with partial reverse space time, which constitutes a multidimensional version of the nonlocal Schrödinger–Boussinesq equation with a parity-time-symmetric potential. By using the Hirota bilinear method, soliton solutions are obtained. Although these soliton solutions may have singularities, generaln-breather solutions and mixed solutions consisting of breathers and periodic line waves can be derived under proper parameter constraints and lead to nonsingular solutions. On taking a long wave limit, nonsingular rational solutions or lumps can be generated. The exact explicit lump solutions up to the third order are presented, and properties of the dynamics of interaction between lumps have been revealed. Furthermore, by taking a coalescence of wavenumbers at a finite value, two subclasses of semirational solutions are derived. One subclass of these semirational solutions describes lumps on a background of periodic line waves. Another one highlights interaction between lumps and breathers. In particular, two solutions of the nonlocal Schrödinger–Boussinesq equation, namely, fundamental rogue waves and mixed modes consisting of a rogue wave and periodic line waves, are obtained as reductions of the corresponding solutions of the partial reverse space-time nonlocal Mel′nikov equation. Finally, special classes of doubly periodic solutions are presented too. We believe that similar techniques can be applied to many nonlocal evolution equations being studied in the literature. As nonlocal differential equations have been amply demonstrated in the literature to be applicable to many fields, e.g., population dynamics [25] and microelectro mechanical system [63], further efforts will likely be fruitful.



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