Based on the generalized dressing method, we propose a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation and derive its Lax pair. Using separation of variables, we have derived the explicit solutions of the equation. With the aid of Matlab, the curves of the solutions are drawn.

National Natural Science Foundation of China11301149Henan Natural Science Foundation of Basic Research1623004100721. Introduction

It is well known that the interactions of long wave and short wave play an important role in fluid dynamics. The model is described by the following equation:(1)i∂tS+icg∂xS-βLS-γ∂xx2S-δS2S=0,∂tL+cl∂yL+α∂xS2=0.The inverse scattering technique proposed in [1] plays an important role in constructing the complete solution of the long wave and short wave resonance equations. The N soliton solution of long wave and short wave has been obtained in [2]. Radha et al. in [3] derived periodic solutions and localized solutions of (1). Lai and Chow in [4] studied positon and dromion solutions of 2+1-dimensional long wave and short wave resonance interaction equations. Researchers have focused on the long wave and short wave equation [5–10] by using different methods. Serkin et al. [11] discussed integrable variable-coefficient nonlinear evolution equations. By utilizing the exp-function method, the generalized solitary solution and periodic solution of soliton equations can be given in [12]. Authors in [13] discussed the interactions of dark soliton and bright soliton in a double-mode optical fiber. In [14, 15], authors (Dai and Jeffrey [14], Jeffrey and Dai [15]) extended the dressing method [16, 17] to a generalized version for solving soliton equations associated with matrix nonspectral problems and variable-coefficient cases. The generalized dressing method was based on the problems of factorization of an integral operator F on the line into the product of two Volterra type integral operators K±, from which the Gel’fand-Levitan-Marchenko (GLM) equation is obtained. These Volterra operators are then used to construct dressed operators (N1,N2) starting from a pair of initial operators (M1,M2). Integrable variable-coefficient nonlinear equations are obtained from the compatibility of the dressed operators. There are some differences between the original dressing method and the generalized dressing method. In the original dressing method, the constant coefficient operators have transformed into different constant coefficient operators. The generalized dressing method transforms the variable-coefficient operators into different variable-coefficient ones. The advantages in the generalized dressing method lie in deriving integrable variable-coefficient nonlinear evolution equation and corresponding Lax pairs. However, the original dressing method is a system way to study constant coefficient nonlinear evolution equation [18–20]. Authors (Dai and Jeffrey [14]) presented the generalized dressing method; we also discussed some integrable variable-coefficient evolution equations [21–25]. In fact, the dressing method can be thought as a rather general formulation of the inverse scattering method, which has the advantage of bypassing the scattering problem. The common point between the two methods is that two methods can deal with the initial boundary value problem.

In the paper, we applied the generalized dressing method to derive a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation:(2)iα2wxy+β2wxx+β2uvx=0,-iα1ut+β1vxx-uxx-uyy+2β1u2v+uwx=0,iα1vt-β1vyy+2β1uv2+vwx=0,where α2 and β2 are functions of t and y. α1 and β1 are functions of t. Particularly, the above equation is reduced to a new 2+1-dimensional integrable variable-coefficient equation:(3)iα1uvt+β1uvxx+uvyy-β1u-vvxx+2uvyy+2uxvx+uyvy=0,in view of α2=β2.

Furthermore, under the transformations α1=1, β1=1/t, (3) can be read as the cylindrical equation:(4)iuvt+1tuvxx+uvyy-1tu-vvxx+2uvyy+2uxvx+uyvy=0.Moreover, (2) are written as a 2+1-dimensional integrable modified long wave-short wave equation for α2=β2 and α1=β1:(5)iwxy+wxx+uvx=0,-iut+vxx-uxx-uyy+2u2v+uwx=0,ivt-vyy+2uv2+vwx=0.The outline of the paper is as follows. In Section 2, we briefly describe the generalized dressing method and its properties. Moreover, we introduce two dressing operators. In Section 3, new integrable variable-coefficient 2+1-dimensional long wave-short wave equations and their Lax pairs are derived with the aid of the generalized dressing method. In Section 4, as an application, we obtain explicit solutions of these equations and draw the curves of the solutions.

2. The Generalized Dressing Method and Dressing Operators

First, we consider three integral operators F(x,z,y), K+(x,z,y), and K-(x,z,y) defined by [16](6)K+ψx≡∫x∞K+x,z,yψzdz,K-ψx≡∫-∞xK-x,z,yψzdz,Fψx≡∫-∞∞Fx,z,yψzdz.We assume that (I+K+)-1 exists and F admits the triangular factorization(7)I+F=I+K+-1I+K-,where I is the identity operator. From (7), a direct calculation shows that F and K+ satisfy the Gel’fand-Levitan-Marchenko (GLM) equation [16]:(8)K+x,z,y+Fx,z,y+∫x∞K+x,s,yFs,z,yds=0,z>x,here it is supposed that K±(x,z,y) and F(x,z,y) satisfy the condition (9)sup∫x0+∞K±x,z,yψzdz<+∞,sup∫x0+∞Fx,z,yψzdz<+∞,x0>-∞.We now introduce two differential operators M1 and M2 defined by(10)M1=α1∂t+iβ1Θ∂xx2,M2=α2∂y-iβ2Θ∂x,with α1 and β1 being matrix functions of t. α2 and β2 are matrix functions of t and y:(11)Θ=1000.Suppose that the operator F commutes with M1 and M2; that is,(12)M1,F=M1F-FM1=0,M2,F=M2F-FM2=0,which together with (10) implies the following equations:(13)α1Ft+iβ1ΘFxx+iFzzβ1Θ=0,(14)α2Fy-iβ2ΘFx-iFzβ2Θ=0.In what follows, we obtain the dressing operators N1 and N2 with the aid of operators M1 and M2. The dressing procedure is accomplished through the relations [14, 15](15a)N1I+K+-I+K+M1=0,(15b)N2I+K+-I+K+M2=0.The difference between the original dressing method and the generalized dressing method lies in the differential operators M1 and M2 which satisfied the relation(16)M1,M2=ϕ1M1+ϕ2M2,with ϕ1 and ϕ2 being arbitrary functions of their arguments. In view of [14, 15], the corresponding dressed operators obey the equation(17)N1,N2=ϕ1N1+ϕ2N2.Letting N1=M1+u1(1)∂x+u0(1), N2=M2+u0(2), from (15a) and (15b), we have(18)u01=u11K^+iβ1ΘK^x+Kxz=x+iK^zz=xβ1Θ,u11=iβ1ΘK^-K^Θ,u02=iβ2K^Θ-ΘK^.In view of (16), it is easy to obtain(19)α1α2t=ϕ2α2,α1β2t=ϕ2β2,ϕ1=0;thus, we have α2=e∫ϕ2/α1dt+c1(y),β2=e∫ϕ2/α1dt+c2(y). Here, ϕ is an arbitrary function of t and y. c1(y) and c2(y) are arbitrary functions of y.

3. A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and Its Lax Pair

In this section, based on the generalized dressing method, we derive a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation. From (17), we have(20)α1u0t2+iβ1Θu0xx2-α2u0y1+iβ2Θu0x1+u01u02-u02u01+u11u0x2=ϕ2u02,(21)2iβ1Θu0x2-α2u1y1+iβ2Θu1x1-iβ2u01Θ+iβ2Θu01+u11u02-u02u11=0.We denote(22)K^=Kx,z,y,tx=z=k^11k^12k^21k^22,k^11=w,k^12=u,k^21=v.In view of (16), we have(23)iα2k^12y+β2k^12k^22+β2k12xz=x=0,iα2k^21y-β2k^21k^11+β2k21zz=x=0.Based on (18) and (22)-(23), we obtain(24)u11=iβ10k^12-k^210=iβ10u-v0,(25)u02=iβ20-k^12k^210=iβ20-uv0,(26)u01=iβ1k^12k^21+2k^11xk^12x+k12xz=x+k^12k^22-k^11k^21+k12zz=x-k^12k^21=iβ1uv+2wxux-iα2β2uy-iα2β2vy-uv.From (20), we derive new integrable variable-coefficient 2+1-dimensional long wave-short wave equation with the aid of (22)–(26):(27)iα2wxy+β2uvx+β2wxx=0,-iα1ut+β1vxx-uxx-uyy+2β1u2v+uwx=0,-iα1β2vt-α2β1vyy+2β2β1uv2+vwx=0.Particularly, the above equations are reduced to the cylindrical form:(28)iwy+wx+uv=0,-iut+1tvxx-uxx-uyy+2tu2v+uwx=0,ivt-1tvyy+2tuv2+vwx=0,where α2=β2, α1=1, β1=1/t, and the integration constant is zero.

The Lax pairs of (27) are N1 and N2. u1(1), u0(1), and u0(2) are presented in (24)–(26).

Particularly, we consider the case for y=x; then (28) is reduced to a new coupled equation:(29)i+1wx+uv=0,-iut+1tvxx-2tuxx+2tu2v+uwx=0,ivt-1tvxx+2tuv2+vwx=0.

4. Explicit Solutions and the Curves of Solutions

In this section, we shall apply the generalized dressing method to construct explicit solutions of these obtained 2+1-dimensional long wave-short wave equation and its reductions. We assume that F and K have solutions in the form of separation of variables:(30)Fx,z,y,t=∑j=1Nfjx,y,tgjz,y,t,(31)Kx,z,y,t=∑j=1Nkjx,y,tgjz,y,t,where fj(x,y,t), gj(z,y,t) are some 2×2 matrices.

Substituting (30) and (31) into the GLM equation (8) yields the following:(32)Kx,x,y,t=∑j=1Nkjx,y,tgjx,y,t=-f1,f2,…,fNL-1g1,g2,…,gNT,where L=(Ljl)2N×2N is defined by (33)Ljl=δjl+∫x∞gjs,y,tfls,y,tds,1≤j,l≤N,and δjl is the Kronecker delta.

We denote F=(Fij)2×2, and, in view of (13) and (14), we obtain(34)α1F11t+iβ1F11xx+F11zz=0,α2F11y-iβ2F11x+F11z=0,α1F12t+iβ1F12xx=0,α1F21t+iβ1F21zz=0,α2F12y-iβ2F12x=0,α2F21y-iβ2F21z=0,α1F22t=0,α2F22y=0.In the what follows, we obtain one soliton solution for the case of N=1 of (30). Let(35)Fx,z,y,t=fx,y,tgz,y,t=e-iβ1/α1l12+m12t+l1x+il1+m1ye-iβ1/α1l22t+l2x+iβ2/α2l2ye-iβ1/α1m12t+l3x+iβ2/α2m1y0em1z00em2z.Then, from (32) we obtain (36)Kx,x,y,t=1Δk^11k^12k^21k^22,with (37)Δ=1-1m1+l1e-iβ1/α1l12+m12t+m1+l1x+iβ2/α2m1+l1y-1m1+l2m2+l3e-iβ1/α1l22+l32t+m1+m2+l2+l3x+iβ2/α2l2+m1y,k^11=1Δe-iβ1/α1l12+m12t+l1+m1x+iβ2/α2y+1/m1+l2e-2iβ1/α1l22t+2m1+l2x+2iβ2/α2y,k^12=1Δe-iβ1/α1l22t+l2x+iβ2/α2l2y+1m2+l3e-iβ1/α1l12+m12+l32t+2m2+l1+l3x+iβ2/α2l1+2m1y-1m1+l1e-iβ1/α1l12+l22+m12t+l1+l2+m1+m2x+iβ2/α2l1+l2+m1y,k^21=1Δe-iβ1/α1l32t+m1+l3x+iβ2/α2m1y,k^22=1Δ1m2+l3e-2iβ1/α1l32t+2m2+l3x+2iβ2/α2m1y.

Using (22), we obtain the solutions of (27). Particularly, for α2=β2, α1=1, and β1=1/t, we derive the solutions of (28). In what follows, we draw the curves of the solutions for α2=β2, α1=1, and β1=t. Figures 1 and 2 describe the imaginary of u and real of u, respectively. From the curves, we can see that the forms are similar. The imaginary of v and real of v are shown by Figures 3 and 4, respectively. From the curves, we can see that the forms are different and with diminishing energy. Figures 5 and 6 construct the imaginary of w and real of w, respectively. In view of the solution curves, we can read the difference between the imaginary of w and real of w. Furthermore, we find that imaginary of v and that of w are similar. At the same time, we find that real of v and that of w are similar. Similarly, in later paper, we will discuss two soliton solutions and N-soliton solutions.

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

x=0.9, l1=0.3, l2=0.5, l3=0.8, m1=1, m2=2, y=3, and t∈[-6,6].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors acknowledge the support by National Natural Science Foundation of China (no. 11301149) and Henan Natural Science Foundation of Basic Research (no. 162300410072).

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