# Nonlocal Symmetry, CRE Solvability, and Exact Interaction Solutions of the (2 + 1)-Dimensional Boussinesq Equation

Authors: Sixing Tao
Journal: Journal of Mathematics
 (2022)

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## Abstract

The (2 + 1)-dimensional Boussinesq equation is considered in this study. Nonlocal symmetries of the (2 + 1)-dimensional Boussinesq equation are obtained by means of the truncated Painlevé expansion. The consistent Riccati expansion (CRE) solvability of the Boussinesq equation is derived. Three special forms of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically.


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## Body

## 1. Introduction

In the research of nonlinear science, more and more attention has been paid to the two-dimensional or three-dimensional nonlinear models [1–3]. Compared with the (1 + 1)-dimensional equations, multidimensional ones would be more realistic in describing the nonlinear phenomena in science and engineering [4–6]. It is known that the (2 + 1)-dimensional Boussinesq equation can be used to describe the propagation of gravity waves on the surface of water, particularly, the head on collision of oblique waves. Usually, the generalized (2 + 1)-dimensional Boussinesq equation can be written as(1)utt−γ1uxx−γ2u2xx−γ3uxxxx−γ4uyy=0,where γ1,γ2,γ3,γ4 are arbitrary constants with γ3γ4≠0. The special cases of equation (1) have been studied by many researchers. Taking γ1=γ2=γ3=γ4=1, then equation (1) is reduced to(2)utt−uxx−u2xx−uxxxx−uyy=0.Abdel Rady et al. [8] obtained some exact traveling wave solutions of equation (2) by applying the repeated homogeneous balance method. Chen et al. [9] obtained many explicit exact solutions of equation (2) by using new generalized transformation in the homogeneous balance method. Zeng and Zhang Hong-Qing [10] gave the Bäcklund transformation of equation (2). Taking γ1=γ2=1,γ3=−3,andγ4=1, then equation (1) is reduced to equation(3)utt−uxx−u2xx+3uxxxx−uyy=0.Johnson [11] obtained some exact and general solitary-wave, two-soliton, and resonant solutions of (3) by using the bilinear method. Zhang et al. [12] studied the periodic solutions of equation (3) by Hirota bilinear method and Riemann theta functions. Taking γ1=γ2=1,γ3=3,andγ4=1, then equation (1) is reduced to equation(4)utt−uxx−u2xx−3uxxxx−uyy=0.Wu Yong-Qi [13] obtained new exact solution of equation (4) by the Hirota bilinear method and Riemann theta functions. Wang [14] gave the triply periodic wave solutions of equation (4) with Riemann theta functions. Ma and Deng [15] obtained the lump solutions of equation (4).As we know, the Painlevé analysis techniques [16] are famous and widely employed in the area of investigating mathematical properties, such as Painlevé integrable property, bilinear forms, and Bäcklund transformation of nonlinear evolution equations. It is found that the residue of the truncated Painlevé expansion for the bosonized supersymmetric KdV equation is a nonlocal symmetry; we call it as the residual symmetry. In 2013, the theory of nonlocal residual symmetry was put forward by Lou [17]. It is proved that, for a given truncated Painlevé expansion of an arbitrary nonlinear Painlevé integrable system, the residue with respect to the singularity manifold is a nonlocal symmetry. By using symmetry analysis, exact and special solutions of a given equation in terms of solutions of lower-dimensional equations can be obtained [18, 19]. Based on the novel results of the symmetry reduction with nonlocal symmetries, Lou [20] further proposed the consistent Riccati expansion (CRE) method. For a given derivative nonlinear polynomial system,(5)Px,t,u=0,x=x1,x2,…,xn,P=P1,P2,…,Pm,u=u1,u2,…,um.One uses the following Riccati expansion:(6)ui=∑j=0Jiui,jRjω,where Jii=1,2,…,m are the positive integers. ω is an arbitrary function of x1,x2,…,xn,t and the function Rω is a solution of the Riccati equation:(7)Rω=l0+l1R+l2R2.Substituting (6) with (7) into (5) and equating the coefficients of Rωj to be zero, an overdetermined equation system will be obtained. Through a large number of detailed calculations, if the obtained overdetermined equation system has a universal property, it is consistent. Then, the expansion (6) is called a consistent Riccati expansion and the nonlinear system (5) is CRE solvable. The CRE method can be used to identify many CRE solvable systems. Since seeking for exact solutions [21–23] and interaction solutions [24] plays an important part in the study of nonlinear science, the CRE method can also be used to find interaction solutions between a soliton and other nonlinear waves, such as the soliton-cnoidal waves [25–27], soliton-perodic waves, and soliton-rational waves [28, 29]. Particular examples of CRE solvable are found for many integrable equations such as the AKNS equation [30], the BKP equation [31], the (1 + 1)-dimensional Boussinesq equation [32], and Kadomtsev–Petviashvili equation [33].The rest of this study is organized as follows. In Section2, the nonlocal symmetry of (4) is given by using the truncated Painlevé method. In Section 3, through the CRE method, we prove that (4) is CRE solvable. In Section 4, soliton-cnoidal wave interaction solutions of (4) are discussed. Finally, some conclusions are given.


## 2. Nonlocal Symmetry from the Truncated Painlevé Expansion

The truncated Painlevé expansion method is proved to be very useful in solving nonlinear partial differential equations. For equation (4), the truncated Painlevé expansion [16] can be written as(8)u=u0+u1ϕ+u2ϕ2,where u0,u1,u2 are functions to be determined later and ϕ=ϕx,y,t is a differentiable function defined on singularity manifold. Substituting expression (8) into equation (4) and equating all the coefficients of the same powers of 1/ϕ to be zero, we get the following non-auto-BT theorem through some detail calculations.Theorem 1 (non-auto-Bäcklund transformation theorem).
If the functionϕ is a solution of the following Schwarz-type equation,(9)Ct+CCx−Ky−KKx−Sx=0.in which(10)C=ϕtϕx,K=ϕyϕx,S=ϕxxxφx−32ϕxx2ϕx2.Then,(11)u=172ϕx4−48ϕx3ϕxxx+36ϕx2ϕxx2+12ϕx2ϕt2−12ϕx4−ϕx2ϕy2+2ϕxxϕ−2ϕx2ϕ2,is a solution of equation (4), where(12)u0=172ϕx4−48ϕx3ϕxxx+36ϕx2ϕxx2+12ϕx2ϕt2−12ϕx4−ϕx2ϕy2,u1=2ϕxx,u2=−2ϕx2.Once the solutionsϕ of (9) are determined, the corresponding expression u for (4) can be obtained. By using the BT Theorem 1, the new solutions of (4) can be derived. Meanwhile, the nonlocal symmetry of equation (4) can also obtained from the BT Theorem 1.Theorem 2 (nonlocal symmetry theorem).
Ifϕ is a solution of the Schwarzian Boussinesq equation (9), then(13)τu=2ϕxx,is a nonlocal symmetry of the Boussinesq equation (4).Theorem2 means the Boussinesq equation (4) is form invariant under the transformation:(14)u⟶u+ετu.It can be verified that nonlocal symmetry (13) exactly satisfies the linearized equation of (4):(15)τttu−τxxu−6uτxxu−6uxxτu−12uxτxu−τxxxxu−τyyu=0.It is invariant for the Schwarzian form (9) under the following Möbious transformation [17]:(16)ϕ⟶a1ϕ+a2a3ϕ+a4,a1a3−a2a4≠0.If we take a special casea1=1,a2=0,a3=ϵ,anda4=1, then equation (16) can be written as(17)ϕ⟶ϕ−ϵϕ2,which means (9) possesses the point symmetry:(18)τϕ=−ϕ2.For the nonlocal symmetry (11), the corresponding initial value problem is(19)du¯ϵdϵ=2ϕ¯xxϵ,u¯ϵ|ϵ=0=u,where ϵ being the infinitesimal parameter. However, it is difficult to solve equation (19) for u¯ϵ due to the intrusion of the function u¯ϵ and its differentiations. To solve the problem equation (19), we prolong (19) such that nonlocal symmetry (13) becomes the local Lie point symmetries for the prolonged system. Therefore, we introduce two new dependent variables:(20)f=ϕx,g=fx.Now, the nonlocal symmetry (11) of the original equation (4) becomes a Lie point symmetry of the prolonged system (4), (9), and (18), namely,(21)τuτϕτfτg=2g−ϕ2−2ϕf−2f2−2ϕg.According to Lie’s first theorem, the corresponding initial value problem of (21) reads(22)du¯ϵdϵ=2g¯ϵ,u¯0=u,dϕ¯ϵdϵ=−ϕ¯2ϵ,ϕ¯0=ϕ,df¯ϵdϵ=−2ϕ¯ϵf¯ϵ,f¯0=f,dg¯ϵdϵ=−2f¯2ϵ−2ϕ¯ϵg¯ϵ,g¯0=g.By solving the initial value problem (22), one can easily obtain the transformation group related to symmetry (21) of the enlarged system as follows.Theorem 3.
Ifu,ϕ,f,g is a solution of the prolonged systems (4), (9), and (18), then u¯,ϕ¯,f¯,g¯ is given by(23)u¯=u+2ϵg1+ϵϕ,ϕ¯=ϕ1+ϵϕ,f¯=f1+ϵϕ2,g¯=g1+ϵϕ2−2ϵf21+ϵϕ3,which provides an effective method to generate new solutions from known ones.


## 3. CRE Solvability of the (2 + 1)-Dimensional Boussinesq Equation

For the (2 + 1)-dimensional Boussinesq equation (4), based on the CRE method [20], the CRE solution for equation (4) reads(24)u=u0+u1Rω+u2Rω2,ω=ωx,y,t,where u0,u1,u2 are functions of x,y,t and Rω satisfies the following Riccati equation:(25)Rω=l0+l1R+l2R2,l2≠0,with l0,l1 are arbitrary constants. Substituting (24) along with (25) into (4), we have seven over determined equations for four undetermined functions u0,u1,u2,ω, vanishing all the coefficients of the powers of Rω; it is fortunate that the over determined system is consistent; thus, we obtain(26)u0=16ωx2−8l0l2ωx4−l12ωx4−6l1ωx2ωxx−ωx2−4ωxωxxx+3ωxx2+ωt2−ωy2,u1=−2l1l2ωx2−2l2ωxx,u2=−2l22ωx2.with the function ω only need to satisfy:(27)C^t+C^C^x−K^y−K^K^x−S^x+σwxxwx=0,with σ=l12−4l0l2 and C^,K^,S^ are in the form as C,K,S in (9). In summary, we have the following theorem.Theorem 4.
If the functionω is a solution of system (27), then(28)u=16ωx2−8l0l2ωx4−l12ωx4−6l1ωx2ωxx−ωx2−4ωxωxxx+3ωxx2+ωt2−ωy2−2l1l2ωx2+2l2ωxxRω−2l22ωx2Rw2,is a non-auto-Bäcklund transformation between ω and the solution u of equation (4) with Rω satisfies the Riccati equation (25).


## 4. Interaction Wave Solutions of the (2 + 1)-Dimensional Boussinesq Equation

From Theorem4, once functions ω and Rω of (24) are determined, the corresponding solutions for(29)u=16ωx2−8l0l2ωx4−l12ωx4−6l1ωx2ωxx−ωx2−4ωxωxxx+3ωxx2+ωt2−ωy2−2l1l2ωx2+2l2ωxxRω−2l22ωx2Rw2,of equation (4) will be obtained. It can be verified that equation (24) has a hyperbolic cotangent function solution:(30)Rw=−12l2δcoth12δω+l1,δ=l12−4l0l2>0.Case 1.
For theω equation (25), we take a trivial expression as follows:(31)ω=k1x+k2y+k3t+ξ0,where k1,k2,k3, and ξ0 are arbitrary constants. Substituting (31) along with (30) into (29) yields the following solution:(32)u=k12δ3−16−k226k12+k326k12−12k12δcoth12δω2.
Obviously, this solution includes two wave functions. One is a plane wave solutionu1=k12δ/3−1/6−k22/6k12. The other one is an inverted bell type of wave u2=−1/2k12δcoth1/2δω2. Therefore, solution (32) is an inverted bell wave solution on a plane. Substituting equations (31) into (27), we find that k1,k2,k3, and ξ0 can be selected as arbitrary constants. So, we select suitable parameters to demonstrate (32), which is shown in Figure 1.
Figure1 demonstrates the evolution of the interaction solution (32). At different times, different number of troughs appear. The depth of the deepest trough first decreases and then increases.Figure 1
Profile of equation (30) by choosing suitable parameters: l0=1,l1=3,l2=2,k1=1,k2=2,k3=−1,andξ0=1 at different times. (a) Perspective view of the wave at t=−6. (b) Perspective view of the wave at t=0. (c) Perspective view of the wave at t=6.
(a)(b)(c)Case 2.
For theω equation (25), we take ω in the following form:(33)ω=m1x+m2y+m3t+G,G=Gθ,θ=n1x+n2y+n3t,where G1=G1θ=Gθ satisfies the following elliptic equation:(34)G1θ2=d0+d1G1+d2G12+d3G13+d4G14,in which di0≤i≤4 are constants. Substituting (33) along with (34) into (27) yields the following results:(35)d0=13n16−36l0l2m14n12+9l12m14n12+3d2m12n14−6d3m13n13−5m12n22+5m12n32+4m1m2n1n2−4m1m3n1n3+m22n12−m32n12,d1=1n15−16l0l2m13n12+4l12m13n12+2d2m1n14−3d3m12n13−2m1n22+2m1n32+2m2n1n2−2m3n1n3,d4=l12−4l0l2,while the other constants remain free. Substituting (33) with (34) into (29) yields(36)u=16m1+n1Gθ2−8l0l2m1+n1Gθ4−l12m1+n1Gθ4−6l1n12m1+n1Gθ2Gθθ−m1+n1Gθ2−4n13m1+n1GθGθθθ+3n14Gθθ2−m3+n3Gθ2−m2+n2Gθ2+l1m1+n1Gθ2+n12Gθθδcoth12δω+l1−12m1+n1Gθ2δcoth12δω+l12.
It is known that different types of Jacobian elliptic functions can be used to express the solution of equation (34).(i)
As the first example, we consider the solution of (34) as(37)G=12arctanhsnθ,k.Substituting (37) into (36), then the exact soliton-cnoidal wave interaction solution can be derived as(38)u=162m1C+n1D2C212n14S2C4k4+32C5Dk2m1n13+16C4D2k2n14+48C4Sk2l1m12n12+48C3DSk2l1m1n13+8C2D2S2k2n14+12C2D2Sk2l1n14−128C4l0l2m14−16C4l12m14−32C3D3m1n13+64C3Dk2m1n13−256C3Dl0l2m13n1−32C3Dl12m13n1−16C2D4n14−48C2D2Sl1m12n12−192C2D2l0l2m12n12−24C2D2l12m12n12−48CD3Sl1m1n13−64CD3l0l2m1n13−8CD3l12m1n13−20D4S2n14−12D4Sl1n14−64CD3Sm1n13−8D4l0l2n14−D4l12n14−16C4m12−16C4m22+16C4m32−16C3Dm1n1−16C3Dm2n2+16C3Dm3n3−4C2D2n12−4C2D2n22+4C2D2n32+l1m1+n1D2C2+n12D2S2C2−n12k2S2δcoth12δω+l1−12m1+n1D2C2δcoth12δω+l12,whereC=cnn1x+n2y+n3t,k,D=dnn1x+n2y+n3t,k,andS=snn1x+n2y+n3t,k. Substituting (33) along with (37) into (27), we can further obtain several types of parameter restrictions for ω. We take the following parameter restrictions for an example:(39)l0=3k2n14n32+l12m22n22−l12m22n324m22n22−n32l2,m1=0,m3=m2n2n3,in whichl1,l2,m2,n1,n2,n3,k are free. In order to demonstrate (38), we choose the following parameters:(40)m1=0,m2=12,m3=14,n1=1,n2=1,n3=2,l0=0,l1=2,l2=1,k=12,which is shown in Figure2.Figure2 demonstrates the evolution of the soliton-cnoidal wave interaction solution (38) with the parameters being fixed at (40). There are two deep troughs in Figure 2(a), five troughs in Figure 2(b), one trough in Figure 2(c). The depth of the deepest trough first decreases and then increases.(ii)
The second special solution of equation (32) is taken as this form:(41)Gθ=μ0+μ1snθ,k,namely,(42)G=μ0θ+μ1∫θ0θsnθ,kdθ,which leads to the soliton-cnoidal wave interaction solution of equation (4):(43)u=16Sμ1n1+μ0n1+m12−8S4l0l2μ14n14−S4l12μ14n14−32S3l0l2μ0μ13n14−4S3l12μ0μ13n14+4C2S2k2μ12n14−6CDS2l1μ13n14−32S3l0l2m1μ13n14−4S3l12m1μ13n14−48S2l0l2μ02μ12n14−6S2l12μ02μ12n14+4C2Sk2μ0μ1n14−12CDSl1μ0μ12n14−96S2l0l2m1μ0μ12n13−12l12m1μ0μ12n13−32Sl0l2μ03μ1n14−4Sl12μ03μ1n14+3n14μ12C2D2+4C2Sk2m1μ1n13−12CDSl1m1μ12n13−6CDl1μ02μ1n14+4D2S2μ12n14−48S2l0l2m12μ12n12−6S2l12m12μ12n12−96Sl0l2m1μ02μ1n13−12Sl12m1μ02μ1n13−8l0l2μ04n14−l12μ04n14−12CDl1m1μ0μ1n13+4D2Sμ0μ1n14−96Sl0l2m12μ0μ1n12−12Sl12m12μ0μ1n12−32l0l2m1μ03n13−4l12m1μ03n13−6l1m12n12μ1CD+4D2Sm1μ1n13−32Sl0l2m13μ1n1−4Sl12m13μ1n1−48l0l2m12μ02n12−6l12m12μ02n12−32l0l2m13μ0n1−4l12m13μ0n1−S2μ12n12−S2μ12n22+S2μ12n32−8l0l2m14−l12m14−2Sμ0μ1n12−2Sμ0μ1n22+2Sμ0μ1n32−2Sm1μ1n1−2Sm2μ1n2+2Sm3μ1n3−μ02n12−μ02n22+μ02n32−2m1μ0n1−2m2μ0n2+2m3μ0n3−m12−m22+m32+S2l1μ12n12+2Sl1μ0μ1n12+n12μ1CD+2Sl1m1μ1n1+l1μ02n12+2l1m1μ0n1+l1m12δcoth12δω+l1−12m1+Sμ1n1+μ0n12δcoth12δω+l12,whereC=cnn1x+n2y+n3t,k,D=dnn1x+n2y+n3t,k,andS=snn1x+n2y+n3t,k. Substituting (33) along with equations (41) into (27), we can further obtain several types of parameter restrictions for ω. Since the calculation process is very complex, we can designate two parameters in advance, such as μ0=0andμ1=1; then, remaining parameter restrictions will be obtained. Here, we take the following parameter restrictions for an example:(44)l0=−k2−l124l2,m2=−Ω1Ω2,n2=Ω3Ω2,where(45)Ω1=k2m14n1−2k2m12n13−2m12n13+3n15−m1m3n3+m32n1,Ω2=3k2m14n12−3k2m1n14−3m12n14+3n16+m12n32−2m1m3n1,n3+m32n12Ω3=2k2m13n12−k2m1n14−m1n14+m1n32−m3n1n3.in whichk,l1,l2,m1,m3,n1,n3 are free. When n1≠0 and n2≠0, we also have difficulties in plotting the profile of equation (43) with the help of Maple. So, one sets Ω2=1andΩ3=0; then, n2=0. Choose the following parameters,(46)k=13,l0=29,l1=1,l2=1,m1=45+1663,m2=−1217753729−1640806243,m3=102727+16069,n1=1,n2=0,n3=16270+96627.The profile of equation (43) with the parameters being fixed at equation (46) is shown in Figure 3.It is clear that the soliton-cnoidal wave interaction solution (43) with the parameters being fixed at equation (46) is a periodic solution. As time changes, its shape remains unchanged.(iii)
The third special solution of equation (32) is taken as(47)G=Eπsnθ,k,μ,k,where Eπη,μ,k denotes the third type of incomplete integral. Equation (47) also leads to the soliton-cnoidal wave interaction solution of equation (4):(48)u=16S2μ−12S2μm1−m1−n12−8S8μ4l0l2m14−S8μ4l12m14+8C2S6k2μ3m1n13+24C2D2S4μ3m1n13−12S5μ3l1m12n12CD+8D2S6μ3m1n13+32S6μ3l0l2m14+32S6μ3l0l2m13n1+4S6μ3l12m14+4S6μ3l12m13n1−16C2S4k2μ2m1n13−8C2S4k2μ2n14−S8μ4m12−S8μ4m22+S8μ4m32−16C2D2S2μ2m1n13−12n14μ2S2C2D2+24S3μ2l1m12n12CD+24S3μ2l1m1n13CD−16D2S4μ2m1n13−8D2S4μ2n14−48S4μ2l0l2m14−96S4μ2l0l2m13n1−48S4μ2l0l2m12n12−6S4μ2l12m14−12S4μ2l12m13n1−6S4μ2l12m12n12+8C2S2k2μm1n13+8C2S2k2μn14+4S6μ3m12+2S6μ3m1n1+4S6μ3m22+2S6μ3m2n2−4S6μ3m32−2S6μ3m3n3−8C2D2μm1n13−8C2D2μn14−12l1m12n12μSCD−24l1m1n13μSCD−12l1n14μSCD+8D2S2μm1n13+8D2S2μn14+32S2μl0l2m14+96S2μl0l2m13n1+96S2μl0l2m12n12+32S2μl0l2m1n13+4S2μl12m14+12S2μl12m13n1+12S2μl12m12n12+4S2μl12m1n13−6S4μ2m12−6S4μ2m1n1−6S4μ2m22−6S4μ2m2n2+6S4μ2m32+6S4μ2m3n3−S4μ2n12−S4μ2n22+S4μ2n32−8l0l2m14−32l0l2m13n1−48l0l2m12n12−32l0l2m1n13−8l0l2n14−l12m14−4l12m13n1−6l12m12n12−4l12m1n13−l12n14+4S2μm12+6S2μm1n1+4S2μm22+6S2μm2n2−4S2μm32−6S2μm3n3+2S2μn12+2S2μn22−2S2μn32−m12−2m1n1−m22−2m2n2+m32+2m3n3−n12−n22+n32+1S2μ−12S4μ2l1m12+2n12μSCD−2S2μl1m12−2S2μl1m1n1+l1m12+2l1m1n1+l1n12δcoth12δω+l1−12S2μm1−m1−n12S2μ−12δcoth12δω+l12,where C=cnn1x+n2y+n3t,k,D=dnn1x+n2y+n3t,k,andS=snn1x+n2y+n3t,k. Substituting (33) along with equations (47) into (27), we designate two parameters in advance, such as μ=1andk=1/2; then, several types of parameter restrictions for remain parameters of ω will be obtained. We take one-parameter restrictions for an example:(49)n1=0,n2=n3,in which l0,l1,l2,m1,m2,m3,n3 are free. In order to demonstrate (46), choose the following parameters:(50)m1=1,m2=2,m3=1,n1=0,n2=32,n3=32,l0=1,l1=3,l2=2,k=12,μ=1.
Profile of equation (48) is shown in Figure 4.
Figure4 demonstrates the evolution of the soliton-cnoidal wave interaction solution (48) with the parameters being fixed at equation (50). It is clear that there are two deep troughs at t = 0 in Figure 4(a). Then, two troughs combine into one at t = 1 in Figure 4(b) and one trough splits into several troughs at t = 5 in Figure 4(c). As time changes, its shape changes a lot.Figure 2
Evolution of equation (38) at different times. (a) Perspective view of the wave at t=−5. (b) Perspective view of the wave at t=0. (c) Perspective view of the wave at t=5.
(a)(b)(c)Figure 3
Evolution of equation (43) at different times. (a) Perspective view of the wave at t=−5. (b) Perspective view of the wave at t=0. (c) Perspective view of the wave at t=5.
(a)(b)(c)Figure 4
Evolution of equation (48). (a) Perspective view of the wave at t=0. (b) Perspective view of the wave at t=1. (c) Perspective view of the wave at t=5.
(a)(b)(c)


## 5. Remarks

In this study, nonlocal symmetries of the Boussinesq equation are studied by means of the truncated Painlevé expansion. Two Bäcklund transformation Theorems1 and 4 for (4) are obtained via the truncated Painlevé expansion and CRE method, respectively. The nonlocal symmetry (13) of equation (4) related to the residual of the truncated Painlevé expansion is presented. Then, the nonlocal symmetry (13) is localized by introducing two new dependent variables and the corresponding nonlocal transformation group is derived. Furthermore, equation (4) is proved integrable under the sense of CRE solvable, and the soliton-cnoidal wave interaction solutions of equation (4) are explicitly given by choosing special forms of Jacobian elliptic function solutions (35), (39), and (45) for (34). Although the calculation process may be very complex, we can designate some parameters in advance; then, the whole results can be worked out with the help of Maple. In addition, the CRE method is a powerful method for dealing with exact interaction solutions to nonlinear evolution equations. Our research results may have a significant contribution to investigations of the dynamics of the distinct nonlinear waves, such as rogue waves, breather solutions, and dark solitary waves, for nonlinear systems in optics, electromagnetic field, and plasma physics. So far, methods for solving nonlocal symmetry depend on the truncated Painlevé expansion; how to solve higher order nonlocal symmetries still needs to investigate in future. For CRE solvability, many examples have been verified to be CRE solvable, but they are almost (1 + 1)-dimensional or (2 + 1)-dimensional equations. For (3 + 1)-dimensional equations, how to verify their CRE solvability is still an interesting problem.



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