# Exact Combined Solutions for the (2+1)-Dimensional Dispersive Long Water-Wave Equations

Authors: Yi Wei; Xing-Qiu Zhang; Zhu-Yan Shao; Lu-Feng Gu; Xiao-Feng Yang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The homogeneous balance of undetermined coefficient (HBUC) method is presented to obtain not only the linear, bilinear, or homogeneous forms but also the exact traveling wave solutions of nonlinear partial differential equations. Linear equation is obtained by applying the proposed method to the (2+1)-dimensional dispersive long water-wave equations. Accordingly, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the (2+1)-dimensional dispersive long water-wave equations are obtained directly. The HBUC method, which can be used to handle some nonlinear partial differential equations, is a standard, computable, and powerful method.


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## Body

## 1. Introduction

Nonlinear partial differential equations (NLPDEs) are used to describe a variety of phenomena not only in physics [1, 2], thermodynamics [3], fluid dynamics [4, 5], and practical engineering [6, 7] but also in several other fields [8]. How to obtain the traveling wave solutions for NLPDEs is very important in the nonlinear phenomena [1, 9, 10]. In recent decades, there are many excellent methods, such as the G′/G-expansion method [11, 12], the homotopy perturbation method [13, 14], the Riccati-Bernoulli sub-ODE method [15, 16], the three-wave method [17, 18], the inverse scattering method [19, 20], the first integral method [21, 22], Hirota’s bilinear method [23, 24], the homogeneous balance method [25, 26], the iteration method [27, 28], the tanh-sech method [29, 30], and the extended homoclinic test method [31, 32], which are applied to obtain the exact traveling wave solutions of some NLPDEs.The above traditional methods can be used to handle some well-known NLPDEs. However, there is no unified approach, which can be dealt with all NLPDEs. To obtain the traveling wave solutions of NLPDEs, Hirota’s bilinear method, the three-wave method, and theG′/G-expansion method are employed to investigate the traveling wave solutions of many NLPDEs. Unfortunately, some exact solutions are omitted by using Hirota’s bilinear method, the three-wave method, and the G′/G-expansion method if the NLPDEs can be linearized. To solve this problem, the HBUC method is proposed to derive the linear forms of NLPDEs.In this paper, the (2+1)-dimensional dispersive long water-wave equations (DLWEs) [33, 34] are investigated as follows:
(1)uyt+uxxy−2vxx−u2xy=0,(2)vt−vxx−2uvx=0,where u=ux,y,t represents the surface velocity of water along the x-direction and v=vx,y,t gives the surface velocity of water along the y-direction.The DLWEs can also be derived from the well-known Kadomtsev-Petviashvili equation using the symmetry constraint. The DLWEs were used to model nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The DLWEs also appear in many scientific applications such as nonlinear fiber optics, plasma physics, fluid dynamics, and coastal engineering. Moreover, the solutions of the DLWEs are very helpful for coastal and engineers to apply the nonlinear water model to coastal and harbor design [33].The DLWEs were investigated where different approaches were exploited. Wu et al. reported that the DLWEs exist of many nonpropagating hydrodynamical solitons both in theory and in experiment, and the DLWEs have no Painleve property, though the system is Lax or inverse scattering transformation integrable [35]. Paquin and Winternitz investigated the DLWEs by the Lie group method [36]. The extended mapping approach [37], the extended projective approach [38], and the tanh-sech method [33] are among many other methods that were used to handle the DLWEs. Much effort has been focused on the existence of propagating solitons [36], multiple soliton solutions, and rational solutions [33].In this paper, the linear equation of the DLWEs is derived by the HBUC method. Then, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the DLWEs are obtained directly.The remainder of this paper is organized as follows: the HBUC method is presented in Section2. In Section 3, the HBUC method is used to obtain N-multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of Equations (1) and (2) directly. In Section 4, some important conclusions are given.


## 2. Description of the HBUC Method

In this section, the following general NLPDE in two variables is considered:(3)Pu,ut,ux,uxx,uxt,⋯=0,where P is a polynomial function of its arguments; the subscripts x and t denote the partial derivatives of u, respectively. The HBUC method consists of three steps as follows:Step 1.
Assume that the Equation (3) has a solution of the following form:
(4)u=amnlnwm,n+∑i,j=0i+j≠0,m+ni=m,j=naijlnwi,j+a00,where u=ux,t, w=wx,t, and lnwi,j=∂i+jlnwx,t/∂xi∂tj, and m,n (balance numbers) and aiji=0,1,⋯,m;j=0,1,⋯,n (balance coefficients) amn≠0 are constants to be determined later.
The balance numbers can be determined by balancing the highest nonlinear terms and the highest order partial derivative terms. A set of algebraic equations for the balance coefficients is obtained by substituting Equation (4) into Equation (3) and balancing the terms with wx/wiwt/wj.Step 2.
If the NLPDEs can be linearized, the linear equation can be obtained by solving the set of algebraic equations and simplifying Equation (3) directly or after integrating some time (generally, integrating times equal to the orders of the lowest partial derivative of Equation (3)) with respect to x and t.Step 3.
Based on Step1 and Step 2, by using traveling wave transformations
(5)wx,t=wξ,and
(6)ξ=x−Vt,Equation (3) can be reduced to a linear partial differential equation
(7)wt+α1wxx+α2wx+α3w,where α1, α2, and α3 are constants. Then, solving the linear partial differential equation (7) yields the exact combined solutions of Equation (3). Next, Equations (1) and (2) are chosen to obtain the combined solutions by applying the HBUC method.


## 3. Application to the (2+1)-Dimensional DLWEs

Assume that the solutions of Equations (1) and (2) are of the forms
(8)u=amnllnwm,n,l+∑i,j,k=0i+j+k≠0,m+n+li=m,j=n,k=laijklnwi,j,k+a000,(9)v=bpqrlnwp,q,r+∑i,j,k=0i+j+k≠0,p+q+ri=p,j=q,k=rbijklnwi,j,k+b000,where u=ux,y,t, w=wx,y,t, and lnwi,j,k=∂i+j+klnwx,y,t/∂xi∂yj∂tk, and m,n,l,p,q,r (balance numbers) and aijki=0,1,⋯,m;j=0,1,⋯,n;k=0,1,⋯,lamnl≠0 and bijki=0,1,⋯,p;j=0,1,⋯,q;k=0,1,⋯,rbpqr≠0 (balance coefficients) are constants to be determined later.Balancinguxxy, vxx, and u2xy in Equation (1) and vxx and uvx in Equation (2), it is required that
(10)m+2=2m+1=p+2,n+1=2n+1=q,l=2l=r,p+2=m+p+1,q=q+n,l+r=l.Solving the above algebraic equations, we getm=1,n=0,l=0,p=1,q=1,r=0. Then, Equations (8) and (9) can be written as
(11)u=a1lnwx+a0,v=b3lnwxy+b2lnwx+b1lnwy+b0,where aii=0,1 and bii=0,1,2,3 are constants to be determined later.Substituting Equation (11) into Equations (1) and (2) and equating the coefficients of wx3wy/w4 on the left-hand side of Equations (1) and (2) to zero yield a set of algebraic equations for a1 and b3 as follows:
(12)−6a12+a1−2b3=0,−6b3a1−1=0.Solving the above algebraic equations and noticinga1b3≠0, we get a1=b3=1. Substituting a1 and b3 back into Equation (11), we get
(13)u=lnwx+a0,v=lnwxy+b2lnwx+b1lnwy+b0.Substituting Equation (13) back into Equations (1) and (2), Equation (1) minus Equation (2) is
(14)A1w+A2w2+A3w3=0,where
(15)A1=b22a0wxx−wxxx−wxt+b12a0wxy−wxxy−wyt+2b0wxx,A3=−6b2wx3−6b1wx2wy,A2=b2wxwt+7wxxwx−2a0wx2+b13wxxwy+wywt+4wxywx−2a0wxwy−2b0wx2.Obviously, settingbi=0i=0,1,2, we find that Equation (1) coincides with Equation (2). According to the above analysis, suppose that the solutions of Equations (1) and (2) are of the forms
(16)ux,y,t=lnwx+a0,vx,y,t=lnwxy,where a0 is an arbitrary constant and w=wx,y,t is a function of x,y,t that will be determined later.Substituting Equation (16) into Equations (1) and (2) yields a single NLPDE
(17)K0+K1+K2+K3=0,where
(18)K0=a0−2wxxyw+2wxxwy+4wxwxyw2−4wx2wyw3,K1=wxyt−wxxxyw,K2=wxwxxy−wxywt−wxtwy+wxxwxy+wxxxwy−wxwytw2,K3=2wxwywt−2wxwxxwyw3.Simplifying Equation (17) and integrating with respect to x once, we get
(19)∂∂xwytw−wywtw2−wxxyw−wxxwyw2−2a0wxyw−wxwyw2=0.Equation (19) is identical to
(20)wytw−wywt−wxxyw−wxxwy−2a0wxyw−wxwy−py,tw2=0.where py,t is an arbitrary function of y,t.Particularly, takingpy,t=0 in Equation (20), the bilinear equation of Equations (1) and (2) is obtained as follows:
(21)wytw−wywt−wxxyw−wxxwy−2a0wxyw−wxwy=0.Equation (21) can be written concisely in terms of D-operator as
(22)Dywt−wxx−2a0wx·w=0,where
(23)DxmDtna·b=∂x−∂x′m∂t−∂t′nax,tbx′,t′x′=x,t′=t.By using the property ofD-operator, Equation (22) is identical to
(24)wt−wxx−2a0wx−qx,tw=0,where qx,t is an arbitrary function of x,t.Particularly, takingqx,t=α=constant in Equation (24), we get a linear equation
(25)wt−wxx−2a0wx−αw=0.Remark 1.
We note that Equation (25) does not depend on the variable y, instead it depends only on the variables x,t.
Using the transformation(26)wx,y,t=wξ=wx+ly−ct,Equation (25) is reduced to
(27)w″+2a0+cw′+αw=0,where ly is an arbitrary function of y and the prime denotes the derivation with respect to ξ.
There are three types of traveling wave solutions of Equation (27) as follows.
WhenΔ=2a0+c2−4α>0,
(28)w=C1e−2a0+c+Δ/2ξ+C2e−2a0+c−Δ/2ξ,where ξ=x+ly−ct; C1,C2,a0,α, and c are arbitrary constants; and ly is an arbitrary function of y.
Generally, noticing the linear property of Equation (25), we can get the exact solution of Equation (25) as follows:
(29)w1=∑i=1n1C1,1ie−2a0+c1i+2a0+c1i2−4α/2ξ1i+C2,1ie−2a0+c1i−2a0+c1i2−4α/2ξ1i,where Δ1i=2a0+c1i2−4α>0 and ξ1i=x+l1iy−c1it; C1,1i,C2,1i,a0,α, and c1i are arbitrary constants; and l1iyi=1,2,⋯,n1 are arbitrary functions of y.
WhenΔ=2a0+c2−4α<0,
(30)w=e−2a0+c/2ξC1cos−Δ2ξ+C2sin−Δ2ξ,where ξ=x+ly−ct; C1,C2,a0,α, and c are arbitrary constants; and ly is an arbitrary function of y.
Generally, noticing the linear property of Equation (25), we can get the exact solution as follows:
(31)w2=∑i=1n2e−2a0+c2i/2ξ2iC1,2icos−Δ2i2ξ2i+C2,2isin−Δ2i2ξ2i,where Δ2i=2a0+c1i2−4α<0 and ξ2i=x+l2iy−c2it; C1,2i,C2,2i,a0,α, and c2i are arbitrary constants; and l2iyi=1,2,⋯,n2 are arbitrary functions of y.
WhenΔ=2a0+c2−4α=0,
(32)w=C1+C2ξe−2a0+c/2ξ,where ξ=x+ly−ct; C1,C2,a0,α, and c are arbitrary constants; and ly is an arbitrary function of y.
Generally, noticing the linear property of Equation (25), we can get the exact solution as follows:
(33)w3=∑i=12C1,3i+C2,3iξ3ie−2a0+c3i/2ξ3i,where Δ3i=2a0+c3i2−4α=0 and ξ3i=x+l3iy−c3it; C1,3i,C2,3i,a0,α, and c3i are arbitrary constants; and l3iyi=1,2 are arbitrary functions of y.
Generally, we can get combined solutions of Equation (25) as follows:
(34)w=w1+w2+w3,where w1,w2,andw3 are given by Equations (29), (31), and (33), respectively.
Accordingly, we can get combined solutions of Equations (1) and (2)
(35)w=w1+w2+w3,ux,y,t=lnwx+a0,vx,y,t=lnwxy=uy.
Choosing the appropriate parameters in Equation (35) can obtain all solutions of DLWEs in Ref. [15]. For example, setting C1,2i=C2,2i=0i=1,2,⋯,n2, C1,3i=C2,3i=0i=1,2, n1=N,C1,1i=1/N,C2,1i=1,α=a0=0, and l1iy=l1iy, we get the N-kink solutions and the N-soliton solutions
(36)ux,y,t=−∑i=1Nc1ie−c1ix+l1iy−c1it1+∑i=1Ne−c1ix+l1iy−c1it,vx,y,t=−∑i=1Nc1ie−c1ix+l1iy−c1it∑i=1Nc1il1ie−c1ix+l1iy−c1it1+∑i=1Ne−c1ix+l1iy−c1it2−∑i=1Nc1il1ie−c1ix+l1iy−c1it1+∑i=1Ne−c1ix+l1iy−c1it,where c1i and l1ii=1,⋯,N are arbitrary constants.
SettingC1,1i=C2,1i=0i=1,2,⋯,n1, C1,3i=C2,3i=0i=1,2, n2=1, ξ20=−arctanC2,21/C1,21, and l2iy=l2iy, we get the singular solution (periodic solutions)
(37)ux,y,t=−4α−2a0+c222tan4α−2a0+c222x+l2y−c2t+ξ20−c22,vx,y,t=uy.where C1,21,C2,21,l2,α,a0, and c2 are arbitrary constants.
SettingC1,1i=C2,1i=0i=1,2,⋯,n1, C1,2i=C2,2i=0i=1,2,⋯,n2,C1,31=C1, C1,32=C2, and C1,32=C2,32=0, we get the rational solutions
(38)ux,y,t=C2C1+C2x+l3y−c3t−c32,vx,y,t=uy,where C1,C2,l3,α,a0, and c32a0+c32−4α=0 are arbitrary constants.Remark 2.
We can deal with Equation (25) by using some assumptions. For example, suppose that w=−tβy+Wx, α=0, and a0≠0, we get
(39)w=−tβy−C1ye−2a0x2a0−βyx2a0+C2y,ux,y,t=βy2a02t+a0x+1−a0C1ye−2a0x−2C2ya02βy2a0t+x+C1ye−2a0x−2a0C2y,vx,y,t=uy,where C1y,C2y, and βy are arbitrary functions of y, and a0≠0 is an arbitrary constant.
Suppose thatw=−βyt+Wx and α=a0=0, we get
(40)w=−tβy−βyx22+xC1y+C2y,ux,y,t=2xβy−2C1yβyx2+2t−2xC1y−2C2y,vx,y,t=uy,where C1y,C2y, and βy are arbitrary functions of y.
Similarly, whenα=a0=0, Equation (25) is reduced to wt−wxx=0. We can get the exact solution
(41)w=C1y−C2yte−x2/4t,ux,y,t=−xC2ye−x2/4t2ttC1y+C2ye−x2/4t,vx,y,t=uy,where C1y and C2y are arbitrary functions of y.Similarly, we can assume thatw=∑i=1npixqit; then, a new solution of Equation (25) can be obtained. Being similar to the above process, we omit it.


## 4. Conclusions

The (2+1)-dimensional dispersive long water-wave equations can be linearized by the HBUC method. Then, the N-multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of Equations (1) and (2) can be obtained. Many well-known NLPDEs, such as the Whitham-Broer-Kaup equations, the Broer-Kaup equations, and the variant Boussinesq equations, can be handled by the HBUC method. The performance of the HBUC method is found to be simple and efficient. The HBUC method is also a standard, computable, and powerful method, which allows us to solve complicated and tedious algebraic calculations by the availability of computer systems like Maple.



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