# Optimal System and Group Invariant Solutions of the Whitham-Broer-Kaup System

Authors: Xiaorui Hu; Yongyang Jin; Kai Zhou
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

From the nonlocal symmetries of the Whitham-Broer-Kaup system, an eight-dimensional Lie algebra is found and the corresponding one-dimensional optimal system is constructed to provide an inequivalent classification. Six types of inequivalent group invariant solutions are demonstrated, some of which reflect the interactions between soliton and other nonlinear waves.


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## Body

## 1. Introduction

For a given nonlinear evolution equation, whether it is integrable or not, the Lie group theory is one of the most effective methods for seeking exact and analytic solutions. Due to the Lie point symmetries, one can obtain new solutions from the old one through the symmetry group and acquire the group invariant solutions by the similar reductions. With the development of symmetry theory, a variety of nonlocal symmetries have been intensely investigated in the literature, which may come from the Darboux transformation [1–5], Bäcklund transformation [6, 7], and Painlevé analysis [8–10]. Using these kinds of nonlocal symmetries, one can obtain new analytic solutions, especially the exact interaction solutions between soliton and cnoidal periodic waves.Another interesting and important problem about the group invariant solutions is to find those inequivalent branches of solutions, which leads to the concept of the optimal system. A method of constructing optimal system was firstly established by Ovsiannikov [11], using a global matrix for the adjoint transformation. Hereafter, the method has received extensive development by authors in [12–19] and others. Chou and Qu [20–22] offer many numerical invariants to address the inequivalences among the elements in the optimal system. More recently, a direct and systematic algorithm that can guarantee both the comprehensiveness and the inequivalence of the optimal system is proposed in [23] to find one-dimensional optimal system.The Whitham-Broer-Kaup (WBK) system [24–26](1)ut+uux+vx=0,(2)vt+uxxx+ux+uvx=0is derived from the Hamiltonian theory of water waves and has been said to be “the oldest, simplest and most widely known set of equations…". Here v(x,t) is the amplitude of a surface wave, propagating along the x-axis with the horizontal velocity u(x,t). A good understanding of the WBK system is very helpful for coastal and civil engineers to apply the nonlinear water wave model in a harbor and coastal design. The multisoliton solutions and the scattering problem are presented by Kaup [26]. A large class of almost periodic solutions are found algebrogeometrically by Matveev and Yavor [27]. Some soliton excitations and periodic waves solutions without dispersion relation are obtained in [28, 29]. The classical symmetries and corresponding one-dimensional optimal system of (1)-(2) are performed in [30]. Recently, Zhou and Lu have obtained the nonlocal symmetries of (1)-(2) in [31]. So a question arises: can we construct the one-dimensional optimal system based on the nonlocal symmetries? The purpose of this paper is to answer this question. Thus, Section 2 is devoted to constructing the corresponding one-dimensional optimal system. To show the interactions between soliton and other nonlinear waves, six types of inequivalent group invariant solutions are demonstrated in Section 3. The last section contains a summary.Here let us give a brief account of the nonlocal symmetries, which are somewhat different from [31], where six dependent variables were considered. We point out that five dependent variables are just enough to constitute a closed prolonged system. By direct calculations, we know that if the function ϕ satisfies(3)Ct+2Cxx-CCx+Sx=0with(4)C≡ϕtϕx,S≡ϕxxxϕx-32ϕxxϕx2,then(5)u=-ϕxx+ϕtϕx,v=-1-S-Cx-12ϕxx2ϕx2is just the solution of the WBK system (1)-(2). Thus one can verify that the WBK system (1)-(2) possesses the following nonlocal symmetries:(6)σu=2ϕx,σv=2ϕxx.That is to say, if {u,v} satisfy (1)-(2), so do {u¯(ϵ),v¯(ϵ)}={u,v}+ϵ{σu,σv}. Because (6) is nonlocal for {u,v}, we need to extend the original system by introducing two other new dependent variables:(7)g1=ϕx,g2=g1x.Furthermore, the compatibility conditions of ϕxt=ϕtx and g1xt=g1tx require(8)ux=v+1,(9)g2t+g2x+ug2+vg1x+g2=0In the following, we investigate the prolonged system including five dependent variables and eight equations (i.e., (1), (2), (5), (7), (8), and (9)), instead of the two original equations, (1) and (2).


## 2. One-Dimensional Optimal System

For the original WBK system (1)-(2), its Lie point symmetries are provided in the following:(10)ϑ=C1ϑ1+C2ϑ2+C3ϑ3+C4ϑ4with(11)ϑ1=x∂∂x+2t∂∂t-u∂∂u-21+v∂∂v,ϑ2=∂∂t,ϑ3=t∂∂x+∂∂u,ϑ4=∂∂x.For the whole prolonged system, we need to consider the following one-parameter Lie group of infinitesimal transformations with a small parameterϵ≪1,(12)x→x+ϵXx,t,u,v,ϕ,g1,g2+Oϵ2,t→t+ϵTx,t,u,v,ϕ,g1,g2+Oϵ2,u→u+ϵUx,t,u,v,ϕ,g1,g2+Oϵ2,v→v+ϵVx,t,u,v,ϕ,g1,g2+Oϵ2,ϕ→ϕ+ϵΦx,t,u,v,ϕ,g1,g2+Oϵ2,g1→g1+ϵG1x,t,u,v,ϕ,g1,g2+Oϵ2,g2→g2+ϵG2x,t,u,v,ϕ,g1,g2+Oϵ2and the vector field associated with the above group of transformations(13)K=X∂∂x+T∂∂t+U∂∂u+V∂∂v+Φ∂∂ϕ+G1∂∂g1+G2∂∂g2.Applying the standard Lie symmetry approach [12–19] to the prolonged system, we have(14)X=c1tx+c2x+c4t+c5,T=c1t2+2c2t+c3,U=c1x-tu-c2u+c4+2c6g1,V=c11-2t-2tv-2c21+v+c4+2c6g2,Φ=-c6ϕ2+c7ϕ+c8,G1=-c1tg1-c2g1-2c6g1ϕ+c7g1,G2=-2c1tg2-2c2g2-2c6g12+ϕg2+c7g2. Thus we can obtain an eight-dimensional Lie algebra represented by the following eight generators:(15)K1=tx∂∂x+t2∂∂t+x-tu∂∂u+1-2t-2tv∂∂v-tg1∂∂g1-2tg2∂∂g2,K2=x∂∂x+2t∂∂t-u∂∂u-21+v∂∂v-g1∂∂g1-2g2∂∂g2,K3=∂∂t,K4=t∂∂x+∂∂u,K5=∂∂x,K6=2g1∂∂u+2g2∂∂v-ϕ2∂∂ϕ-2g1ϕ∂∂g1-2g12+ϕg2∂∂g2,K7=ϕ∂∂ϕ+g1∂∂g1+g2∂∂g2,K8=∂∂ϕ.Clearly seen from (15), all K1-K8 are point symmetries to the eight equations of the prolonged system, while K6 is a nonlocal symmetry of the WBK system (1)-(2). The corresponding transformation groups of (15) are obtained directly by solving the initial value problems, saying(16)G1:x,t,u,v,ϕ,g1,g2→x1-ϵt,t1-ϵt,u+ϵx-ut,v+1-2t-2vtϵ+tt+vt-1ϵ2,ϕ,g11-ϵt,g21-ϵt2,G2:x,t,u,v,ϕ,g1,g2→xeϵ,te2ϵ,ue-ϵ,-1+v+1e-2ϵ,ϕ,g1e-ϵ,g2e-2ϵ,G3:x,t,u,v,ϕ,g1,g2→x,t+ϵ,u,v,ϕ,g1,g2,G4:x,t,u,v,ϕ,g1,g2→x+ϵt,t,u+ϵ,v,ϕ,g1,g2,G5:x,t,u,v,ϕ,g1,g2→x+ϵ,t,u,v,ϕ,g1,g2,G6:x,t,u,v,ϕ,g1,g2→x,t,u+2ϵg11+ϵϕ,v+2ϵg21+ϵϕ-2ϵ2g121+ϵϕ2,ϕ1+ϵϕ,g11+ϵϕ2,g21+ϵϕ2-2g12ϵ1+ϵϕ3,G7:x,t,u,v,ϕ,g1,g2→x,t,u,v,ϕeϵ,g1eϵ,g2eϵ,G8:x,t,u,v,ϕ,g1,g2→x,t,u,v,ϕ+ϵ,g1,g2.The general Lie point symmetries of the prolonged system read(17)σu=c1tx+c2x+c4t+c5ux+c1t2+2c2t+c3ut-c1x-tu-c2u+c4+2c6g1,σv=c1tx+c2x+c4t+c5vx+c1t2+2c2t+c3vt-c11-2t-2tv-2c21+v+c4+2c6g2,σϕ=c1tx+c2x+c4t+c5ϕx+c1t2+2c2t+c3ϕt--c6ϕ2+c7ϕ+c8,σg1=c1tx+c2x+c4t+c5g1x+c1t2+2c2t+c3g1t--c1tg1-c2g1-2c6g1ϕ+c7g1,σg2=c1tx+c2x+c4t+c5g2x+c1t2+2c2t+c3g2t--2c1tg2-2c2g2-2c6g12+ϕg2+c7g2.A meaningful and interesting job is to list those inequivalent group invariant solutions, which can be reduced to give a classification of the corresponding Lie algebra. We concentrate on constructing an optimal system of the eight-dimensional Lie algebra spanned by the basis{K1,K2,…,K8} in (15). Denote the symmetry algebra generated by {K1,K2,…,K8} as G and the corresponding group as G. The commutation relations between Ki and Kj are shown in Table 1, where the entry in row i and column j represents [Ki,Kj]=KiKj-KjKi. Then the adjoint representation relations between Ki and Kj, that is,(18)Adexp⁡ϵKiKj=e-ϵKiKjeϵKi=Kj-ϵKi,Kj+12!ϵ2Ki,Ki,Kj-⋯,are directly provided in Table 2 with the help of Table 1.Table 1
Commutator table ofG.
K 1 K 2 K 3 K 4 K 5 K 6 K 7 K 8 K 1 0 - 2 K 1 - K 2 0 - K 4 0 0 0  K 2 2 K 1 0 - 2 K 3 K 4 - K 5 0 0 0  K 3 K 2 2 K 3 0 K 5 0 0 0 0  K 4 0 - K 4 - K 5 0 0 0 0 0  K 5 K 4 K 5 0 0 0 0 0 0  K 6 0 0 0 0 0 0 - K 6 2 K 7  K 7 0 0 0 0 0 K 6 0 - K 8  K 8 0 0 0 0 0 - 2 K 7 K 8 0Table 2
Adjoint representation table ofG.
A d K 1 K 2 K 3 K 4 K 5 K 6 K 7 K 8 K 1 K 1 K 2 + 2 ϵ K 1 K 3 + ϵ K 2 + ϵ 2 K 1 K 4 K 5 + ϵ K 4 K 6 K 7 K 8  K 2 e - 2 ϵ K 1 K 2 e 2 ϵ K 3 e - ϵ K 4 e ϵ K 5 K 6 K 7 K 8  K 3 K 1 - ϵ K 2 + ϵ 2 K 3 K 2 - 2 ϵ K 3 K 3 K 4 - ϵ K 5 K 5 K 6 K 7 K 8  K 4 K 1 K 2 + ϵ K 4 K 3 + ϵ K 5 K 4 K 5 K 6 K 7 K 8  K 5 K 1 - ϵ K 4 K 2 - ϵ K 5 K 3 K 4 K 5 K 6 K 7 K 8  K 6 K 1 K 2 K 3 K 4 K 5 K 6 K 7 + ϵ K 6 K 8 - 2 ϵ K 7 - ϵ 2 K 6  K 7 K 1 K 2 K 3 K 4 K 5 e - ϵ K 6 K 7 e ϵ K 8  K 8 K 1 K 2 K 3 K 4 K 5 K 6 + 2 ϵ K 7 - ϵ 2 K 8 K 7 - ϵ K 8 K 8A real functionϕ on the Lie algebra G is called an invariant if ϕ(Adg(K))=ϕ(K) for all K∈G and all g∈G. If two vectors Υ and Ψ are equivalent to each other under the adjoint action, it is necessary that ϕ(Υ)=ϕ(Ψ) for any invariant ϕ. To compute the invariants of Υ=∑i=18‍aiKi, applying any Ψ=∑j=18‍bjKj to it yields(19)Adexp⁡ϵΨΥ=a1K1+⋯+a8K8-ϵΘ1K1+…+Θ8K8+Oϵ2with(20)Θ1=2a1b2-2b1a2,Θ2=a1b3-b1a3,Θ3=2a2b3-2b2a3,Θ4=a1b5-a5b1+a4b2-a2b4,Θ5=a2b5-a5b2+a4b3-a3b4.Θ6=a6b7-a7b6,Θ7=2a8b6-2a6b8,Θ8=a7b8-a8b7.For any bi(i=1⋯8), it requires(21)Θ1∂ϕ∂a1+Θ2∂ϕ∂a2+⋯+Θ8∂ϕ∂a8=0.Extracting the coefficients of all bi in (21) and setting them to be zeros, two basic invariants of G are obtained, saying(22)Δ1=a72+4a6a8,Δ2=a22-a1a3.Moreover, applying the separate adjoint actions of K1,K2,…,K8 to Υ=∑i=18‍aiKi, respectively, and with the help of Table 2, the corresponding adjoint matrices A1,A2,…,A8 are obtained: (23)A1=100000002ϵ11000000ϵ12ϵ110000000010000000ϵ11000000001000000001000000001,A2=e-2ϵ200000000100000000e2ϵ200000000e-2ϵ200000000eϵ2000000001000000001000000001,A3=1-ϵ3ϵ320000001-2ϵ300000001000000001-ϵ300000001000000001000000001000000001,A4=10000000010ϵ400000010ϵ40000001000000001000000001000000001000000001,A5=100-ϵ500000100-ϵ5000001000000001000000001000000001000000001000000001,A6=10000000010000000010000000010000000010000000010000000ϵ61000000-ϵ62-2ϵ61.A7=100000000100000000100000000100000000100000000e-ϵ700000000100000000eϵ7,A8=10000000010000000010000000010000000010000000012ϵ8-ϵ820000001-ϵ800000001.Then the general adjoint transformation matrix A is selected as(24)A=A6A7A8A4A5A1A2A3=e-2ϵ2-ϵ3e-2ϵ2ϵ32e-2ϵ2-ϵ5e-ϵ2ϵ3ϵ5e-ϵ200102ϵ1e-2ϵ21-2ϵ1ϵ3e-2ϵ22ϵ3ϵ1ϵ3e2-ϵ2-1ϵ4-ϵ1ϵ5e-ϵ2ϵ1ϵ5-ϵ4ϵ3e-ϵ2-ϵ5eϵ2000ϵ12e-2ϵ2ϵ11-ϵ1ϵ3e-2ϵ2eϵ2-ϵ1ϵ3e-ϵ22ϵ1ϵ4e-ϵ2ϵ4eϵ2-ϵ1ϵ3e-ϵ2000000e-ϵ2-ϵ3e-ϵ2000000ϵ1e-ϵ2eϵ2-ϵ1ϵ3e-ϵ200000000e-ϵ72ϵ8e-ϵ7-ϵ82e-ϵ700000ϵ6e-ϵ71+2ϵ6ϵ8e-ϵ7-ϵ81+ϵ6ϵ8e-ϵ700000-ϵ62e-ϵ7-2ϵ61+ϵ6ϵ8e-ϵ7ϵ6ϵ8e-1/2ϵ7+e1/2ϵ72.That is to say, after the adjoint action of (25)Adexp⁡ϵ6K6Adexpϵ7K7Adexp⁡ϵ8K8Adexp⁡ϵ4K4Adexp⁡ϵ5K5Adexp⁡ϵ1K1Adexp⁡ϵ2K2Adexp⁡ϵ3K3,Υ=∑i=18‍aiKi is transformed into Υ~=∑i=18‍a~iKi, shown by(26)a~1,a~2,a~3,a~4,a~5,a~6,a~7,a~8=a1,a2,a3,a4,a5,a6,a7,a8A.Now, by virtue of (26) and all the invariants, one can start to construct the optimal system of G step by step. Since the degrees of Δ1 and Δ2 are both two, we just need to consider the following five cases:(27)Δ1=1,Δ2=c;Δ1=-1,Δ2=c;Δ1=0,Δ2=1;Δ1=0,Δ2=-1;Δ1=0,Δ2=0.For each case, select a corresponding representative element in the simplest form named Υ~=∑i=1n‍a~iKi and solve (26). If (26) has the solution with respect to ϵ1,⋯ϵ8, it signifies that the selected representative element is right. If the chosen representative element makes (26) be incompatible, we need to adopt a new proper one. Repeat the process until all the cases in (27) are finished. Omitting the tedious but regular operational processes, the one-dimensional optimal system of eight-dimensional Lie algebra (15) is found to be (28)ω1=K1-cK3+K6+14K8,ω2=K1-cK3+K6-14K8,ω3=K1-cK3-K6+14K8,c∈Rω4=K1+K5+K6+14K8,ω5=K1-K5+K6+14K8,ω6=K1+K5+K6-14K8,ω7=K1-K5+K6-14K8,ω8=K1+K5-K6+14K8,ω9=K1-K5-K6+14K8,ω10=K4+K6+14K8,ω11=K4+K6-14K8,ω12=K1+K3+K6,ω13=K1+K3-K6,ω14=K1-K3+K6,ω15=K1+K5+K6,ω16=K1+K5-K6,ω17=K1-K5+K6,ω18=K1-K5-K6,ω19=K1+K6,ω20=K1-K6,ω21=K4+K6,ω22=K1+K3,ω23=K1-K3,ω24=K1+K5,ω25=K1-K5,ω26=K1,ω27=K4,ω28=K6,ω29=K6+14K8,ω30=K6-14K8.


## 3. Group Invariant Solutions

For any given subgroup, an original nonlinear system can be reduced to a system with fewer independent variables, which may be easily solved to provide group invariant solutions. As Olver said, since there is almost always an infinite amount of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. In this section, some types of inequivalent group invariant solutions which correspond to the elements in the optimal system (28) are listed.( 1 ) For ω1 with c=0, we consider an equivalent case K6+K8+K3+αK5, where α is an arbitrary constant. The group invariant solution is(29)ϕ=2tanh⁡22t+22P,u=-2G1tanh⁡22t+22P+U,v=-G12tanh2⁡22t+22P+2G12+2G2tanh⁡22t+22P+V,g1=-2G11+cosh⁡2t+2P,g2=2G2+42G121+e2t+P-11+cosh⁡2t+2P.Here U, V, P, G1, and G2 are all group invariant functions of X(=x-αt). Substituting the group invariant solutions (29) into the prolonged system leads to the symmetry reduction equations for U, V, P, G1, and G2:(30)P′=-G1,G2=-G1′-2G12,U=-G1′-αG1-1G1,V=-12U2+αU+c0with(31)G1′′=32G1′2G1-2G1′G1+G13-α0G1+12G1,α0=1+12α2+c0.Here, and in the latter of the paper, the primes ′,′′ on the functions with only one independent variable denote derivatives. Thus by the transformation(32)G1=R24R′,R≡R24X,(31) becomes(33)R′′′=R′′22R′-R22R′+22α0R′.For different values of α0, (33) is exactly solvable. For example, if we take α0=3/2, this equation has a solution(34)R=c1e2X+c2eX+c3e-X+c2c32c1e2X.Then, by solving (30), one can directly obtain an exact solution of the WBK system (1)-(2):(35)u=-22c2c3eX+c12e-X+c1c2e2X+c1c3e-2X22c2c3eX-2c12e-X+2c1c2e2X-2c1c3e-2Xtanh⁡22t+12ln⁡c1e2X+2c3eXc1-2c2e1+2X+c4+22c2c31+αeX-2c12α-1e-X+2c1c2α+2e2X-2c1c32-αe-2X22c2c3eX-2c12e-X+2c1c2e2X-2c1c3e-2X,v=ux-1.( 2 ) One equivalent case of ω5 is K6+K8+K3-K4 and the group invariant solution in this case is(36)ϕ=2tanh⁡22t+22P,u=-2G1tanh⁡22t+22P+U+t+P,v=-G12tanh2⁡22t+22P+2G12+2G2tanh⁡22t+22P+V,g1=-2G11+cosh⁡2t+2P,g2=2G2+42G12+2G2e2t+P1+e2t+P1+cosh⁡2t+2P.Here U, V, P, G1, and G2 are all group invariant functions of Z(=x-t2/2), which should satisfy(37)P′=-G1,G2=-G1′-2G12,U=1-G1′G1-P,V=Ux+G12-G1-1.And(38)G1′′=32G1′2G1-2G1′G1+G13+X-c0G1+12G1.Thus by means of the transformation(39)G1=W24W′,W≡WY,Y=24X,we can transform (38) into(40)W′′′=W′′22W′-W22W′+122c0-24YW′.On differentiation, (40) becomes linear and of the fourth order:(41)W′′′′=-242W′+2c0-24YW′′-W.Thus, the general solution of (38) is a rational function of the constants of integration [32].Remark 1.
It is known that the solitary wave is a very important phenomenon in the nature and physics. In fact, these solitary waves must interact with other waves or spread on some nonconstant background. Here, the forms of solutions (29) and (36) directly embody this kind of interactions between the soliton and other nonlinear waves.( 3 ) For ω10 and ω11, it is not difficult to obtain the following group invariant solutions:(42)u=±22ttanh⁡24x+c1t+c2t+x+c2t,v=ux-1.( 4 ) Instead of studying ω15, consider an equivalent case: v3-v4+v6. The corresponding group invariant solution of the WBK system reads(43)u=-t+UX-G1Xt+2PX,v=VX-2-UXG1Xt+2PX-G1X22t+2PX2,X=x+12t2,where(44)UX=43c1Ai1,X1+Bi1,X1c1AiX1+BiX1,VX=X-1+23c1Ai1,X1+Bi1,X1c1AiX1+BiX12,G1X=2X-223c1Ai1,X1+Bi1,X12+c2c1AiX1+BiX12,PX=-14X2+232∫X0Xc1Ai1,X1+Bi1,X12+c2c1AiX1+BiX12dX,X1=X23.Here, Ai(x) and Bi(x) are the Airy wave functions, which are linearly independent solutions for w′′-xw=0.( 5 ) For ω19, it is equivalent to K3+αK5+K6 with α being an arbitrary constant. The group invariant solution is presented as(45)u=UX-G1Xt+2PX,v=VX-G2Xt+2PX-G1X22t+2PX2,X=x-αtwith(46)UX=α+2c1tanhc1X+c2,VX=2c12sech2⁡c1X+c2-1,PX=-X4c1+2c1X-c34c121+e2c1X+2c2+c4,G1X=2c1X+sinh⁡2c1X+2c2-c3c11+cosh⁡2c1X+2c2,G2X=4+4cosh⁡2c1X+2c2+2c3-2c1Xsinh⁡2c1X+2c21+cosh⁡2c1X+2c22.( 6 ) By solving ω21, one can easily give a simple exact solution:(47)u=2x-c1t+2c2,v=-2x-c1t+2c22-1.


## 4. Summary

In the beginning of [31], the authors also discussed the nonlocal symmetry (6) of the WBK system. To realize the localization of the nonlocal symmetry, they referred to six dependent variables. In fact, we point that five dependent variables are enough in this paper. Based on a Bäcklund transformation, [31] constructed some special exact solutions, which have no direct connection with the given nonlocal symmetry. The purpose of our article is to give a classification of the eight-dimensional Lie algebra (15) and then to find out some inequivalent group invariant solutions with respect to the nonlocal symmetry (6).In this paper, we firstly localize the nonlocal symmetries of the (1+1)-dimensional WBK system (1)-(2) to Lie point symmetries by prolonging the original system to a larger one. Then, all the Lie point symmetries of the whole prolonged system are presented, which constitute an eight-dimensional Lie algebra (15). The one-dimensional optimal system (28) is constructed to give a classification of the elements in the Lie algebra. The adjoint transformation matrix and all the invariants of the Lie algebra are also displayed. Finally, we demonstrate six kinds of inequivalent group invariant solutions, some of which reflect the interactions between soliton and other nonlinear waves.



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