Xi-zhong Liu(刘希忠) and Jun Yu(俞军)

Institute of Nonlinear Science,Shaoxing University,Shaoxing 312000,China

Keywords: nonlocal Boussinesq equation,N-soliton solution,periodic waves,symmetry reduction solutions

1. Introduction

Nonlocal equations have attracted much attention in nonlinear field since Ablowitz and Musslimani[1]introduced and investigated a PT symmetric nonlocal Schr¨odinger equation

with* denoting complex conjugate andqbeing a complex valued function of the real variablesxandt. Equation (1) is integrable in the sense of having infinite number of conservation laws and some soliton solutions of it are obtained by using the inverse scattering transform method in Ref. [1]. Since then, many nonlocal equations with certain space time symmetries have been introduced and investigated,such as the nonlocal Korteweg–de-Vrise(KdV)systems,[2]the nonlocal Davey–Stewartson systems,[3]nonlocal Schr¨odinger equation,[4–7]and so on. Among the methods of solving nonlinear systems the Hirota’s bilinear method[8,9]plays an important role,which is developed recently to getN-soliton solutions for both(1+1)-dimensional integrable equations[10]and(2+1)-dimensional integrable equations.[11–14]For nonlocal systems, the Hirota’s bilinear method can also be used to getN-soliton solutions.[6,15,16]In the same time various methods for solving nonlocal nonlinear systems have also been developed,such as inverse scattering transformation,[1,17]Darboux transformations,[18,19]symmetry analysis,[20]etc.

In recent years, Lou proposed a concept of Alice–Bob(AB) system, in which two variables are linked byB= ˆfAwith ˆfbeing a combination of parity(ˆP), time reversion(ˆT),and conjugation (ˆC)-related operator, to describe two events intertwined with each other and introduced ˆP- ˆT- ˆCprinciple and ˆf= ˆf-1equivalence principle to construct and solve nonlocal AB systems from known local ones.[21]Many nonlocal AB systems, including AB–KdV system,[2,22]AB–mKdV[23,24]system,AB–KP system,[25]AB–Toda system,[2]AB–Schr¨odinger equation,[26]etc., have been introduced and their integrable properties and exact solutions have also been studied a lot.

It is known that the Boussinesq equation plays an important role in describing long wave propagation in shallow water.[27–29]Here, we take the Boussinesq equation in the form

The paper is organized as follows. In Section 2,we give multiple soliton solutions of the nonlocal Boussinesq equation(3)using known soliton solutions of the local Boussinesq equation. In Section 3,we obtain periodic solutions and traveling solutions of Eq.(3)directly from known solutions of the local Boussinesq equation. In Section 4,the symmetry reduction solutions of Eq.(3)are derived by using classical symmetry method. The last section devotes to a summary.

2. Multiple soliton solutions of the nonlocal Boussinesq equation(3)

According to the consistent correlated bang method in Ref.[30],the functionsAandBin Eq.(3)can be written as

withνi=0,1 (i=1,...,N), the summations of it should be done for all permutations and

where the summation ofνshould be done for all non-dual permutations ofνi=1,-1,(i=1,...,N)and

with arbitrary constantsc1andc2. It can be seen from the expression ofuin Eq. (14) that it is an even function under the operator of ˆPsˆTd,which meansvin Eq.(19)being an odd function, so the condition of Eq. (6) is satisfied. Now we have a special form of solution of Eq.(3)

with Eq.(14).

Substituting Eqs.(16)and(17)into Eq.(20)with Eq.(14)we get 2-soliton and 3-soliton solutions of the nonlocal Boussinesq equation(3), which are shown in Figs.1(a)–1(c).Similarly, we can get any arbitraryNsoliton solutions of Eq. (3), Fig. 1(d) is a plot of 4-soliton solution. In Fig. 1,the parameters are fixed as follows:k1=1,k2=4,x0=t0=η10=η20=0,c1=1,c2=1 for Fig. 1(a);k1=1,k2=3,k3= 4,x0=t0=η10=η20=η30= 0,c1= 1,c2= 2 for Figs. 1(b) and 1(c); andk1=-1,k2=-2,k3=4,k4=6,x0=t0=η10=η20=η30=η40=0,c1=4,c2=-2 for Fig.1(d). From Fig.1,we see that for any 2-soliton,3-soliton,and 4-soliton solutions, the interaction between solitons are elastic with phases changed.

Fig.1. The density plots of N soliton solutions of Eq.(3)with(a)N=2,(b)N=3,(d)N=4,panel(c)is a three-dimensional(3D)plot of N=3 case.

3. Periodic wave solutions and traveling wave solutions of the nonlocal Boussinesq equation(3)

To solve the Boussinesq equation (7), the author of Ref. [27] obtained some kinds of exact solutions, including soliton solutions, traveling wave solutions, plane periodic solutions,etc.Here, we use these known solutions, which all satisfy the parity condition(6), to obtain corresponding solutions of the nonlocal Boussinesq equation(3).

(i)Soliton solution 1

When we take

Figure 3 is density plots of two plane periodic solutions of the nonlocal Boussinesq equation(3): Fig.3(a)for Eq.(30)with parameters being fixed as Eq.(38);Fig.3(b)for Eq.(32)with parameters being fixed as Eq.(37).

Fig. 2. The 3D plot of two single soliton solutions of Eq. (3): panel (a) is for Eq.(22)with parameters being fixed as Eq.(38);panel(b)is for Eq.(24)with parameters being fixed as Eq.(37).

Fig.3.The density plot of two plane periodic solutions of Eq.(3):panel(a)is for Eq.(30)with parameters being fixed as Eq.(37);panel(b)is for Eq.(32)with parameters being fixed as Eq.(38).

4. Symmetry reduction solutions of the nonlocal Boussinesq equation(3)

Symmetry analysis plays an important role in solving nonlinear equations, in this section we seek Lie point symmetry of the nonlocal Boussinesq equation (3) by using the classic Lie group method.[31,32]To this end, we assume that the Lie point symmetry of Eq.(3)is in the form

To solve forX,T,A,B, in Eq. (39), by substituting Eqs. (41a) and (41b) into Eqs. (42a) and (42b) and eliminatingAtt,Bttby Eqs. (3) and its ˆPsˆTdcounterpart, respectively,after vanishing all the coefficients of the independent partial derivatives of variablesA,B,a system of over determined linear equations are obtained. After solving these equations using software like Maple and considering the relation(42c),we obtain

The invariant solutions of the nonlocal Boussinesq equation(3)can be obtained by assumingσA=σB=0 in Eqs.(43a)and (43b), which is equivalent to solving the characteristic equation

which can be verified easily.

5. Conclusion

In summary, a nonlocal Boussinesq equation is investigated by converting it into two local equations.Using a knownN-soliton solutions of the local Boussinesq equation,a general form of multiple soliton solutions is obtained, among which theN= 2, 3, 4 soliton solutions are plotted and analyzed.Some kinds of single soliton solutions, travelling wave solutions,and plane periodic solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation.The symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by applying the classic Lie symmetry method on it.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos. 11975156 and 12175148) and the Natural Science Foundation of Zhejiang Province of China(Grant No.LY18A050001).