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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

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Volume 72, Issue 7

Issues

Rational Solutions and Lump Solutions of the Potential YTSF Equation

Hong-Qian Sun / Ai-Hua Chen
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  • College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, P.R. China
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Published Online: 2017-06-26 | DOI: https://doi.org/10.1515/zna-2017-0137

Abstract

By using of the bilinear form, rational solutions and lump solutions of the potential Yu–Toda–Sasa–Fukuyama (YTSF) equation are derived. Dynamics of the fundamental lump solution, n1-order lump solutions, and N-lump solutions are studied for some special cases. We also find some interaction behaviours of solitary waves and one lump of rational solutions.

Keywords: Bilinear Form; Lump Solution; Potential YTSF Equation; Rational Solution

PACS: 02.30.Ik; 02.30.Jr

1 Introduction

Solving high-dimensional nonlinear partial differential equation is an important topic in soliton theory and has attracted much attention in recent years. In [1], Yu extended the Bogoyavlenskii–Schiff equation

wt+Φ(w)wz=0,   Φ(w)=x2+4w+2wx1(1)

to the (3+1)-dimensional Yu–Toda–Sasa–Fukuyama (YTSF) equation

(Φ(w)wz4wt)x+3wyy=0,   Φ(w)=x2+4w+2wx1.(2)

In this article, we consider rational solutions of the potential form (w=ux) of the (3+1)-dimensional YTSF equation (2), that is, the potential YTSF equation

4uxt+uxxxz+4uxuxz+2uxxuz+3uyy=0.(3)

For the potential YTSF equation (3), in [2], [3], [4], [5], the soliton solutions have been obtained by the extended homogeneous balance method, generalised G′/G expansion, and Hirota’s method. In [6], [7], [8], [9], [10], [11], the solitary and periodic solutions were found using tanh-function method, exp-function method, homoclinic test technique, and three-wave method. In [12], [13], the nontravelling wave solutions were found using auto-Bäcklund transformation method and some ansatz. In [14], by homoclinic (heteroclinic) breather limit method (HBLM), authors got the rogue wave solution. In [15], the rogue wave and the lump solutions were obtained by using special test function in bilinear form.

The lump solution and the rogue wave solution exist in the special form of rational solutions, which may occur in ocean and optical fibre. The lump solution is localised in all directions in the space. The rogue wave solution can be at least two times higher than the average wave crests and accompanied by deep troughs. In recent years, the lump solution and rogue wave solution are frequently presented in nonlinear partial differential equations, such as the nonlinear Schrödinger equation [16], [17], [18], [19], [20], the Davey–Stewartson equations [21], [22], the Kadomtsev–Petviashvili equation [23], [24], [25], [26], and so on [27], [28], [29], [30], [31].

In this article, by using bilinear forms, we obtain a type of rational solutions whose crest and trough are the same. And then the obtained rational solutions are lump solutions. In Section 2, general rational solutions of the potential YTSF equation are deduced with the obtained bilinear form. In Section 3, we study the dynamics of the fundamental lump solution, whose form is more general than the lump solutions obtained in [26], [29], [30]. In Sections 4 and 5, the higher order-lump (i.e. n1-order-lump) solution and multilump (i.e. N-lump) solution are obtained for the potential YTSF equation (3), and their dynamics are analysed. Finally, we give some conclusions in Section 6.

2 Rational Solutions of the Potential YTSF

In this section, through appropriate transformation, we deduce the bilinear form and rational solutions of the potential YTSF equation (3).

First by letting

xzx,   yy,   tt,(4)

the potential YTSF equation (3) is transformed into

4uxtuxxxx6uxuxx+3uyy=0.(5)

Substituting the transformation

u(x,y,t)=2(lnf(x,y,t))x(6)

into (5), we have

ffxxxx4fxfxxx+3fxxfxx3ffyy+3fyfy+4ffxt4fxft=0.

Then, we obtain the bilinear form of (5)

(Dx43Dy2+4DxDt)ff=0.(7)

Theorem 2.1 Equation (5) has rational solution (6) with f given by N×N determinant f=τ0, where

τn=det1i,jN(mij(n)),(8)

mij(n)=k=0njcjk(pjpj+ξ^j+n)njkl=0nidil(qiqi+η^in)nil1pj+qi,(9)

ξ^j=pjx+2ipj2y3pj3t,η^i=qix2iqi2y3qi3t,cik=d¯ik,qj=p¯j,(10)

ni, nj are arbitrary positive integers, pj, cik are arbitrary complex constants.

By scaling of f, we can normalise ci0=1. Thus, without the loss of generality, hereafter we set ci0=1. We call the abovementioned solution the N-rational solution of order (n1, n2, ℒ, nN).

To prove Theorem 2.1, we introduce the following lemma. This lemma can be proved using the similar method in [20] or the ideas in [32] with determinants and Pfaffians. Thus, its proof is omitted here.

Lemma 2.1 Let mij(n), φi(n), and ψj(n) be functions of x1, x2, and x3 satisfying the following differential and difference relations

mij(n)x1=φi(n)ψj(n),   mij(n)x2=φi(n1)ψj(n)+φi(n)ψj(n+1),mij(n)x3=φi(n2)ψj(n)+φi(n1)ψj(n+1)+φi(n)ψj(n+2),φi(n)xk=φi(nk),   ψj(n)xk=ψj(n+k),   (k=1,2,3).

Then the determinant

τn=det1i,jN(mij(n))

satisfies the bilinear equation

(Dx14+3Dx224Dx1Dx3)τnτn=0.(11)

Proof of Theorem 2.1: In order to prove Theorem 2.1, the functions mij(n), φi(n), and ψj(n) satisfy

ψj(n)=Ajpjneξj,φi(n)=Bi(qi)neηi,mij(n)=AjBi1pj+qipjqineξj+ηi,(12)

where ξj=pjx1+pj2x2+pj3x3, ηi=qix1qi2x2+qi3x3, and Aj and Bi are differential operators of order nj and ni with respect to pj and qi, respectively, which are defined by

Aj=k=0njcjk(pjpj)njk,   Bi=l=0nidil(qiqi)nil.(13)

Substituting Aj and Bi into (12), we have

(pjpj)pjneξj=eξjpjn(pjpj+pjpjξj+n),(qiqi)(qi)neηi=eηi(qi)n(qiqi+qiqiηin).

For simplicity, the function mij(n) can be rewritten as

mij(n)=k=0njcjk(pjpj)njkl=0nidil(qiqi)nil1pj+qi(pjqi)neξj+ηi=eξj+ηi(pjqi)nk=0njcjk(pjpj+ξ^j+n)njkl=0nidil(qiqi+η^in)nil1pj+qi.

Since the solution τn can be scaled by an arbitrary constant and the complex conjugate condition of (10), we see that τn with matrix elements (9) also satisfies the bilinear equation (11). Via the variable transformations x1=x, x2=iy, and x3=−t, the bilinear equation (11) is reduced to the bilinear form (7) of the potential YTSF equation (3).■

In the following three sections, we consider special cases for mij(n) in (9) with N=1, n1=1 as fundamental lump solution, N=1, n1=2, 3, ℒ as n1-order-lump solutions, n1=1, N>2 as N-lump solutions, respectively. For simplicity, we normalise ci0=1 and cij=0 (i·j≠0) without the loss of generality.

3 Fundamental Lump Solution of the Potential YTSF Equation

In this section, taking N=1 and n1=1 in (9), we consider the fundamental lump solution of the potential YTSF equation (3). From Theorem 2.1 and the transformation (4), we have

f=(p1p1+ξ^1)(q1q1+η^1)1p1+q1 =12a1((ξ^1p12a1)(η^1q12a1)+a12+b124a12)(14)

with

p1=a1+b1i,q1=a1b1i,   ξ^1=p1(xz)+2ip12y3p13t,η^1=q1(xz)2iq12y3q13t.

Then, the fundamental lump solutions of the potential YTSF equation (3) can be written as

u(x,y,z,t)=2(lnf(x,y,z,t))x=4a1(2a1Δ1)2a12Δ22a1Δ+8a14(y3b1t)2+1(15)

and

Δ=xz2b1y3(a12b12)t.(16)

Given y=y0 and z=z0, if a1≠0, b1≠0, the stationary points of u are

((a12+b12)y0b1+z0,y03b1),   ((a12+b12)y0b1+z0+1a1,y03b1),(17)

the extreme values are ±4a1 (see Fig. 1a). This is a fundamental lump solution since the lump solutions obtained in the following discussions consist of this type of lump solution. We note that this lump solution is the general case of the lump solutions obtained in [15].

Plots of the solution (15) of the potential YTSF equation (3) with a1=−1, y0=1, z0=−1.
Figure 1:

Plots of the solution (15) of the potential YTSF equation (3) with a1=−1, y0=1, z0=−1.

If a1≠0, b1=0, the stationary points of u stay on two straight lines

x=3a12t+2a1z0+11+16a14y022a1,x=3a12t+2a1z0+1+1+16a14y022a1.(18)

The extreme values are ±4a1/1+16a14y02 (see Fig. 1b).

4 n1-Order-Lump Solutions of the Potential YTSF Equation

In this section, taking N=1, in (9), with different values of n1>1, we consider n1-order-lump solutions of the potential YTSF equation (3). From Theorem 2.1, we have

f=(p1p1+ξ^1)n1(q1q1+η^1)n11p1+q1.(19)

We study the dynamics of n1-order-lump solutions according to the analysis of n1=2 and n1=3 vividly via the help of Mathematica.

As n1=2, from (19), we have

f=12a1(ξ^12η^12p1η^1+q1ξ^1a1ξ^1η^1p1η^12+q1ξ^122a1+p12η^12+q12ξ^122a12+p1η^12ξ^1+q1ξ^12η^1a12+b12a1(ξ^1η^1+η^1ξ^1)+2(a12+b12)a12ξ^1η^1+(a12+b12)ξ^1η^1a12+b122a1(ξ^1+η^1)+a12+b122a12(q1ξ^1+p1η^1)+a12+b12a12(ξ^1+η^1)3(a12+b12)2a13(q1ξ^1+p1η^1)a12+b12a12+3(a12+b12)22a14)

and

p1=a1+b1i,   q1=a1b1i,   ξ^1=p1(xz)+2ip12y3p13t,η^1=q1(xz)2iq12y3q13t,   ξ^1=xz+4ip1y9p12t,η^1=xz4iq1y9q12t.

Then the second-order-lump solution of the potential YTSF equation (3) can be written as

u(x,y,z,t)=2(lnf(x,y,z,t))x=4a1N2D2(20)

with

N2=4a13(a12+b12)Δ36a12b12Δ2+2a1[(a12+3b12)4a13(3a12t+b1y)+8a14(a12+b12)(y3b1t)2]Δ+8a14(2a12+b12)(y3b1t)2+12a13b12t(a12+3b12),D2=2a14(a12+b12)Δ44a13b12Δ3+2a12[(a12+3b12)4a13(3a12t+b1y)+8a14(a12+b12)(y3b1t)2]Δ2+2a1[8a14(2a12+b12)(y3b1t)2+12a13b12t(a12+3b12)]Δ+32a18(a12+b12)(y3b1t)432a17b1(y3b1t)3+16a16(1+6a1(a12+b12)t)(y3b1t)212a13b1(4a13y+b1)t+(72a16t2+1)(a12+3b12),

where Δ is defined by (16).

For given y=y0 and z0=−1, if a1=1/2, b1=1, the solution u has two lumps and their interaction is on the symmetric axis when y0<0. The two lumps get together on the symmetric axis when y0=0. While, when y0>0, two lumps split into the left and the right of the symmetric axis. Then this type of interaction has a rotation (see Fig. 2).

Plots of the second-order-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, z0=−1.
Figure 2:

Plots of the second-order-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, z0=−1.

As n1=3, the third-order-lump solution of the potential YTSF equation (3) can also be obtained, whose exact expression is very long and complex, and for simplicity, we only give the plot of this solution. The interaction behaviours are similar with the case of n1=2. For given y=y0 and z0=−1, if a1=1, b1=1, the solution u has three lumps and the interaction of the three lumps is on the symmetric axes when y0<0. The three lumps get together on the symmetric axes when y0=0. While, when y0>0, two lumps split into the left and the right of the symmetric axes and one lump still stays on the symmetric axes (see Fig. 3).

Plots of the third-order-lump solution of the potential YTSF equation (3) with a1=1, b1=1, z0=−1.
Figure 3:

Plots of the third-order-lump solution of the potential YTSF equation (3) with a1=1, b1=1, z0=−1.

If a1≠0, b1=0, the solution u is an even function about y0. One solitary wave split into two and three solitary waves for n1=2 and n1=3, respectively. Also, we find one lump interacting with one solitary wave as y0≠0 (see Fig. 4).

Plots of the rational solution of the potential YTSF equation (3) with b1=0, z0=−1.
Figure 4:

Plots of the rational solution of the potential YTSF equation (3) with b1=0, z0=−1.

5 N-lump Solutions of the Potential YTSF Equation

In this section, taking ni=nj=1 in (9), we consider N-lump solutions of the potential YTSF equation (3). From Theorem 2.1 and the transformation (4), we have

mij=(pjpj+ξ^j)(qiqi+η^i)1pj+qi =2pjqi(pj+qi)3pjη^i(pj+qi)2qiξ^j(pj+qi)2+ξ^jη^ipj+qi,(21)

with

pj=aj+bji,   pj=q¯j,   ξ^j=pj(xz)+2ipj2y3pj3t,η^i=qi(xz)2iqi2y3qi3t.(22)

We study the dynamics of N-lump solutions according to the analysis of N=2 and N=3 with Mathematica.

As N=2, the 2-lump solution of the potential YTSF equation (3) is

u(x,y,z,t)=2(lnf(x,y,z,t))x(23)

with

f=(a12+b12)(a22+b22)16a13a23(2a12Δ122a1Δ1+8a14(y3b1t)2+1)×(2a22Δ222a2Δ2+8a24(y3b2t)2+1)(a12+b12)(a22+b22)[(a1+a2)2+(b1b2)2]3[2(a1+a2i(b1b2))(Θ1+Θ¯2)+(a1+a2i(b1b2))2Θ1Θ¯2]×[2(a1+a2+i(b1b2))(Θ¯1+Θ2)+(a1+a2+i(b1b2))2Θ¯1Θ2]

and

Δ1=xz2b1y3(a12b12)t,   Θ1=Δ12ia1(y3b1t),Δ2=xz2b2y3(a22b22)t,   Θ2=Δ22ia2(y3b2t).

Given y=y0 and z0=−1, if a1=1/2, b1=1, a2=1 and b2=−1, the solution u has two lumps whose interaction behaviours are different from those of the second-order-lump solution (see Fig. 5).

The 2-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, a2=1, b2=−1, z0=−1.
Figure 5:

The 2-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, a2=1, b2=−1, z0=−1.

As N=3, the 3-lump solution of the potential YTSF equation (3) is

u(x,y,z,t)=2(lnf(x,y,z,t))x,   f=|m11m12m13m21m22m23m31m32m33|,(24)

where mij (i, j=1, 2, 3) are defined by (21).

Given y=y0 and z0=−1, if a1=1/2, b1=1, a2=1, b2=−1, a3=3/2 and b3=1, the solution u has three lumps whose interaction behaviours are different from those of the third-order-lump solution (see Fig. 6).

Plots of the 3-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, a2=1, b2=−1, a3=3/2, b3=1, z0=−1.
Figure 6:

Plots of the 3-lump solution of the potential YTSF equation (3) with a1=1/2, b1=1, a2=1, b2=−1, a3=3/2, b3=1, z0=−1.

If ai≠0, bi=0, i=1, 2 or i=1, 2, 3, the solution of u is an even function about y0. There are two or three solitary waves interacting with one lump (see Fig. 7).

Plots of the rational solution of the potential YTSF equation (3) with y0=−2, z0=−1. (a)(c): a1=1/2, a2=1, b1=b2=0; (b)(d): a1=1/2, a2=1, a3=3/2, b1=b2=b3=0.
Figure 7:

Plots of the rational solution of the potential YTSF equation (3) with y0=−2, z0=−1. (a)(c): a1=1/2, a2=1, b1=b2=0; (b)(d): a1=1/2, a2=1, a3=3/2, b1=b2=b3=0.

6 Conclusions

In this article, we obtain explicit rational solutions of the (3+1)-dimensional potential YTSF equation by using of the bilinear form. For different values of N and n1, we obtain different kinds of lump solutions such as the fundamental lump solution, n1-order-lump solutions and N-lump solutions. We also analyse their dynamics containing interactions, extreme values, and stationary points. If the parameters bi≠0 (i=1, 2, 3, ℒ), n1 or N lumps interact with each other and various interaction behaviours are shown in Figures 2, 3, 5, and 6. If the parameters bi=0 (i=1, 2, 3, ℒ), we get rational solutions containing solitary waves and one lump which are shown in Figures 4 and 7.

Acknowledgements

We are most grateful to the anonymous referees for the help in improving the original manuscript. The work described in this paper was supported by the National Natural Science Foundation of China (11471215).

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About the article

Received: 2017-04-23

Accepted: 2017-05-25

Published Online: 2017-06-26

Published in Print: 2017-07-26


Citation Information: Zeitschrift für Naturforschung A, Volume 72, Issue 7, Pages 665–672, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2017-0137.

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