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ORIGINAL RESEARCH article

Front. Phys., 08 December 2022
Sec. Interdisciplinary Physics
Volume 10 - 2022 | https://doi.org/10.3389/fphy.2022.1055998

The coupled Boussinesq equation and its Darboux transformation on the time–space scale

www.frontiersin.orgXiaoqian Huang www.frontiersin.orgYong Zhang www.frontiersin.orgHuanhe Dong*
  • College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China

Gel’fand-Dikii (GD) formalism is an approach for generating integrable systems in terms of fractional powers of the δ differential operator. In this paper, it extends the GD formalism associated with the third-order δ differential operator L to the time scale. Then, the coupled Boussinesq equation on the time–space scale is given by taking special values, and it can be reduced on different time–space scales. Moreover, the exact solutions of the coupled Boussinesq equation on the time–space scale and the classical Boussinesq equation are constructed via employing the extensions of the Darboux theorem and Crum theorem on the time scale.

1 Introduction

There are many methods to produce soliton equations, such as the Ablowitz–Kaup–Newel–Segur (AKNS) method [1], Ablowitz–Ladik (AL) method [2], and GD formalism [3]. Compared with the other two methods, the difference of GD formalism is that differential operators, shift operators, and their inverses are regarded as essential tools to generate integrable systems. In this formalism, arbitrary Lax pairs can be constructed from the calculation of fractional powers of the operator L [4]. In addition, it is also quite significant in studying the symmetries, bi-Hamiltonian formulations, and construction of recursive operators for nonlinear PDE [57].

Nevertheless, earlier studies mostly centered on continuous systems or discrete systems. Since there are some complex problems that can only be solved exactly when considering the parallel analysis of continuous and discrete cases, the time scale was initiated by Stefan Hilger in 1988 [810]. Recently, soliton equations on time scales have been studied widely [1114]. Agarwal et al. defined hyperbolic functions on the time scale to solve evolution equations [15]. Gürses et al. extended the GD formalism on the time scale to obtain more universal integrable nonlinear dynamic equations and generated integrable equations in terms of integers Z and q-numbers particularly. On this basis, the Δ-Burgers hierarchy and its recursion operator were derived by taking appropriate δ differential operators [7]. Ahlbrandt et al. investigated the Hamiltonian system on the time scale and its symplectic flow properties [16].

Darboux theorem was first proposed by Darboux in 1882 from the eigenfunction of the Schrödinger equation and the covariant transformation of potential and applied to surface theory [17]. In 1995, Crum obtained a popularization of the Darboux theorem via a recurrent relation from the N-fold Darboux transformation [18]. The notable merit of DT (Darboux transformation) is that solutions of the soliton equation may be obtained by means of finite iteration, which results it in a research focus of integrable systems in continuous or discrete form [1923]. A Lax pair for the modified Boussinesq equation and its DT was derived by Geng [24]. Nimmo constructed the DT for the discrete KP equation and reduced to the discrete BKP equation resulting from an invariance of the binary transformation [25]. At present, the DT has been considered on the time scale, and several research results have been acquired. Hovhannisyan et al. obtained a solution of the KdV equation hierarchy from extensions of the Darboux theorem and Crum theorem on the time scale calculus [13]. By using gauge transformation, Dong et al. obtained the DT of the Gerdjikov–Ivanov equation on the time–space scale [26]. However, the DT of integrable systems has still rarely been studied under the unified analysis of continuous systems and discrete systems.

The Boussinesq equation is an important soliton equation [2729], alongside other important research achievements in physics and mathematics such as one-dimensional nonlinear lattice waves [30, 31], vibrations in a nonlinear string [27], and ion sound waves in plasma [32]. A normative approach was presented to obtain the DT for the classical Boussinesq system by Zhang et al. [33]. In [34], the new solutions of the Boussinesq–Burgers equation were achieved via applying a DT with multi-parameters [34]. In order to better promote the research of the Boussinesq equation on the time scale, the coupled Boussinesq equation and its exact solution will be obtained on the time–space scale by extending the GD formalism and Darboux theorem to the time scale in this paper.

This paper is organized as follows. In Section 2, through employing the GD formalism associated with the third-order δ differential operator L, the coupled Boussinesq equation on the time–space scale is produced from its spectral problem. Section 3 presents the exact solutions of the coupled Boussinesq equation on the time–space scale, and the single-soliton solution of the classical Boussinesq equation is achieved on the special time scale R. The last section is the conclusion.

2 Coupled Boussinesq equation on the time–space scale

In this section, taking special values, we will establish the coupled Boussinesq equation on the time–space scale from the extension for GD formalism on the time scale. Primarily, we will introduce several notions connected to the time scale [8, 9, 35].

Definition 1. Let T be a time scale. For sT, forward and backward jump operators ξ, ζ: TT are specified as

ξs=infxT:x>s,ζs=supxT:x<s.(1)

Definition 2. Forward and backward jump distance functions μ, ν: T[0,+) are specified as

μs=ξss,νs=sζs,(2)

which are also called graininess functions.

Lemma 1. When T=R, T=Z, forward jump operators and graininess functions become

ξs=infs,=s,ξs=infx+1,x+2,=x+1,μs=ξss=0,μs=ξss=1,(3)

respectively.

Theorem 1. Let Tk represent Hilger’s truncated set which is composed of T removing the largest left-scattered point in T, and f:TR,sTk,(1) when f is Δ differentiable at s, f is continuous at s;(2) when f is continuous at s and s is right-scattered, f is Δ differentiable at s

fss=fξsfsμs;

(3) when s is right-dense, f is Δ differentiable at s and the limit

limysfsfysy

exists as a finite number. In this case, f(s)(s) is equal to this limit;(4) when f is Δ differentiable at s,

fξsfξs=fs+μsfss.

Remark: when T=R, T=hZ, the Δ − derivative f(s)(s) in (2) becomes

fss=fss,fss=fs+hfsh=Effh,

where E is the shift operator.Afterward, the general integrable nonlinear evolution equations on the time–space scale are generated via considering Theorem 2 and Theorem 3 [7].

Theorem 2. Let the Nth-order δ differential operator

L=uNδN+uN1δN1+uN2δN2++u0,(4)

where uj (j = 0, …, N) are Δ-smooth functions with the time variable(t) and space variable(s). Here, the time variable is taken as continuous. By using the Lax equation

dLdt=A,L,A=LnN0,n=1,2,(5)

hierarchies of integrable nonlinear evolution equations are produced, where n is a positive integer not divisible by N.

Theorem 3. Let

L13=a1δ+a0+a1δ1+a2δ2+,

then, through employing the Gel’fand-Dikii formalism on the time scale, the operator A is obtained:

A=L230=a1a1ξδ2+a1a1s+a1a0ξ+a1a0δ+a1a0s+a1a1ξ+a02+a1a1ξ1.(6)

When N = 3, Eq. 4 becomes

L=u3δ3+u2δ2+u1δ+u0.(7)

Then, by substituting Eqs. 6, 7 into Eq. 5, we get

dLdt=L230,L,(8)

and the following equations

a1a1ξu3ξξa1a1ξξξξu3=0,(9)
a1a1ξu3sξ+u3ξsu3b2+a1a1ξu2ξξξu2a1a1ξξξ+b1u3ξu3b1ξξξ=0,(10)
a1a1ξu3ss+a1a1ξu1ξξu1a1a1ξξ+b1u3s+b1u2ξu2b1ξξu3b3+u3b4=u3t,(11)
u1ξs+u1sξu3a1a1ξsssu1a1a1ξs+a1a1ξu2ss+u0ξξu0+b1u2s+b1u1ξu1b1ξu2a1a1ξss+b1sξ+b1ξsu3b5+u2b6=u2t,(12)
a1a1ξu1ss+u0ξs+u0sξ+b1u1su1b1s+b1u0ξu0b1u3b7u2b8+u1b9=u1t,(13)
a1a1ξu0ss+b1u0su3b10sssu2b10ssu1b10s=u0t.(14)

where

b1=a1a1s+a1a0ξ+a1a0,b2=a1a1ξξξs+a1a1ξξsξ+a1a1ξsξξ,b3=a1a1ξssp+a1a1ξsξs+a1a1ξξss+b1ξξs+b1ξsξ+b1sξξ,b4=a1a0sa1a0sξξξ+a1a1sa1a1sξξξ+a1a1ξ1a1a1ξ1ξξξ+a02a0ξξξ,
b5=a1a0ssξξ+a1a0sξξs+a1a0sξsξ+b1ssξ+b1sξs+b1ξss+a1a1ξξξs+a1a1ξξsξ+a1a1ξsξξ+a02ξξs+a02ξsξ+a02sξξ,b6=a1a0sa1a0sξξ+a1a1ξa1a1ξξξ+a02a0ξξ2+a1a1ξ1a1a1ξξξ,b7=b1sss+a1a0sssξ+a1a0ssξs+a1a0sξss+a1a1ξssξ+a1a1ξsξs+a1a1ξξss+a02ssξ+a02sξs+a02ξss+a1a1ξ1ssξ+a1a1ξ1sξs+a1a1ξ1ξss,b8=b1ss+a1a0ssξ+a1a0sξs+a1a1ssξ+a1a1sξs+a02sξ+a02ξs+a1a1ξ1sξ+a1a1ξ1ξs,b9=a1a1ξa1a1ξξ+a1a0sa1a0sξ+a02a0ξ2+a1a1ξ1a1a1ξ1ξ,b10=a1a0s+a1a1ξ+a02+a1a1ξ1.

Next, by taking a1 = 1, a0 = u3 = u2 = 0, the coupled Boussinesq equation is obtained on the time–space scale:

u1t=u1ssu0s+a1u1+a1ssa1ξξssu1a1ξξ,u0t=u0ssu1sssu1u1s+u1a1ξξs+a1ξξsss.(15)

In the following, the classical Boussinesq equation and coupled semi-discrete Boussinesq equation are obtained, respectively.(1) In the case T=R, (t,s)R×R, we get the coupled Boussinesq equation

u1t=u1ssu0s,u0t=u0ss23u1sss23u1u1s,(16)

then when u1 = u0s, the classical Boussinesq equation is obtained:

u1tt+2u1ss13u12ss53u1ssss=0.(17)

(2) In the case T=Z, (t,s)R×Z, we have μ(t) = 0, μ(s) = 1, and

fξt,s=ft,s,fξt,s=Eft,s=ft,s+E1ft,s.

Therefore, the coupled semi-discrete Boussinesq equation is obtained

u1t=1Eu0+E2+E+11u12u1E2E2+E+11u1+E121+E2+E+11E2E2+E+11u1,u0t=E12u0+E13E2E2+E+111u1+u1E1E2E2+E+11+1Eu1.(18)

3 The exact solutions of the coupled Boussinesq equation on the time–space scale

In the following, the exact solutions of the coupled Boussinesq equation on the time–space scale and classical Boussinesq equation will be obtained based on the extension of the DT on the time scale.

Theorem 4. [13] If functions ψ1 (t, s), ψ(t, s) satisfy

Lψ1=λ1ψ1,Lψ=λψ,

then the function

ψ1=Q1ψ=Wψ1,ψψ1,Q1δd1,d1ψ1sψ1

is a solution of

L1ψ1=λψ1,

where

L1=k=0Nuk1δk,up1=upξ+Q1up+1+k=p+1nuk1MSp+1kdM,p=0,,N,

with Sjk being specified as the set containing the whole probable strings of length k that include j ξs and k-j δs exactly.Considering Lax pairs

L0ψt,s=λψt,s,A0ψt,s=ψtt,s,(19)

where

L0=u30δ3+u20δ2+u10δ+u00,A0=a10a1ξ0δ2+a10a1s0+a10a0ξ0+a10a00δ+a10a0s0+a10a1ξ0+a020+a10a1ξ10.

Let a−1[0] = a0[0] = 0, u2[0] = u1[0] = u0[0] = 0. Taking eigenfunctions ψ and ψ1 of Eq. 19 with eigenvalues λ and λ1, we get

u30δ3ψ=λψ,a10a1ξ0δ2ψ=ψt,(20)
u30δ3ψ1=λ1ψ1,a10a1ξ0δ2ψ1=ψ1t.(21)

By using Theorem 4, the one-fold DT is constructed:

ψ1=Q1ψ,Q1=δd1,d1=ψ1sψ1,

then

L1ψ1=λψ1,A1ψ1=ψt1,(22)

where

L1=u31δ3+u21δ2+u11δ+u01,A1=a11a1ξ1δ2+a11a1s1+a10a0ξ1+a11a01δ+a11a0s1+a11a1ξ1+a021+a11a1ξ11,(23)

with

u31=u3ξ0,u21=Q1u30+u3ξ0d1ξξξ,u11=u21d1ξξ+u31d1sξξ+d1ξsξ+d1ξξs,u01=u11d1ξ+u21d1sξ+d1ξs+u31d1ssξ+d1sξs+d1ξss.(24)

Thus, Eq. 24 is the solution to Eqs. 1114:

u3t1=a11a1ξ1u3ss1+a11a1ξ1u1ξξ1u11a11a1ξ1ξ+b11u3s1+b11u2ξ1u21b1ξξ1u31b31+u31b41,(25)
u2t1=u31a11a1ξ1sssu11a11a1ξ1s+b11u2s1+b11u1ξ1u11b1ξ1+a11a1ξ1u2ss1+u1ξs1+u1sξ1+u0ξξ1u01u31b51+u21b61u21a11a1ξ1ss+b1sξ1+b1ξs1,(26)
u1t1=a11a1ξ1u1ss1+u0ξs1+u0sξ1u21b81u11b1s1u01b11+b11u0ξ1u31b71+b11u1s1+u11b91,(27)
u0t1=a11a1ξ1u0ss1+b11u0s1u31b10sss1u21b10ss1u11b10s1,(28)

where

b11=a11a1s1+a11a0ξ1+a11a01,b21=a11a1ξ1ξξs+a11a1ξ1ξsξ+a11a1ξ1sξξ,b31=a11a1ξ1ssξ+a11a1ξ1sξs+a11a1ξ1ξss+b1ξξs1+b1ξsξ1+b1sξξ1,b41=a11a0s1a11a0s1ξξξ+a11a1s1a11a1s1ξξξa0ξξξ1+a11a1ξ11a11a1ξ11ξξξ+a021,b51=a11a0s1sξξ+a11a0s1ξξs+a11a0s1ξsξ+b1ss1ξ+a11a1ξ1ξξs+a11a1ξ1ξsξ+a11a1ξ1sξξ+a021ξξs+a021sξξ+b1sξs1+b1ξss1+a021ξsξ,b61=a11a0s1a11a0s1ξξ+a11a1ξ1a11a1ξ1ξξ+a021a11a1ξ1ξξ+a11a1ξ11a0ξξ21,b71=b1sss1+a11a0s1ssξ+a11a0s1sξs+a11a0s1ξss+a11a1ξ1sξs+a11a1ξ1ξss+a021ssξ+a021sξs+a11a1ξ11ssξ+a11a1ξ11sξs+a11a1ξ11ξss+a021ξss+a11a1ξ1ssξ,b81=b1ss1+a11a0s1sξ+a11a0s1ξs+a11a1s1sξ+a11a1s1ξs+a11a1ξ11sξ+a11a1ξ11ξs+a021ξs+a021sξ,
b91=a11a1ξa11a1ξξ+a11a0sa11a0s1ξ+a021a0ξ12+a11a1ξ11a11a1ξ11ξ,b101=a11a0s1+a11a1ξ1+a021+a11a1ξ11.

Taking the eigenfunction ψ2 of Eq. 19 with the eigenvalue λ2 and applying Theorem 4., we get the two-fold DT:

ψ2=Q2ψ1=Q2Q1ψ,Q2=δd2,d2=ψ2s1ψ21,

then

u32=u3ξ1,u22=u2ξ1+Q2u31+u3ξ0d2ξξξ,u12=u1ξ1+Q2u21+u22d2ξξ+u32d2sξξ+d2ξsξ+d2ξξs,u02=u0ξ1+Q2u11+u12d2ξ+u32d2ssξ+d2sξs+d2ξss+u22d2sξ+d2ξs.(29)

Similarly, we take the eigenfunction ψN of Eq. 19 with the eigenvalue λN. The N-fold DT is constructed:

ψN=QNQ1ψ=Wψ1,,ψN,ψWψ1,,ψN,(30)

and so

u3N=u3ξN1,u2N=u2ξN1+Q2u3N1+u3ξN2dNξξξ,u1N=u1ξN1+Q2u2N1+u2NdNξξ+u3N×dNsξξ+dNξsξ+dNξξs,u0N=u0ξN1+Q2u1N1+u1NdNξ+u3N×dNssξ+dNsξs+dNξss+u2NdNsξ+dNξs.(31)

In particular, taking seed solutions of Eqs. 1114

a10=a00=0,a10=1,u00=u10=u20=0,u30=1,(32)

we obtain

u31=u3ξ0=1,(33)
u21=νd1+d1ξ+d1ξξs,(34)
u11=νd1+d1ξ+d1ξξsd1ξξ+d1sξξ+d1ξsξ+d1ξξs,(35)
u01=νd1+d1ξ+d1ξξsd1ξξd1ξ+d1sξ+d1ξs+d1sξξ+d1ξsξ+d1ξξs1+d1ξ.(36)

When ν ≠ 0, Eqs, 35, 36 are exact solutions of the coupled Boussinesq equation on the time–space scale Eq. 15:

u1t1=u1ss1u0s1+a11u11+a1ss1a1ξξss1u11a1ξξ1,u0t1=u0ss1u1sss1u1u1s1+u11a1ξξs1+a1ξξsss1.(37)

When ν = 0, we get the eigenfunction ψ1 of Eq. 17 with the eigenvalue λ1 = k3

ψ1=eα1t+β1s+eα2t+β2s,(38)

where

α1=k2i+33+i,α2=k2i33+i,β1=2ik3+i,β2=2ik3+i,

then the single-soliton solution of the classical Boussinesq Eq. 17 is obtained:

u11=3d1s=3β12+β222eα1+α2t+β1+β2seα1t+β1s+eα2t+β2s2.(39)

The dynamics image of the single-soliton solution of Eq. 39 is shown in Figure 1.

FIGURE 1
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FIGURE 1. Single-soliton solution Eq. 39 with k = 0.12 + 0.26 * i.

4 Conclusion

The general coupled nonlinear integrable evolution equations and their N-soliton solutions on the time–space scale are formulated via employing the extensions of the GD formalism and standard DT on the time scale. By taking special values, we derive the Boussinesq equation on the time–space scale, and then, the classical Boussinesq equation and coupled semi-discrete Boussinesq equation are obtained by considering the continuous time scale R and discrete time scale Z, respectively. Afterward, the exact solutions of the Boussinesq equation on the time–space scale are acquired via applying its DT, and the single-soliton solution of the classical Boussinesq equation is obtained as the special case. The extensions may supply more nonlinear integrable models and facilitate solving some practical problems unifying continuous and discrete cases. In the next, we will use the generalization of the GD formalism and DT to construct different integrable systems on time–space scales with other the higher-order δ differential operator so as to enrich the research of integrable systems on arbitrary time–space scales.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

XH, YZ, and HD contributed to the conception and design of the study. XH and YZ wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11975143, 12105161, and 61602188), the Natural Science Foundation of Shandong Province (Grant No. ZR2019QD018), the CAS Key Laboratory of Science and Technology on Operational Oceanography (Grant No. OOST2021-05), and the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (Grant Nos. 2017RCJJ068 and 2017RCJJ069).

Acknowledgments

The authors would like to express their thanks to the editors and the reviewers for their kind comments to improve our paper.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: Gel’fand-Dikii formalism, Darboux transformation, time–space scale, Boussinesq equation, soliton solution

Citation: Huang X, Zhang Y and Dong H (2022) The coupled Boussinesq equation and its Darboux transformation on the time–space scale. Front. Phys. 10:1055998. doi: 10.3389/fphy.2022.1055998

Received: 28 September 2022; Accepted: 17 November 2022;
Published: 08 December 2022.

Edited by:

Yunqing Yang, Zhejiang Ocean University, China

Reviewed by:

Wenjun Liu, Nanjing University of Information Science and Technology, China
Xiaoyong Wen, Beijing Information Science and Technology University, China

Copyright © 2022 Huang, Zhang and Dong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Huanhe Dong, donghuanhe@sdust.edu.cn

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