Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense

Yong Tang; Zhao Li

doi:10.1515/phys-2023-0130

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Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense

Yong Tang and Zhao Li

Published/Copyright: November 28, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

The main aim of this article is to focus on the dynamics and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense. First, the considered model is converted to the nonlinear ordinary differential equations by using traveling wave transformation. Secondly, two-dimensional phase portrait of the nonlinear ordinary differential equation and its periodic perturbation system are drawn by using Maple software. Finally, the traveling wave solutions of the investigated equation are obtained via the planar dynamic system method. Moreover, three-dimensional graphs of some obtained solutions are drawn.

Keywords: stochastic coupled Konno–Oono equation; phase portrait; traveling wave solution; planar dynamic system method

1 Introduction

Stochastic partial differential equation (SPDE) [1,2] plays a very important role in the fields of physics, life science, nonlinear optics, engineering technology, and control science. More and more models from natural and social sciences need to be simulated by SPDE. Therefore, the study of SPDE is particularly important. At present, the main problems of SPDE include the existence of solutions, the uniqueness of solutions, the stability of solutions, martingale representation theory, numerical solutions, and exact solutions [3–8]. However, in real life and scientific calculation, it is more likely to use numerical or analytical solutions of SPDE. However, due to the complexity of stochastic problems, there are still many challenging problems in finding the exact solutions of SPDE. As is well known, the problem of traveling wave solutions for nonlinear partial differential equation (NLPDE) has received widespread attention [9–19], and many classic methods have been proposed for constructing traveling wave solutions for NLPDE [20–32]. In some recent studies, Mohammed and his collaborators [33,34] can simplify some special SPDE into nonlinear ordinary differential equations by means of traveling wave transformation and mathematical analysis, which is very helpful for us to study SPDE.

In this article, we consider the stochastic coupled Konno–Oono (K–O) equation [35]:

(1.1) v x t − 2 v u = σ F ( v ) , u t + 2 v v x = 0 ,

where v = v ( t , x ) , u = u ( t , x ) . σ is the noise strength. F ( v ) = v x ∘ β t represents the noise term in the Stratonovich sense. β ( t ) stands for the standard Wiener process. When σ = 0 , Eq. (1.1) describes a current-fed string interacting with an external magnetic fed [36]. Stratonovich integral and Itô integral are as follows [37]:

(1.2) ρ ∫ 0 t φ ∘ d β ( s ) = ρ ∫ 0 t φ d β ( s ) + ρ 2 2 ∫ 0 t φ d s .

Mohammed et al. [35] studied the exact solutions of Eq. (1.1) by using the generalized G ′ G -expansion method. However, Mohammed et al. [35] did not obtain a Jacobian function solution and did not discuss the dynamic and chaotic behavior of Eq. (1.1). The main purpose of this article is to discuss the dynamic behavior and exact traveling wave solutions of Eq. (1.1).

The rest of this article is arranged as follows. In Section 2, two-dimensional phase portrait of the nonlinear ordinary differential equation and its periodic perturbation system are drawn. Moreover, the traveling wave solutions of systems (1.1) are obtained. Finally, in Section 3, a conclusion is given.

2 Phase portraits and traveling wave solutions of Eq. (1.1)

2.1 Traveling wave transformation

First, the following transformation is considered

(2.1) v ( t , x ) = V ( ξ ) e σ β ( t ) − σ 2 t , u ( t , x ) = U ( ξ ) , ξ = x − k t ,

where V ( ξ ) and U ( ξ ) are deterministic real function, σ stands for the noise strength, and k is the nonzero constant.

Substituting Eq. (2.1) into Eq. (1.1), we have

(2.2) − k V ″ + σ V ′ β ( t ) − σ 2 V ′ − 2 V U = σ V ′ ∘ β ( t ) ,

(2.3) − k U ′ + 2 V V ′ e 2 σ β ( t ) − 2 σ 2 t = 0 .

Applying (1.2) to Eq. (2.2), we can obtain

(2.4) k V ″ + 2 V U = 0 .

Next, taking mathematical expectation on both sides of Eq. (2.3) at the same time, we can obtain

(2.5) − k U ′ + 2 V V ′ = 0 .

Integrating both sides of Eq. (2.5) with respect to ξ , we have

(2.6) U = 1 k ( V 2 + c ) ,

where c is the integration constant.

Substituting Eq. (2.6) into Eq. (2.4), we have

(2.7) V ″ + 2 k 2 V 3 + 2 c k 2 V = 0 .

2.2 Phase portraits of Eq. (2.7)

Suppose that d V ( ξ ) d ξ = y , then we obtain the following two-dimensional plane system:

(2.8) d V ( ξ ) d ξ = y , d y d ξ = − ϒ 1 V 3 + ϒ 2 V ,

which has the first integral

(2.9) H ( V , y ) = 1 2 y 2 + ϒ 1 4 V 4 − ϒ 2 2 V 2 = h ,

where ϒ 1 = 2 k 2 and ϒ 2 = − 2 c k 2 .

When k = 2 2 and c = − 1 , system (2.8) has three equilibrium points, namely ( 0 , 0 ) , ( 0 , 1 ) , and ( 0 , − 1 ) . Its phase portrait is shown in Figure 1a. When k = 2 2 and c = 1 , system (2.8) has only one equilibrium point, namely ( 0 , 0 ) . Its phase portrait [38] is shown in Figure 1b.

$Figure 1 The phase portraits of system (2.8). (a) ϒ 1 > 0 {\Upsilon }_{1}\gt 0 , ϒ 2 > 0 {\Upsilon }_{2}\gt 0 and (b) ϒ 1 > 0 {\Upsilon }_{1}\gt 0 , ϒ 2 < 0 {\Upsilon }_{2}\lt 0 .$

Figure 1

The phase portraits of system (2.8). (a) ϒ 1 > 0 , ϒ 2 > 0 and (b) ϒ 1 > 0 , ϒ 2 < 0 .

The periodic perturbation of system (2.8) is expressed as follows:

(2.10) d V ( ξ ) d ξ = y , d y d ξ = − ϒ 1 V 3 + ϒ 2 V + A cos ( B ξ ) ,

where A stands for the periodic perturbation and B represents the frequency. It can be seen from the comparison between Figures 1 and 2 that when a disturbance term is added to the system, the system will produce chaotic behavior [39].

$Figure 2 The phase portraits of system (2.10). (a) ϒ 1 > 0 {\Upsilon }_{1}\gt 0 , ϒ 2 > 0 {\Upsilon }_{2}\gt 0 , A = 0.6 A=0.6 , B = 0.8 B=0.8 and (b) ϒ 1 > 0 {\Upsilon }_{1}\gt 0 , ϒ 2 < 0 {\Upsilon }_{2}\lt 0 , A = 0.6 A=0.6 , B = 0.8 B=0.8 .$

Figure 2

The phase portraits of system (2.10). (a) ϒ 1 > 0 , ϒ 2 > 0 , A = 0.6 , B = 0.8 and (b) ϒ 1 > 0 , ϒ 2 < 0 , A = 0.6 , B = 0.8 .

2.3 Traveling wave solutions of Eq. (1.1)

Let h 1 = H ( ± ϒ 1 ϒ 2 , 0 ) = − ϒ 1 2 4 ϒ 2 .

(i) When h ∈ ( h 1 , 0 ) , system (2.6) can be rewritten as

(2.11) y 2 = ϒ 2 2 − V 4 + 2 ϒ 1 ϒ 2 V 2 + 4 h ϒ 2 = ϒ 2 2 ( δ 1 2 − V 2 ) ( V 2 − δ 2 2 ) ,

where δ 1 2 = 1 ϒ 2 ( ϒ 1 + ϒ 1 2 + 4 ϒ 2 h ) and δ 2 2 = 1 ϒ 2 ( ϒ 1 − ϒ 1 2 + 4 ϒ 2 h ) .

Substituting (2.11) into d V d ξ = y and integrating it, we have

(2.12) ∫ V δ 2 d ϕ ( δ 1 2 − ϕ 2 ) ( ϕ 2 − δ 2 2 ) = ∓ ϒ 2 2 ( ξ − ξ 0 ) .

(2.13) ∫ − δ 2 V d ϕ ( δ 1 2 − ϕ 2 ) ( ϕ 2 − δ 2 2 ) = ± ϒ 2 2 ( ξ − ξ 0 ) .

Integrating Eqs (2.12) and (2.13), we obtain

(2.14) v 1 ( t , x ) = ± δ 2 dn δ 2 ϒ 1 2 ( x − k t − ξ 0 ) , δ 1 2 − δ 2 2 δ 1 e σ β ( t ) − σ 2 t .

(2.15) u 1 ( t , x ) = 1 k c + δ 2 2 dn 2 δ 2 ϒ 1 2 ( x − k t − ξ 0 ) , δ 11 2 − δ 21 2 δ 1 e σ β ( t ) − σ 2 t .

(ii) When h = 0 , so δ 1 2 = 2 ϒ 1 ϒ 2 and δ 2 2 = 0 . Thus, the solution of Eq. (1.1) is presented as

(2.16) v 2 ( t , x ) = ± 2 ϒ 2 ϒ 1 sech ( ϒ 2 ( x − k t − ξ 0 ) ) e σ β ( t ) − σ 2 t .

(2.17) u 2 ( t , x ) = 1 k c + 2 ϒ 2 ϒ 1 sech 2 ( ϒ 2 ( x − k t − ξ 0 ) ) e σ β ( t ) − σ 2 t .

(iii) When h ∈ ( 0 , + ∞ ) , system (2.7) is recorded as

(2.18) y 2 = ϒ 2 2 ( V 2 + δ 3 2 ) ( δ 4 2 − V 2 ) ,

where δ 3 2 = 1 ϒ 2 ( − ϒ 1 + ϒ 1 2 + 4 ϒ 2 h ) and δ 4 2 = 1 ϒ 2 ( ϒ 1 + ϒ 1 2 + 4 ϒ 2 h ) .

Plugging (2.18) into d V d ξ = y and integrating it along the periodic orbits, then, we have

(2.19) ∫ 0 V d ϕ ( ϕ 2 + δ 3 2 ) ( δ 4 2 − ϕ 2 ) = ± ϒ 2 2 ( ξ − ξ 0 ) .

So, we have

(2.20) v 3 ( t , x ) = ± δ 3 2 cn ϒ 1 ( δ 3 2 + δ 4 2 ) 2 ( x − k t − ξ 0 ) , δ 4 δ 3 2 + δ 4 2 e σ β ( t ) − σ 2 t .

(2.21) u 3 ( t , x ) = 1 k c + δ 3 2 cn 2 ϒ 1 ( δ 3 2 + δ 4 2 ) 2 ( x − k t − ξ 0 ) , δ 4 δ 3 2 + δ 4 2 e σ β ( t ) − σ 2 t .

Remark 2.1

Figure 3(a) and (c) shows solutions to a Jacobian function, which are periodic solutions. Figure 3(b) shows a bell shaped solitary wave.

$Figure 3 The traveling wave solution of system (1.1) for k = 2 2 k=\frac{\sqrt{2}}{2} , c = − 1 c=-1 , σ = 0 \sigma =0 . (a) h = − 3 4 h=-\frac{3}{4} , (b) h = 0 h=0 , and (c) h = 1 4 h=\frac{1}{4} .$

Figure 3

The traveling wave solution of system (1.1) for k = 2 2 , c = − 1 , σ = 0 . (a) h = − 3 4 , (b) h = 0 , and (c) h = 1 4 .

3 Conclusion

In this article, the dynamics and traveling wave solution of stochastic coupled K–O equation with multiplicative noise is studied. Two-dimensional phase portrait of stochastic coupled K–O equation with multiplicative noise and its periodic perturbation system are drawn by using the Maple software. Compared with the existing literature [35], this article also obtains Jacobian elliptic function solutions, which are new solutions. In the future, our research will still focus on the dynamic behavior and traveling wave solutions of SPDEs.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-03-12

Revised: 2023-09-25

Accepted: 2023-10-12

Published Online: 2023-11-28

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https://doi.org/10.1515/phys-2023-0130

Keywords for this article

stochastic coupled Konno–Oono equation; phase portrait; traveling wave solution; planar dynamic system method

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