Qualitative analysis and new traveling wave solutions for the stochastic Biswas-Milovic equation

1.   Introduction

The study of the exact solutions and dynamical behavior of nonlinear equations not only helps to understand the essential properties and algebraic structure of soliton theory^[1,2,3], but also plays an important role in the rational explanation and practical application of corresponding natural phenomena ^[4,5,6]. For instance, the Biswas-Milovic equation (BME) has attracted widespread attention for its ability to describe nonlinear wave dynamics and mode formation in various physical systems ^[7,8,9]. Meanwhile, the stochastic BME is an extended form that incorporates the effects of random perturbations or noise, providing a more accurate reflection of dynamic behaviors in complex environments in practical applications ^[10,11]. In the field of optics and photonics, the stochastic BME can be used to describe the propagation behavior of optical pulses in fiber communication systems^[12], especially in the presence of random noise such as thermal noise or phase noise. Within fluid mechanics, this equation can also be employed to simulate the nonlinear evolution of water surface waves ^[13,14,15], particularly when the waves are influenced by random factors such as stochastic wind fields or other external random effects.

The addition of multiplicative white noise further enriches the descriptive power of the equation. Unlike additive noise, which assumes that external random fluctuations uniformly affect the system, multiplicative noise implies that the strength of randomness depends on the state of the system itself. This feature is particularly relevant when fluctuations are inherently related to the size of the variables they affect, thus capturing important aspects of real-world complexity and uncertainty.

In recent years, numerous experts in physics and mathematics have investigated the solutions of the BME employing a variety of classic and intriguing methods ^[16,17,18]. However, there remains limited analysis concerning the bifurcation and chaotic behavior of the BME^[19,20,21]. Our research work is significant because we have not only enriched the solutions of the BME applying the polynomial complete discriminant system method, but also conducted a thorough analysis of the planar dynamical system applying the planar dynamical system method. The findings we have obtained provide insight into the chaotic nature of the framework under examination, thereby enhancing our understanding of the dynamics that underlie it. While our methods demonstrate systematicness, precision, broad applicability, and deep theoretical underpinnings, they do possess certain limitations, particularly when addressing high-dimensional problems, which often necessitate the integration of other approaches, such as numerical simulation methods. Notwithstanding this limitation, current research suggests that the methods employed in this work are highly reliable, versatile, and effective for a variety of nonlinear models across different scientific disciplines.

The stochastic BME with dual-power law nonlinearity and multiplicative white noise in the Itô sense was introduced by Elsayed et al. in ^[22]. It can be expressed as follows:

where $i^{2} = -1$,

is a complex-valued function related to wave profile, and the independent variables $x$ and $t$ denote the distance and time, respectively. Meanwhile, $m$ is the maximum intensity parameter, and $a, b$ and $c$ are real-values constants, where $a$ represents generalized chromatic dispersion, while $b$ and $c$ represent double-power nonlinearity. Besides, $n$ is the parameter of the power law nonlinearity, $W(t)$ is the standard Wiener process, and $\frac{\partial W(t)}{\partial t}$ is the formal derivative of the Wiener process with respect to time, which represents multiplicative white noise used to identify the process where the excitation phase is interrupted. Notably, $\sigma$ is a positive constant denoting the noise strength coefficient, and the well-known BME with dual-power law nonlinearity is the case where $\sigma = 0$ in Eq (1.1).

The interaction between double power nonlinearity and multiplicative white noise poses serious challenges for analysis and numerical research. The purpose of this article is to study the planar dynamical system and new traveling wave solution of Eq (1.1) in order to reveal the complex network of stability, bifurcation, and pattern formation under random influence, and to elucidate the mechanisms driving unpredictable behavior in natural and engineering systems. To the best our knowledge, no literature has used this method before. Therefore, the research findings of this article contribute to make a comprehensive and systemic study on the propagation dynamics of Eq (1.1) with different methods.

This article consists of the several parts as follows: In Section 2, Eq (1.1) is first transformed into a nonlinear ordinary differential equation (NLODE) with taking traveling wave transformation, and then a second-order NLODE is obtained by applying the trial method of polynomial for rank homogeneous equations and the homogeneous equilibrium principle of polynomials. In Section 3, the dynamical bifurcation conditions and phase diagrams of the equation considered are obtained through the method of planar dynamical systems. Moreover, the phase portrait of the dynamical system is given. Meanwhile, two-dimensional (2D) and three-dimensional (3D) phase portrait, sensitivity analyses and Poincaré section of its perturbation system are plotted by Maple. In Section 4, the optical soliton solutions of the BME with dual-power law nonlinearity and multiplicative white noise in the Itô sense are constructed by using the complete discrimination system method. Finally, a brief conclusion is given in Section 5.

2.   Mathematical analysis

To solve Eq (1.1), we consider the solution in the following form:

where $\kappa$, $\omega$, $V$ are nonzero real values, while $Q(\xi)$ and $\eta(x, t)$ are real-valued functions. More precisely, $Q(\xi)$ and $\eta(x, t)$ represent the amplitude and phase components of the soliton, respectively. In Eq (2.1), $\xi$ is the wave variable, $\omega$ is the wave number, $\kappa$ is the frequency, and $V$ is the velocity of the soliton.

Substituting (2.1) into (1.1), and setting the real and imaginary parts separately equal to zero, we can obtain

and

From (2.3), we derive the velocity

Remark 2.1. Due to $\kappa$, $V$ being nonzero real values and satisfying

the case when $a < 0$, $m > 0$, and $\kappa > 0$ is our main focus in the following part.

Next, setting

we can reduce (2.2) to the following form of integer balance number:

Then, with the trial method ^[23,24,25] of polynomial for rank homogeneous equations and the principle of homogeneous balance, we can get the trial equation as follows:

where $\alpha_j(j = 1, 2, 3)$ are all constants. After that, multiplying both sides of (2.5) by $H^{'}$, integrating once, and converting the coefficient of $(H^{'})^2$ to integer, we yield

where $d_0$ is an integral constant.

Substituting (2.5) and (2.6) into (2.4), an algebraic equation about $H$ is obtained. By setting the coefficients to zero, we arrive at

in the case of $m\neq n$.

Remark 2.2. If $m = n$, apart from $\alpha_{0}\in R$ differing from the results obtained previously, the values of the other parameters $\alpha_{1}$–$\alpha_{3}$, and $d_0$ are calculated by substituting $m = n$ into the corresponding coefficient expressions that were established in the case of $m\neq n$, that is,

Remark 2.3. If

which reduces (2.4) to

that is, the result calculated by substituting $m = 2n$ into the corresponding coefficient expressions in the case of $m\neq n$.

3.   Qualitative analysis of Eq (2.5)

In this section, we consider the dynamical behavior of Eq (2.5) in the case of $m\neq n$. When $m\neq n$, the Eq (2.5) becomes

First, we set

Second, we convert Eq (3.1) into the two-dimensional Hamiltonian system

The Hamiltonian function is

Here, $d_0 = 0$, i.e., the value of $d_0$ is fixed and unchanged, which means that (3.3) determines the family of invariant curves for system (3.2).

Third, we consider the phase diagram branches of Hamiltonian system (3.2). Supposing that

and assuming that the matrix

where $H_{i}$ ($i\in \{0, 1, 2, 3, \cdots\}$) is the real root of

It is obvious that

Further, set

Then, on the basis of the theory of planar dynamical systems, we classify the trajectories near the equilibrium point and draw the corresponding phase diagram using the Maple software, which are listed in Figures 1–3. Here, symbols $(+, -, +)$ in these figures denote $\alpha_1 > 0$, $\alpha_2 < 0$, and $\alpha_3 > 0$, while, $(+, 0, +)$ in these figures denote $\alpha_1 > 0$, $\alpha_2 = 0$, and $\alpha_3 > 0$. The conclusion is

Proposition 3.1. Assume that $\mathbf{E}_{i}(H_{i}, 0)$ is the equilibrium point of Eq (3.2).

Case Ⅰ. $\Delta < 0$.

When $\alpha_{1} > 0$, we obtain $\det(M(0, 0)) < 0$, then $\mathbf{E}_{0}(0, 0)$ is the saddle point. In addition, the colors of the branch trajectories are the same, and it indicates that the system (3.2) has a doubly-periodic function solution (see Figure 1a, b).

When $\alpha_{1} < 0$, we obtain $\det(M(0, 0)) > 0$, then $\mathbf{E}_{0}(0, 0)$ is the center point. In addition, the closed curve appears, which indicates the path of the system in phase space during periodic repetitive motion and the system (3.2) has a family of periodic solutions (see Figure 1c, d).

Case Ⅱ. $\Delta > 0$.

When $\alpha_{1} > 0$, we obtain $\det(M(0, 0)) < 0$, then $\mathbf{E}_{0}(0, 0)$ is the saddle point.

When $\alpha_{1} < 0$, we obtain $\det(M(0, 0)) > 0$, then $\mathbf{E}_{0}(0, 0)$ is the center point.

When $\alpha_{2} < 0$, $\alpha_{3} > 0$, or $\alpha_{1} < 0$, $\alpha_{2} > 0$, $\alpha_{3} > 0$, or $\alpha_{1} < 0$, $\alpha_{2} > 0$, $\alpha_{3} < 0$, and we obtain

then $\mathbf{E}_{1}(\frac{-\alpha_{2}+\sqrt{\Delta}}{2\alpha_{3}}, 0)$ is the saddle point.

When $\alpha_{2} > 0$, $\alpha_{3} < 0$, or $\alpha_{1} > 0$, $\alpha_{2} > 0$, $\alpha_{3} < 0$, or $\alpha_{1} > 0$, $\alpha_{2} > 0$, $\alpha_{3} > 0$, and we obtain

then $\mathbf{E}_{1}(\frac{-\alpha_{2}+\sqrt{\Delta}}{2\alpha_{3}}, 0)$ is the center point.

When $\alpha_{2} > 0$, $\alpha_{3} > 0$, or $\alpha_{1} < 0$, $\alpha_{2} < 0$, $\alpha_{3} < 0$, or $\alpha_{1} < 0$, $\alpha_{2} < 0$, $\alpha_{3} > 0$, and we obtain

then $\mathbf{E}_{2}(\frac{-\alpha_{2}-\sqrt{\Delta}}{2\alpha_{3}}, 0)$ is the saddle point.

When $\alpha_{2} > 0$, $\alpha_{3} < 0$, or $\alpha_{1} > 0$, $\alpha_{2} < 0$, $\alpha_{3} > 0$, or $\alpha_{1} > 0$, $\alpha_{2} < 0$, $\alpha_{3} < 0$, and we obtain

then $\mathbf{E}_{2}(\frac{-\alpha_{2}-\sqrt{\Delta}}{2\alpha_{3}}, 0)$ is the center point.

In addition, Figure 2b, c indicates that the system (3.2) has a kink wave, while Figure 2a, d indicates that the system (3.2) has the homoclinic orbits, the periodic orbits, and the bounded open orbits.

Case Ⅲ. $\Delta = 0$.

When $\alpha_{1} > 0$, we obtain $\det(M(0, 0)) < 0$, then $\mathbf{E}_{0}(0, 0)$ is the saddle point.

When $\alpha_{1} < 0$, we obtain $\det(M(0, 0)) > 0$, then $\mathbf{E}_{0}(0, 0)$ is the center point.

When

and the Poincaré index of it is equal to zero, then $\mathbf{E}_{3}(-\frac{\alpha_{2}}{2\alpha_{3}}, 0)$ is the cusp point. For instance, $\mathbf{E}_{3}(1, 0)$ in Figure 3a, $\mathbf{E}_{3}(-2, 0)$ in Figure 3b, and $\mathbf{E}_{3}(-1, 0)$ in Figure 3c.

Considering that periodic interference often exists in real environments, here we add periodic perturbations and small perturbations to system (3.2), respectively,

where

or

is the perturbed term, with $A$ and $\iota$ being two real constants. In Figures 4–7, we consider the sensitivity of Eq (3.4) in different initial values and the impact of $A, \iota$ on the solution behavior. Setting

unchanged, the values of $\alpha_1$–$\alpha_3$ change with the vary value of $m$ and $\sigma$. Here, $m = 2, \sigma = 1$ correspond to

while, $m = 5, \sigma = 2$ correspond to

Under the condition of changes in parameters, we use 2D trajectories (see Figures 4a–7a), 3D trajectories(see Figures 4b–7b), sensitivity analyses (see Figures 4c–7c), and Poincaécross-sections (see Figures 4d–7d) to demonstrate the impact of perturbations on solution behavior sepatately. As shown in the grahs, chaos behavior, periodic behavior, and quasi periodic behavior are present by adding perturbation terms. The variation in noise intensity may lead to the transformation of the system's chaotic behavior into periodic motion, while minor variations in the intensity and frequency of external interference may have little impact on the system.

4.   Traveling wave solutions of Eq (1.1)

In this section, we seek for the new traveling wave solutions of Eq (1.1) by applying the complete discriminant method.

4.1.   The case of $m\neq n$

In this scenario, Eq (2.6) becomes

With the condition of

by calculation, Eq (4.1) can be rewritten as following using elementary integration:

where

$\xi_0$ is the integral constant.

Remark 4.1. If

the solution of Eq (1.1) is

where $\varkappa$ is an arbitrary constant.

Setting

In the rest part, with the complete discriminant system of the second order polynomial, by the root of $F(H)$, the traveling wave solution of Eq (1.1) has the following implicit solutions for three situations:

Case 1. When

namely,

also if

the traveling wave solution of Eq (4.1) is given as

provided

Remark 4.2. If

that is,

then the solution of Eq (1.1) is

Case 2. When

namely,

(Ⅰ) If

there are three forms of solutions to Eq (4.1) as below:

(Ⅱ) If

there are another three forms of solutions to Eq (4.1) as below:

where assuming that the internal scores of the radical are all greater than zero in (4.4)–(4.9), and

Case 3. When

the equality

has no real root; also if

the traveling wave solution of Eq (4.1) is given as

provided $p_1, p_3 > 0$.

The new traveling wave solutions of Eq (1.1) constructed above are all implicit solutions in the caes of $m\neq n$.

4.2.   The case of $m=n$

In this scenario, Eq (2.6) becomes

If

i.e., $\frac{c}{a} < 0$, taking the transformation

and inserting into (4.11), we yield

where

If

i.e., $\frac{c}{a} > 0$, taking the transformation

and inserting into (4.11), we yield

where

For simplicity and convenience in symbolic representation, we denote $\varepsilon = 1$, if

while, $\varepsilon = -1$, if

Then, (4.11) can be expressed as below by representing the above two situations in a combined manner:

Furthermore, Eq (4.12) can be rewritten as

where $\zeta_0$ is an integral constant.

As we know, the fourth-order polynomial

has the following complete discrimination system:

Thus, by applying the complete discriminant method, we have the subsequent new families of traveling wave solutions for Eq (1.1):

Case 1. The rational functional solutions.

When

moreover, $\varepsilon = 1$, and Eq (4.12) has the rational functional solutions

Therefore, the solution of Eq (1.1) is

Setting parameters

the real part diagram of solution $q_{1}$ with and without random term can be intuitively shown in Figure 8. Then, changing the parameter value $m = 3, \sigma = 2,$ and other parameters values remain unchanged, the real part diagram of solution $q_{1}$ with and without random term can be intuitively shown in Figure 9. As shown in the grahs, an increase in noise intensity may cause waveform distortion, typically leading to a reduction in wave stability. Additionally, variations in noise intensity may also modify the statistical properties of waves, including the distribution of wave height, trough depth, and wave period.

Case 2. The periodic solutions of triangle function.

When

moreover, $\varepsilon = 1$, and Eq (4.12) has the periodic solution of the triangle function

where $\lambda_1$ and $\lambda_2$ are two real roots of

satisfying $\lambda_2 > 0$.

Therefore, the solution of Eq (1.1) is

Case 3. The hyperbolic function solutions.

When

Moreover, there are two real numbers $\lambda_3$ and $\lambda_4$($\lambda_3 > \lambda_4$), which satisfy

And then, when $\varepsilon = 1$,

(Ⅰ) If $\Upsilon > \lambda_3$ or $\Upsilon < \lambda_4$, Eq (4.12) has the solution

Therefore, the solution of Eq (1.1) is

(Ⅱ) If $\lambda_3 < \Upsilon < \lambda_4$, Eq (4.12) has the solution

Therefore, the solution of Eq (1.1) is

Case 4. The implicit solutions.

When

Moreover, there are three real numbers $\lambda_5$, $\lambda_6$ and $\lambda_7$($\lambda_6 > \lambda_7$), which satisfy

Then, when $\varepsilon = 1$, if $\lambda_5 > \lambda_6$, and $\Upsilon > \lambda_6$, or $\lambda_5 < \lambda_7$ and $\Upsilon < \lambda_7$, Eq (4.12) has the solution:

If $\lambda_5 > \lambda_6$ and $\Upsilon < \lambda_7$, or $\lambda_5 < \lambda_7$ and $\Upsilon < \lambda_6$, Eq (4.12) has the solution:

If $\lambda_6 > \lambda_5 > \lambda_7$, Eq (4.12) has the solution:

When $\varepsilon = -1$, if $\lambda_5 > \lambda_6$ and $\Upsilon > \lambda_6$, or $\lambda_5 < \lambda_7$ and $\Upsilon < \lambda_7$, Eq (4.12) has the solution:

If $\lambda_5 > \lambda_6$ and $\Upsilon < \lambda_7$, or $\lambda_5 < \lambda_7$ and $\Upsilon < \lambda_6$, Eq (4.12) has the solution:

If $\lambda_6 > \lambda_5 > \lambda_7$, Eq (4.12) has the solution:

Case 5. The the solitary wave solutions.

When

Moreover, there are three real numbers $\lambda_{8}$–$\lambda_{10}$, which satisfy

If $\varepsilon = 1$, Eq (4.12) has the solitary wave solution

where

Therefore, the solution of Eq (1.1) is

Setting parameters $a = -4, b = -7.032292869, c = 3, m = n = \sigma = \kappa = 1, \omega = 6, V = 8, \xi_0 = 0$, the real part diagram of solution $q_{5}$ with and without random term can be intuitively shown in Figure 10.

In the rest part, based on the definition of the Jacobian elliptic sine-cosine function, we derived the Jacobian elliptic function solutions of Eq (1.1) for three cases, designated as Cases 6–8, through trigonometric transformations.

Case 6. There are four real numbers $\lambda_{11}$, $\lambda_{12}$, $\lambda_{13}$, and $\lambda_{14}$($\lambda_{11} > \lambda_{12} > \lambda_{13} > \lambda_{14}$), which satisfy

For $D_2 > 0, D_3 > 0, D_4 > 0$, when $\varepsilon = 1$, if $\Upsilon > \lambda_{11}$ or $\Upsilon < \lambda_{14}$, we consider transformation

if $\lambda_{13} < \Upsilon < \lambda_{12}$, we consider transformation

then we can acquire from (4.13) that

where

Thus, the corresponding solutions of (4.12) are

Here,

Therefore, the corresponding solutions of Eq (1.1) are

When $\varepsilon = -1$, if $\lambda_{11} > \Upsilon > \lambda_{12}$, we consider transformation

if $\lambda_{13} > \Upsilon > \lambda_{14}$, we consider transformation

similarly, the corresponding solutions of (4.12) are

where

Therefore, the corresponding solutions of Eq (1.1) are

Case 7. There are four real numbers $\lambda_{15}$–$\lambda_{17}$, and $\lambda_{20}$, which satisfy

and $\lambda_{15} > \lambda_{16}, \lambda_{17} > 0, \lambda_{18} > 0$. For

applying the transformation

where

then we can acquire from (4.13) by this process:

where

Thus, (4.12) has the elliptic double periodic function solutions as below:

Here, $"\mp"$ means that $"-"$ corresponding to $\varepsilon = 1$, and $"+"$ corresponding to $\varepsilon = -1$,

Therefore, the corresponding solutions of Eq (1.1) are

Case 8. There are four real numbers $\lambda_{19}$–$\lambda_{22}$, which satisfy

and $\lambda_{20}\geq\lambda_{22} > 0$. Then, for $D_4 > 0, D_2D_3\leq0$,

When $\varepsilon = 1$, applying the transformation

where

we can acquire from (4.13) that

where

Thus, Eq (4.12) has the elliptic double periodic function solutions as below:

where

Therefore, the solution of Eq (1.1) is

5.   Conclusions

In this study, we consider the dynamical behavior and new traveling wave solution of the stochastic BME with dual-power law nonlinearity and multiplicative white noise in the Itô sense. The qualitative analysis of the dynamical system of Eq (1.1) and its perturbation system are considered though the principle of homogeneous balance and the planar dynamical system method. Then, we draw 2D phase portraits of the dynamical system and easily observe the orbital properties of the dynamical system. Considering that periodic interference often exists in real environments, we add the periodic disturbance term to system (3.2), and then the 2D and 3D phase portrait, sensitivity analyses, and Poincaré section of its perturbation system are plotted by the Maple software. Moreover, by using the complete discrimination system method, the new traveling wave solutions of Eq (1.1) are successfully constructed with the effect of the Wiener process implicated. Compared with existing literature, the research findings of this article contribute to make a comprehensive and systemic study on the propagation dynamics of the stochastic BME with dual-power law nonlinearity and multiplicative white noise in the It$\acute{o}$ sense Eq (1.1) with different methods. Specifically, we have not only enriched the solutions of the BME but also conducted a thorough analysis of the planar dynamical system. The findings we have obtained provide insight into the chaotic nature of the framework under examination, thereby enhancing our understanding of the dynamics that underlie it. The research methods employed in this work are highly reliable, versatile, and effective for a variety of nonlinear models across different scientific disciplines.

Author contributions

Dan Chen: writing–original draft, writing–review & editing; Da Shi: supervision; Feng Chen: software. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This project is supported by funding of Visual Computing and Virtual Reality Key Laboratory of Sichuan Province via grant (No. SCVCVR2023.06VS).

Conflict of interest

The authors declare no conflicts of interest.