Dynamics and embedded solitons of stochastic quadratic χ (2) and cubic χ (3) nonlinear susceptibilities with multiplicative white noise in the Itô sense

The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic χ (2) and cubic χ (3) nonlinear susceptibilities in the Itˆo sense. Firstly, two-dimensional dynamics system and its perturbation system are obtained by using travelling wave transformation. Secondly, phase por-traits of two-dimensional dynamics system are plotted. What’s more, chaotic behaviors, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed by using the Maple software. Finally, the embedded solitons of stochastic quadratic χ (2) and cubic χ (3) nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimension of stochastic quadratic χ (2) and cubic χ (3) nonlinear susceptibilities are plotted.


Introduction
The concept of "embedded soliton"(ES) was introduced at the end of the nineties.After that, Yang et al. [1] found ESs in continuous model, unstable model and discrete model, and further explained ESs.
Generally, ES [2][3][4] is a new type of solitary waves, which exists in the continuous spectrum of the nonlinear wave system and is limited in the continuous spectrum of the nonlinear system [5][6][7].ESs are usually used to describe the solutions of nonlinear partial differential equations from hydrodynamics, nonlinear optics and liquid crystal theory [8][9][10].
In recent years, the analysis of soliton solutions and dynamic behavior of stochastic partial differential equations(SPDEs) [11][12][13][14] have attracted great attention of many experts and scholars.In [11], Han et al. studied the exact solutions and bifurcation of the stochastic fractional long-short wave equation by using the dynamical system method.In [12], Zayed et al. obtained the dispersive optical solitons of stochastic perturbed generalized Schrödinger-Hirota equation by the extended simplest equation algorithm and the Φ 6 -model expansion method.In [13], He and Wang studied the soliton solutions of stochastic nonlinear Schrödinger equation using the bilinear method.In [14], Li and Tao derived the soliton solutions of the stochastic Benjamin-Ono equation by using the Hirota method.Based on the analysis of the above references, we can find that the research results in recent years mainly focus on the discussion of soliton solutions of SPDEs.Although some papers have reported the discussion of the dynamic behavior of partial differential equations [15,16], there are few studies on the dynamic behavior, chaotic behavior and sensitivity of SPDEs and its perturbation.The main purpose of this paper is to discuss the dynamic behavior and embedded soliton solutions of a class of SPDEs and its perturbed system.
The stochastic quadratic χ (2) and cubic χ (3) nonlinear susceptibilities with multiplicative white noise in the Itô sense is a kind of very important SPDEs, which is usually described as follows [17] where u = u(t, x) and v = v(t, x) are the complex-valued functions.a j , b j , c j , d j (j = 1, 2), δ and σ stand for real-valued constants.a j stands for the chromatic dispersion.b j stands spatio-temporal dispersion.c j represents group velocity mismatch.d j is the self phase modulation.dW (t) dt is the white noise.W (t) is the standard Wiener process.σ is the noise strength.
Here, we add the real-valued function of periodic perturbation g 1 and g 2 , then, Eq.(1.1) with periodic perturbation is written as below The format of this article is organized as follows: In section 2, the dynamics of (1.1) and (1.2) are analysed.In section 3, the embedded solitons of (1.1) are constructed by using the complete discrimination system method.In section 4, a brief conclusion is presented.

Mathematical derivation
Assume that the main solutions of Eq.(1.1) is as follows where U 1 (ξ) and U 2 (ξ) are real functions, which are used to represent the soliton amplitude.k stands for the soliton frequency.w is soliton wave number.c stands for the soliton velocity.
Substituting (2.1) into Eq.(1.1), we can obtain the real parts as while, we can get the imaginary parts as From Eq.( 2.3), we can obtain the soliton velocity Moreover, we can get the wave number from (2.4) (2.5) The Eq.(2.2) can be transformed into where .

Phase portraits of system (2.7)
Firstly, two-dimensional dynamics system of Eq.(2.7) can be written as below then, the Hamiltonian function of system (2.8) is defined by (2.9) Let E(U j , 0) be the coefficient matrices of (2.9) at the equilibrium point U j and J(P j ) = det(E(P j , 0)) = −f ′ (P j ), where U j is the root of the function f Then, we can draw the phase portraits of system (2.9) as shown in Figure 1.
Let be g . Then, we consider two-dimensional disturbance system with perturbation term as below where f 0 is the amplitude of (2.11).κ is the frequency of (2.11).

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In Fig. 2, Fig. 3, Fig. 4 and Fig. 5, two-dimensional phase portrait, three-dimensional phase portrait and sensitivity of system (2.11) are presented for giving different initial values and parameters, respectively.
Obviously, in Fig. 2, when the initial value of system (2.11) changes, two-dimensional phase diagram of system (2.11) shows chaotic behavior compared with Fig. 1b.Moreover, as shown in Fig. 3 and Fig. 4, when the initial value changes, the three-dimensional phase diagram and sensitivity of system (2.11) further verify the existence of chaotic behavior.Multiplying both sides of Eq.(2.7) by U ′ j and integrating again yields where D j (j = 1, 2) is the integral constant.
According to the second order polynomial complete discrimination system method, the solution of equation (3.1) can have the following three cases.
Case 1.1 ∆ = 0 From formula (3.2), it can be obtained Case 1.2 ∆ > 0 When A j > 0, it can be obtained from formula (3.2) when A j < 0, it can be obtained from formula (3.2) where where C j > 0. can be written as
When ρ 1j < 0, A j > 0 and C j < 0, the embedded solitons of Eq.(1.1) can be given as Here, we plot the diagrams of the solution u 1 (t, x) of Eq.(1.1) as shown in Figure 6.When ρ 1j > 0, A j > 0 and C j > 0, the embedded solitons of Eq.(1.1) can be given as

Declaration of Competing Interest
The author declares no conflict of interest.