These methods, with dispersion and optic solitons in the focus, are the modified Kudryashov’s algorithm , Laplace–Adomian decomposition , improved Adomian decomposition , the ansatz method , Riccati equation and F-expansion , exp-function method , G ′ / G expansion scheme, the extended auxiliary equation approach, sine–Gordon , stationary solitons , Lie group analysis , Hamiltonian perturbations , traveling wave hypothesis , semi-inverse variational principle , unified Riccati equation, new mapping scheme, addendum to the Kudryashov’s method , Riccati–Bernoulli approach , the extended Kudryashov method , the generalized Jacobi elliptic expansion and many others.
However, some situations in nature cause stochastic losses in data transmission. Examples of these natural situations are sound effects, magnetic and electric field effects, material quality and ambient temperature. This causes the use of stochastic terms in order to obtain more effective results in the mathematical models established for the examination of adverse environmental effects. Researchers have developed stochastic models of nonlinear evolution equations in many studies. Some of these studies are as follows: In , Belic et al. modeled optical soliton solutions of the higher order nonlinear Schrödinger equation with variable coefficients. examined the stochastic nonlinear Shrödinger equation using two different approach. proposed exact solutions to some stochastic equation systems for ion sound and Langmuir waves. used the exp-function method to obtain optical solitons in birefringent fibers with Kerr nonlinearity. presented soliton solutions of the Biswas-Arshed equation with multiplicative noise using the Itô calculus method.
In , Aydin, suggested the stochastic nonlinear Schrödinger equation and examined noise effect on optical soliton solutions of this equation.
Similarly, in ,, ,, ,, ,, ,, ,, ,, ,, researchers obtained optical soliton solutions of different forms of models under different conditions. In this article, we introduce stochastic dispersive Schrödinger–Hirota equation with parabolic law nonlinearity with multiplicative white noise via Ito calculus in the following form: i ∂ u ∂ t + α ∂ 2 u ∂ x 2 + β 1 | u | 2 + β 2 | u | 4 u + i λ ∂ 3 u ∂ x 3 + β 3 | u | 2 + β 4 | u | 4 ∂ u ∂ x + σ ∂ W ∂ t u = 0 ,in which u = u x, t, λ and α are solution function, third-order dispersion (3OD), group velocity dispersion (GVD) terms coefficients and β 1, β 2, β 3, β 4 are the coefficients of the nonlinear terms, respectively. Moreover, i = − 1, σ is noise strength which is constant in the space and W t is standard Wiener process and it has been defined as in the following form , : W t = ∫ 0 t A η d W ( η ), η < t ,where η is stochastic variable.
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