Different types of analytical solutions of the fifth-order KdV equation under the influence of Gaussian white noise and Brownian motion

Hai-Yan Wang^a, Ying Shi^a, Song-Lin Zhao^b, Lu Yan<sup>a

a</sup> School of Science, Zhejiang University of Science and Technology, Hangzhou, Zhejiang, China
^b School of Applied Mathematics, Zhejiang University of Technology, Hangzhou, Zhejiang, China

Abstract: In this paper, we consider the stochastic fifth-order KdV equation, along with its Lax pair, under the influence of Gaussian white noise and Brownian motion. One new result in this paper is that the soliton-periodic mixed solution can be viewed as a novel tool for generating rogue waves when the soliton solution is in the dominant position. By applying the classical Darboux transformation, we obtain analytic solutions to this equation in determinant form. Through detailed analysis of spectral parameters, we construct soliton solutions, periodic solutions, and their mixed solutions for the stochastic fifth-order KdV equation, which incorporates noise terms. We also consider the generalized Darboux transformation and obtain rational solutions to the stochastic fifth-order KdV equation.

Keywords: stochastic fifth-order KdV equation, soliton solutions, periodic solutions, rational solutions, Darboux transformation.

MSC: 35Q51, 35Q53, 37K40

Received: 30.09.2025
Revised: 30.09.2025

DOI: 10.4213/tmf11090

 English version: Theoretical and Mathematical Physics, 2026, 226:2, 257–283