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Exact Solutions of Stochastic Burgers-KdV Equation with variable coefficients

Kolade Adjibi, Allan Martinez, Miguel Mascorro, Carlos Montes, Tamer Oraby, Rita Sandoval, Erwin Suazo

Abstract

We will present exact solutions for three variations of stochastic Korteweg de Vries-Burgers (KdV-Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV-Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries-Burgers (KdV-Burgers) equations.

Exact Solutions of Stochastic Burgers-KdV Equation with variable coefficients

Abstract

We will present exact solutions for three variations of stochastic Korteweg de Vries-Burgers (KdV-Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV-Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries-Burgers (KdV-Burgers) equations.
Paper Structure (4 sections, 4 theorems, 60 equations, 3 figures)

This paper contains 4 sections, 4 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Burgers-KdV Equation
  4. Conclusion

Key Result

Lemma 1

Figures (3)

  • Figure 1: Two realizations of the stochastic process in equation \ref{['solnex01']}.
  • Figure 2: Four realizations of the stochastic process in equation \ref{['ex2_eq']} when $\sigma(t)=1$ (top) and $\sigma(t)=t^2$ (bottom).
  • Figure 3: Two realizations of the stochastic process in equation \ref{['solnex01_2']}.

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 1
  • Proposition 1
  • Example 1
  • Example 2
  • Example 3