New insights into the diversity of stochastic solutions and dynamical analysis for the complex cubic NLSE with <i>δ</i>-potential through Brownian process
Muhammad Naveed Rafiq, Muhammad Hamza Rafiq, Huda Alsaud
Abstract
Abstract The nonlinear Schrödinger equation (NLSE) is a key tool for modeling wave propagation in nonlinear and dispersive media. This study focuses on the complex cubic NLSE with δ -potential, explored through the Brownian process. The investigation begins with the derivation of stochastic solitary wave solutions using the modified <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>exp</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mo mathvariant="normal">Ψ</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> expansion method. To illustrate the noise effects, 3D and 2D visualizations are displayed for different non-negative values of noise parameter under suitable parameter values. Additionally, qualitative analysis of both perturbed and unperturbed dynamical systems is conducted using bifurcation and chaos theory. In bifurcation analysis, we analyze the detailed parameter analysis near fixed points of the unperturbed system. An external periodic force is applied to perturb the system, leading to an investigation of its chaotic behavior. Chaos detection tools are employed to predict the behavior of the perturbed dynamical system, with results validated through visual representations. Multistability analysis is conducted under varying initial conditions to identify multiple stable states in the perturbed dynamical system, contributing to chaotic behavior. Also, sensitivity analysis of the Hamiltonian system is performed for different initial conditions. The novelty of this work lies in the significance of the obtained results, which have not been previously explored for the considered equation. These findings offer noteworthy insights into the behavior of the complex cubic NLSE with δ -potential and its applications in fields such as nonlinear optics, quantum mechanics and Bose–Einstein condensates.