Bifurcation analysis of small amplitude unidirectional waves for nonlinear Schrödinger equations with fractional derivatives

Abstract

Abstract This study explores bifurcation phenomena in the nonlinear time-dependent Schrödinger equation and related models, applying Kudryashov’s methods to find exact solutions. It extends the analysis to the nonlinear time fractional Schrödinger equation and the space-time modified Benjamin-Bona-Mahony equation with beta derivatives for fractional dynamics. The paper derives analytical solutions, highlighting the impact of fractional derivatives on wave propagation. A bifurcation analysis shows how parameter changes affect system behavior, with visual representations of wave solutions illustrating the influence of parameters on wave stability and morphology. The work enhances the understanding of fractional differential equations in fields like fluid dynamics, nonlinear optics, and quantum mechanics.

Keywords

• Schrödinger equation
• Kudryashov techniques
• Exact solutions
• Benjamin Bona Mahony equation
• Beta derivative