Authors: Nate Lovett
Mentors: Harish Bhatt
Insitution: Utah Valley University
Coupled nonlinear Schrödinger equations (CNLSEs) are an extension of the nonlinear Schrödinger equation (NLSE) that applies to multiple interacting wave systems. They occur naturally in many physical systems, including nonlinear optics, multi-component Bose-Einstein condensates, and shallow water waves. Solitons, which are self-contained, localized wave packets that preserve their shape and speed during propagation, are a significant application of CNLSEs. Solitons are prevalent in nonlinear systems and play a critical role in long-distance information transmission in telecommunications. Despite their widespread use in various fields, solving CNLSEs analytically is challenging, and numerical approximations are necessary. However, solving CNLSEs numerically is a difficult task because of their high nonlinearity.
To overcome this challenge, in this presentation, we will introduce, analyze, and implement an established fourth-order Exponential Time Differencing scheme in combination with the Fourier spectral method for simulating one-dimensional CNLSEs. In order to check the performance of this method in terms of accuracy, efficiency, and stability, we will present simulation results on CNLSEs. Our results will consider single, two, and four soliton interactions for homogeneous Neumann, homogeneous Dirichlet, and periodic boundary conditions. The numerical results will show that the proposed method is able to preserve energy and mass for a long time simulation in soliton interactions, as well as preserve the expected order of convergence for the proposed method.