Existence and numerical approximation of solutions of a Schr\"odinger equation with derivative in the nonlinear term

Abstract

In this paper, we study a Schr\"odinger-type equation featuring a derivative in the nonlinear term and incorporating diffusion effects. This type of equation arises in various physical applications, such as modeling low-order magnetization in ferromagnetic nanocables and describing the collision of ferromagnetic solitons in weakly ferromagnetic media. We establish a local well-posedness result for the Cauchy problem associated with this model and analyze the convergence and error order of a Fourier spectral numerical scheme for approximating its solutions in the periodic setting. Additionally, we investigate the behavior of solutions in certain asymptotic regimes, both analytically and through numerical experiments, by examining limiting cases of the model's parameters.