Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

The Cauchy problem for the Kadomtsev–Petviashvili–I equation without the zero mass constraint

Existence and regularity results are obtained for the Cauchy problem for the Kadomtsev–Petviashvili–I equation by the inverse spectral method. The initial data are small Schwartz functions which are not assumed to satisfy the zero mass constraint ∫ℝdxq(x, y, 0)=0.

Matrix spectral problem with multiple-order jumps and poles

The inverse spectral method for a general N × N spectral problem for solving nonlinear evolution equations in one spacial and one temporal dimension is extended to include multi-boundary jumps and high-order poles and their explicit representations. It therefore provides a formalism to generate soliton solutions that correspond to higher-order poles of the spectral data.