Spectral Analysis and Stability of Wave Equations with Dispersive Nonlinearity

This study employs spectral methods to capture the behaviour of wave equation with dispersive-nonlinearity. We describe the evolution of hump initial data and track the conservation of the mass and energy functionals. The dispersive-nonlinearity results to solution in an extended Schwartz space via analytic approach. We construct numerical schemes based on spectral methods to simulate soliton interactions under Schwartzian initial data. The computational analysis includes validation of energy and mass conservation to ensure numerical accuracy. Results show that initial data from the Schwartz space decompose into smaller wave-packets due to the weaker dispersive-nonlinearity but leads to wave collapse as a result of stronger dispersive-nonlinearity. We conjecture that the hyperbolic equation with a positive nonlinearity and exponent greater or equal 2 admits global solutions, while lower exponents lead to localized solutions. A stability analysis of solitonic solutions of the equation is provided via the perturbation approach.