Dynamics near ring solitons of asymmetrical Nizhnik-Novikov-Veselov equation: Integrability and blow-up

Abstract

This work presents an in-depth analytical and numerical investigation of a special class of ring soliton and dromion solutions to the asymmetrical Nizhnik-Novikov-Veselov (ANNV) equation. We introduce the radius r of the closed curve on which the ring soliton lies to parameterize a family of solutions connecting the ring soliton and the dromion, where $r=0$ corresponds to the dromion solution. By fixing the external potential function, we reduced the ANNV equation and analyzed the integrability of the simplified model. Subsequently, we designed a discrete conservation-preserving algorithm to efficiently solve and simulate it. Numerical results verified the conservation and the effectiveness of the algorithm. Furthermore, we explore the rich dynamical behavior of the ring soliton and dromion solutions under various perturbations. Our results show that small perturbations applied to the initial condition can trigger blow-up, and the likelihood of blow-up increases with the radius r of the closed curve.