Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schr'{o}dinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schr"{o}dinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schr"{o}dinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.

💡 Research Summary

This paper investigates a one‑dimensional nonlinear Schrödinger equation (NLS) whose cubic nonlinearity coefficient g(t) varies explicitly with time. Such a model is motivated by recent advances in Bose‑Einstein condensate (BEC) manipulation, where Feshbach‑resonance techniques allow the interaction strength to be tuned dynamically. The authors first establish the complete integrability of the time‑dependent NLS by constructing a Lax pair (L, M) that satisfies the zero‑curvature condition L_t − M_x +