On alleged solutions of the cubically nonlinear Schrödinger equation

TL;DR

This paper interrogates the consistency of a widely used ansatz for the cubically nonlinear Schrödinger equation (CNLSE) by transforming the basic system into a dynamical system and solving via Weierstrass elliptic functions. The authors derive a pair of coupled relations for $z(t)=d^2(t)$ and $Q(x,t)$, leading to explicit elliptic-function representations. They then demonstrate inconsistency in the generic case by showing a nonzero value for the real-part consistency condition, $P(x,t)=0$, under a concrete parameter choice (e.g., $P(1,1;c_j,q,1,1)=0.113$). The result cautions against unverified application of these ansatz-based solutions in physical problems and emphasizes that prior checks may have overlooked essential consistency requirements, while leaving open questions about nongeneric compatibility and alternative solution forms.

Abstract

On the basis of analytical results, we present a numerical example that indicates inconsistency of a widely used ansatz with cubically nonlinear Schrödinger equation.