Universal Stability of Multipower Nonlinear Schrödinger Solitary Waves: A Sharp Exponent Threshold

We study one‐dimensional nonlinear Schrödinger equations with finitely many lower defocusing and higher focusing powers. For any fixed exponent lists, we determine when every positive‐frequency solitary wave is orbitally stable for all positive choices of the coefficients. Writing for the largest defocusing power and for the largest focusing power, the sharp condition is . Under this condition the mass is strictly increasing with frequency, with an explicit positive lower bound on its derivative; equality is included. When the inequality fails, every prescribed exponent list admits an open set of coefficients and frequencies with negative mass slope and unstable waves. The proof decomposes the slope into two‐power contributions and controls their integrals through a concavity argument involving a ratio of gamma functions. As an application, we establish the all‐frequency triple‐power assertion of Morrison and Tsai. We also show that mass provides a global parameter for the stable branch. Numerical calculations quantify the slope bound and illustrate the transition beyond the threshold.