Meromorphic travelling waves for pure-power equations: one pole orbit and a two-wave bound
Year: 2026
Type: preprint
Abstract: Meromorphic travelling waves for pure-power equations: one pole orbit and a two-wave bound The equation $P(d/dz)v+v^n/n=0$, where $P$ is a polynomial of degree $p$ with complex coefficients and $n\ge2$, describes the travelling waves of evolution equations with a pure-power nonlinearity and arbitrary linear dispersion and dissipation. We prove that, when $p\ge3$, every nonconstant solution meromorphic in the whole complex plane has exactly one pole modulo its twisted periods, the translations that multiply it by a root of unity. Hence it is rational, rational in an exponential, or elliptic, and lies in an explicit finite-dimensional space. No condition on the coefficients is imposed; in particular, the equation need not determine the Laurent series at a pole. Up to translation and multiplication by a root of unity, one equation has at most two such solutions, and two do occur. It has at most one unless $n\le3$ and $P$ is even. Once the twisted periods of a solution are known, its power $v^{n-1}$ is determined by $P$; for elliptic solutions this gives an explicit necessary condition on the coefficients.
Author: Migita, Alexander
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215030526