A direct approach for solving the cubic Szeg\"o equation

TL;DR

This paper introduces a novel, direct method for solving the cubic Szeg"o equation, enabling explicit construction of multiphase, multisoliton, and periodic solutions without relying on spectral analysis.

Contribution

It presents a bilinearization approach and determinant theory to derive solutions, offering an alternative to spectral methods and explicitly solving the eigenvalue problem for periodic solutions.

Findings

Derived explicit multiphase and multisoliton solutions

Established a bilinearization framework for the equation

Solved the eigenvalue problem for periodic solutions

Abstract

We study the cubic Szeg\"o equation which is an integrable nonlinear non-dispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szeg\"o equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szeg\"o equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.