The Kawahara equation on star graphs

TL;DR

This paper proves local well-posedness for the Kawahara equation on star graphs by establishing boundary conditions and deriving integral formulas, extending methods used for simpler domains to complex network structures.

Contribution

It introduces a framework for analyzing the Kawahara equation on star graphs, combining boundary condition identification and integral formula derivation, which can be applied to other fifth-order dispersive equations.

Findings

Established local well-posedness for the Kawahara equation on star graphs.

Derived integral formulas using the forcing operator and Fourier restriction methods.

Potential for extending the approach to other nonlinear dispersive equations on networks.

Abstract

In this paper, we establish local well-posedness for the Cauchy problem associated with the Kawahara equation on a general metric star graph. Initially, we identify suitable boundary conditions that produce a well-behaved dynamics for the linear equation. Subsequently, we derive the integral formula using the forcing operator method, previously applied to the Kawahara equation on the half-line by Cavalcante and Kwak (NoDEA 2020), and the Fourier restriction method of Bourgain (GAFA 1993). This work has the potential to be extended to other fifth-order nonlinear dispersive equations on star graphs.