A complexity approach to the soliton resolution conjecture

TL;DR

This paper explores the soliton resolution conjecture in nonlinear dispersive equations through the lens of complexity theory, linking it to thermodynamic principles and providing a novel perspective on the conjecture.

Contribution

It introduces a complexity-based approach to the soliton resolution conjecture, establishing an equivalence with a thermodynamic-like law for solution complexity.

Findings

Complexity of solutions relates to the number of solitons and radiation.

The conjecture is equivalent to a second law of thermodynamics for complexity.

Provides a new framework connecting complexity theory and dispersive PDEs.

Abstract

The soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations. Roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term. In this paper we use the complexity of a finite object, a notion introduced in Algorithmic Information Theory, to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation.