An index theorem for the stability of periodic traveling waves of KdV type

TL;DR

This paper develops an index theorem for the stability of periodic KdV-type waves, linking eigenvalues to wave profile zeros and conserved quantities, extending classical oscillation and solitary wave stability theories.

Contribution

It introduces a novel index theorem that precisely counts unstable eigenvalues for periodic KdV waves, generalizing classical stability results and connecting to algebraic geometry.

Findings

The index theorem accurately predicts the number of unstable eigenvalues.

The index relates to zeros of the wave profile derivative and conserved quantities.

Explicit examples demonstrate the theorem's application.

Abstract

We consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention the theory is much less developed than the analogous theory for solitary wave stability, and appears to be mathematically richer. We prove an index theorem giving an exact count of the number of unstable eigenvalues of the linearized operator in terms of the number of zeros of the derivative of the traveling wave profile together with geometric information about a certain map between the constants of integration of the ordinary differential equation and the conserved quantities of the partial differential equation. This index can be regarded as a generalization of both the Sturm oscillation theorem and the classical stability theory for solitary wave solutions for equations of Korteweg-de Vries type. In the case of a polynomial nonlinearity this index, together with a related one introduced earlier by Bronski and Johnson, can be expressed in terms of derivatives of period integrals on a Riemann surface. Since these period integrals satisfy a Picard-Fuchs equation these derivatives can be expressed in terms of the integrals themselves, leading to an expression in terms of various moments of the solution. We conclude with some illustrative examples.