Stability in solitary waves

Professor Erik Wahlén will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Centre for Mathematical Sciences, Lund University.

The aim of the project is to develop a theory for the stability of solitary water waves. One such wave was discovered almost two hundred years ago, by engineer and boat builder John Scott Russell, when he followed a solitary water wave for several kilometres along a Scottish canal, without it changing either its shape or speed. 

For solitary waves to travel long distances, they must be stable enough that they do not collapse when disturbed. Russell’s waves are two-dimensional – they do not vary perpendicular to the direction of travel – and, if surface tension is disregarded, they are stable. If there a is strong surface tension, the waves become unstable when there are perturbations perpendicular to the direction of travel. However, they are stable in their direction of travel.

One of the project’s two main objectives is to investigate the stability of two-dimensional waves when the surface tension is weak. This causes small periodic ripples behind the solitary waves, in their far field. The stability of the ripples varies with the surface tension; but what is the stability of an object made up of a localised pulse with periodic ripples in the far field?

The second main objective of the project is to develop stability theory for three-dimensional solitary waves with surface tension. These waves are known to be stable if the surface tension is strong enough, or if the water is shallow enough. However, when the surface tension is weak, or the water is very deep, solitary waves with an oscillating shape form in the direction of travel. These waves are believed to be unstable, but there is no mathematical proof. The project aims to fill this gap by using knowledge from a simpler approximate model.

Photo: Stefano Pasquali