Two-dimensional particle solution of the extended Hamilton-Jacobi equation

Abstract

In classical mechanics the Hamilton-Jacobi equation for a free particle has the property of reducing a perturbation of spatially uniform solution into a point. In the late 1970s Sivashinsky proposed an extension of the equation so that it takes the form of the Kuramoto-Sivashinsky equation under which a smooth soliton is formed instead of the point. The soliton was proposed as a model for spatially extended elementary particle. However, this solution is unstable. Developing the Sivashinsky’s idea further, we propose a different extension which ensures stability. We performed two-dimensional computational experiments demonstrating the soliton formation and stability.