Nonlinear Wave Processes in a Medium of Interacting Particles without Collisions

Abstract

   The Fermi–Pasta–Ulam model including the fourth and fifth terms in the potential of interaction between neighboring particles has been considered. A passage to a continuum limit has been performed when the distance between the particles approaches zero and the number of particles tends to infinity. It has been shown that, a nonlinear partial differential equation of the sixth order is obtained instead of the well-known Korteweg–de Vries equation taking into account the quadratic interaction between particles. The fifth-order evolution partial differential equation has been obtained. The analytical properties of the resulting equations have been investigated. It has been shown that the general solution of the fifth-order differential equation obtained during the passage to traveling wave variables has four branches in the expansion into a Laurent series. In the second step of the Painlevé test, Fuchs indices two of which are complex have been found. It has been shown that the fifth-order nonlinear partial differential equations found from the Fermi–Pasta–Ulam model do not pass the Painlevé test. The exact solutions of the fifth-order evolution equations have been obtained using the simplest equation method. The chart of the solutions has been constructed.