We consider traveling wave phenomena for a viscoelastic generalization of Burgers’ equation. For asymptotically constant velocity profiles we find three classes of solutions corresponding to smooth traveling waves, piecewise smooth waves, and piecewise constant (shock) solutions. Each solution type is possible for a given pair of asymptotic limits, and we characterize the dynamics in terms of t...

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of t...

We construct traveling wave solutions with vortex helix structures for the Schrödinger map equation ∂m ∂t = m× (∆m−m3~e3) on R × R, of the form m(s1, s2, s3 − δ| log | t) with traveling velocity δ| log | along the direction of s3 axis. We use a perturbation approach which gives a complete characterization of the asymptotic behavior of the solutions.

We consider traveling wave solutions for nonlinear lattices of particles with nearest neighbour interaction. By using variational methods we prove the existence of monotone traveling waves for nite, periodic lattices and show that in the limit, as the period goes to innnity, these waves converge to corresponding solutions of the innnite lattice.

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed n...

We study the traveling wave front solutions for a two-dimensional periodic lattice dynamical system with monostable nonlinearity. We first show that there is a minimal speed such that a traveling wave solution exists if and only if its speed is above this minimal speed. Then we prove that any wave profile is strictly monotone. Finally, we derive the convergence of discretized minimal speed to t...