We construct traveling wave solutions with vortex helices for the Schrödinger map equation ∂m ∂t = m× (∆m−m3~e3) in 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.

Some explicit traveling wave solutions to a Kolmogorov-PetrovskiiPiskunov equation are presented through two ansätze. By a Cole-Hopf transformation, this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear equation and further two solutions to describe nonlinear interaction of traveling waves are generated. Bäcklund transformations of the linear form and some special cases are...

We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy l...

case II diffusion of a penetrant through a polymer matrix is characterized by constant front speed. Hence, a traveling-wave analysis is appropriate for the model equations. For the previously validated model analyzed here, conditions on the molecular and stress diffusion coefficients are obtained which guarantee the existence of a traveling wave. Conditions are derived under which an interior m...

In this paper, we derive a delayed reaction-diffusion equation to describe a two-species predator-prey system with diffusion terms and stage structure. By coupling the uniformly approximate approach with the method of upper and lower solutions, we prove that the traveling wave fronts exist, which connect the zero solution with the positive steady state. Finally, we draw a conclusion that the ex...

We present a KdV-like 2-parameter equation ut + (3(1− δ)u+ (δ + 1)xx ux )ux = εuxxx. By using the dynamical system method, existence of different traveling wave solutions are discussed, including smooth solitary wave solution of with bell type, solitary wave solutions of valley type and peakon wave solution of valley type. Numerical integration are used to shown the different types of solutions...