We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium u ≡ 1 is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is “above and away” from the intermediate equilibrium. For focusing kernels we ...

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equation Φtt −Φxx + aΦxxxx − bΦxxtt +ΦtΦxx + 2ΦxΦxt = 0. Periodic travelling-wave solutions with an arbitrary fundamental period T0 will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with period T0 will ...

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth th...

A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is developed. The complete system of equations reduces to two coupled differential equations for the transverse electron speed u and a phase variable φ=φp − φe representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are o...

We consider a nonlinear system of parabolic equations describing the transport of contaminant in porous media. For this system travelling wave solutions are known to exist. We prove existence and uniqueness of general solutions of the system and convergence to travelling waves.

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein–Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full...

1. Introduction. In this paper we are interested in travelling wave solutions u = u(x; t)

We propose a model for transport of solutes in a porous medium participating in a dissolu-tion/precipitation reaction, in general not in equilibrium. For an unbounded spatial domain, travelling wave solutions exist, if and only if the charge distribution is constant and we deal with a dissolution situation. The travelling wave in fact exhibits a sharp dissolution front. The wave is given in a n...

In this paper, we established traveling wave solutions of the nonlinear evolution equation. The sine-cosine method was used to construct travelling wave solutions of the Fitzhugh-Nagumo equation and Cahn-Allen equation. Graphical interpretation shows that obtained results include periodic and Soliton wave solutions. It is also shown that for α =−1, solutions of Fitzhugh-Nagumo equation coincide...

Based on our earlier theoretical work on tracking periodic patterns into spatiotemporal chaotic regimes in spatially extended systems, we present in this article a case study of a high-aspectratio three-level laser. Following a detailed investigation into travelling wave solutions of the laser system and their stability conditions, we discuss ways of stabilising these solutions by local feedbac...