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This study employs spectral methods to capture the behaviour of wave equation with dispersive-nonlinearity. We describe the evolution of hump initial data and track the conservation of the mass and energy functionals. The…
We analyse a generalised Fokker-Planck equation by making essential use of its linearisability through a Cole-Hopf transformation. We determine solutions of travelling wave and multi-kink type by resorting to a geometric construction in the…
A generalization of the Hopf-Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is…
We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We present our analytical results on the existence, uniqueness (up…
We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in…
We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes…
We investigate the properties of traveling wave solutions to hyperbolic conservation laws augmented with diffusion and dispersion, and review the existence and qualitative properties of the associated kinetic functions, which characterize…
We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional,…
Nonlinear fronts between spatially extended traveling wave convection (TW) and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated…
We study the existence of monotone heteroclinic traveling waves for a general Fisher-Burgers equation with nonlinear and possibly density-dependent diffusion. Such a model arises, for instance, in physical phenomena where a saturation…
The structure of steady plane-parallel radiative shock waves propagating through the hydrogen gas undergoing partial ionization and excitation of bound atomic states is investigated in terms of the self-consistent solution of the equations…
We present new analytical solutions to the hyperbolic generalization of Burgers equation, describing interaction of the wave fronts. To obtain them, we employ a modified version of the Hirota method.
Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time…
We examine a collection of particles interacting with inverse-square two-body potentials in the thermodynamic limit. We find explicit large-amplitude density waves and soliton solutions for the motion of the system. Waves can be constructed…
I present a review of the recent advancements in scattering theory, which provides a unified approach to studying dispersive and hyperbolic equations with general interaction terms and data. These equations encompass time-dependent…
A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the…
The paper addresses linear hyperbolic systems in one space dimension with random field coefficients. In many applications, a low degree of regularity of the paths of the coefficients is required, which is not covered by classical stochastic…
The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it…
We study travelling wave solutions of a generalised Korteweg-de Vries-Burgers equation with a non-local diffusion term and a concave-convex flux. This model equation arises in the analysis of a shallow water flow by performing formal…
Reaction-nonlinear diffusion (RND) partial differential equations are a fruitful playground to model the formation of sharp travelling fronts, a fundamental pattern in nature. In this work, we demonstrate the utility and scope of…