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We study the symmetry reduction of nonlinear evolution and wave type differential equations by using operators of non-point symmetry. In our approach we use both operators of classical and conditional symmetry. It appears that the…
We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the…
Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics (which is a…
We discuss the theory of spin waves in non-degenerate ultra-cold gases, and compare various methods which can be used to obtain appropriate kinetic equations. We then study non-hydrodynamic situations, where the amplitude of spin waves is…
This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations…
Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Yi et. al. (2021) has demonstrated the existence of such waves for two classes of regularisations, including…
The general method to obtain solutions of the Maxwellian equations from scalar representatives is developed and applied to the diffraction of electromagnetic waves. Kirchhoff's integral is modified to provide explicit expressions for these…
A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input-output passive. We propose…
Traveling wave solutions of degenerate coupled $\ell$-KdV equations are studied. Due to symmetry reduction these equations reduce to one ODE, $(f')^2=P_n(f)$ where $P_n(f)$ is a polynomial function of $f$ of degree $n=\ell+2$, where $\ell…
In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all…
This paper is concerned with the traveling wave solutions for integro-difference systems of higher order. By using Schauder fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and…
This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an $n$-dimensional ball. By combining the variational methods and saddle point reduction technique, we prove there exist at least…
Some classes of the so called "travelling wave" solutions of Einstein and Einstein - Maxwell equations in General Relativity and of dynamical equations for massless bosonic fields in string gravity in four and higher dimensions are…
In this work we study wave propagation in dissipative relativistic fluids described by a simplified set of the 2nd order viscous conformal hydrodynamic equations. Small amplitude waves are studied within the linearization approximation…
We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on…
A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here…
We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic…
Extending results of Oh and Zumbrun in dimensions $d\ge 3$, we establish nonlinear stability and asymptotic behavior of spatially-periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions $d=1,2$,…
A method for solving a quasilinear nonelliptical equation of the second order is developed and we give classification and parametrization of simple elements of the equation.We find exact solutions of an equation for potential stationary…
We present a conservation formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean…