Related papers: Traveling wave solutions for the FPU chain: a cons…
We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small…
We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations \nu u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on a bounded spatial domain.…
Traveling wave solutions, in the form $u(x,t)=f(x+ct)$, to the generalized Burgers-Fisher equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k>0$, are classified with…
Lucas and Moll have proposed a system of forward-backward partial differential equations that model knowledge diffusion and economic growth. It arises from a microscopic model of learning for a mean-field type interacting system of…
In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction…
We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of the Kawahara equation. These solutions exhibit high-frequency instabilities when subject to bounded perturbations on the whole real line. We…
The Penrose-Fife Phase Field Model is now a well-established model in the theory of phase transitions. In the course of study of this model both the rigorous mathematical results and approximate solutions were obtained. However, to the best…
We study the Muskat problem for one fluid in arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force…
The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series…
We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schr\"odinger equations in one space dimension.
An analysis of traveling wave solutions of pure cross-diffusion systems, i.e., systems that lack reaction and self-diffusion terms, is presented. Using the qualitative theory of phase plane analysis the conditions for existence of different…
In this paper we consider traveling waves for the Gross-Pitaevskii equation which are T-periodic in each variable. We prove that if T is large enough, there exists a solution as a global minimizer of the corresponding action functional. In…
Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in $L^2$-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum…
We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of $n$-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying…
In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{R} \\…
Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} + f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ Assume that the system has a sequence of regular and internal layers occurring alternatively along the…
Superconducting parametric amplifiers have great promise for quantum-limited readout of superconducting qubits and detectors. Until recently, most superconducting parametric amplifiers had been based on resonant structures, limiting their…
We construct a two-parameter family of explicit solutions to the cubic wave equation on $\mathbb{R}^{1+3}$. Depending on the value of the parameters, these solutions either scatter to linear, blow-up in finite time, or exhibit a new type of…
A set of travelling wave solutions to a hyperbolic generalization of the convection-reaction-diffusion is studied by the methods of local nonlinear alnalysis and numerical simulation. Special attention is paid to displaying appearance of…
We study waves in a chain of dispersively coupled phase oscillators. Two approaches -- a quasi-continuous approximation and an iterative numerical solution of the lattice equation -- allow us to characterize different types of traveling…