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We propose an explicit numerical method to solve Milne's phase-amplitude equations. Previously proposed methods solve directly Milne's nonlinear equation for the amplitude. For that reason, they exhibit high sensitivity to errors and are…
This paper is concerned with the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. It first proves that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems…
Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These…
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially…
We derive a universal model for atom pairs interacting with non-resonant light via the polarizability anisotropy, based on the long range properties of the scattering. The corresponding dynamics can be obtained using a nodal line technique…
We consider the development of instabilities of homogeneous stationary solutions of discrete time lattice maps. Under some generic hypothesis we derive an amplitude equation which is the space-time continuous Ginzburg-Landau equation. Using…
We consider nonlinear effects in scattering of light by a periodic structure supporting optical bound states in the continuum. In the spectral vicinity of the bound states the scattered electromagnetic field is resonantly enhanced…
In this paper, He's frequency-amplitude formulation with some choice of location points that improve accuracy is applied to determine the periodic solution for the nonlinear oscillations of a punctual charge in the electric field of charged…
This work investigates spectrum and root functions (that is, eigen- and associated functions) of a Sturm-Liouville problem involving an abstract linear operator (nonselfadjoint in general) in the equation together with supplementary…
A sampling method by using scattering amplitude is proposed for shape and location reconstruction in inverse acoustic scattering problems. Only matrix multiplication is involved in the computation, thus the novel sampling method is very…
By using a generalization of the multiple scales technique we develop a method to derive amplitude equations for zero--dimensional forced systems. The method allows to consider either additive or multiplicative forcing terms and can be…
We use a multi-scale similarity analysis which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations, whose exact solutions are generally difficult to obtain.
This paper constructs solutions to linear and nonlinear Schr\"odinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$,…
The aim of this paper is to provide a comprehensive study of some linear nonlocal diffusion problems in metric measure spaces. These include, for example, open subsets in $\mathbb{R}^N$, graphs, manifolds, multi-structures or some fractal…
We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u &=|x|^\alpha |u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on…
This paper is devoted to study the asymptotic stability of wave equations with constant coefficients coupled by velocities. By using Riesz basis approach, multiplier method and frequency domain approach respectively, we find the sufficient…
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of…
In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering…
We study an intricate mechanism of pattern formation in globally coupled heterogeneous oscillatory media. In anodic electrochemical etching of silicon, the electrode surface splits into two amplitude-phase regions, while all oscillators…
A new nonlinear equation governing asymptotic dynamics of ripples is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system. It admits peakon solutions with amplitude, velocity and width in…