arXiv subjects

Rik W. S. Westdorp

Publications and source records attributed to Rik W. S. Westdorp.

6 recordsLinked to original sources

arXiv2025

Radiating Solitary Waves in an FPUT Lattice with Random Coefficients

We study the propagation of solitary waves in a Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with small random heterogeneity in the linear spring force. Perturbed by the random environment, solitary waves lose energy through a radiative tail, resulting in gradual amplitude attenuation. As long as the wave remains coherent, we track its position and amplitude via a modulation approach. An expansion of the resulting modulation equations provides explicit predictions for the slow average amplitude decay, which we verify through numerical simulations.

math.DS↗

arXiv2025

Stability of Stochastically Forced Solitons in the Korteweg-de Vries Equation

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation in the presence of noise and deterministic forcing. The noise is space-dependent and statistically translation-invariant. We show that, for small forcing, solitons remain close to the family of traveling waves in a weighted Sobolev norm, with high probability. We study the effective dynamics of the soliton amplitude and position via their variational phase, for which we derive explicit modulation equations. The stability result holds on a time scale where the deterministic forcing induces significant amplitude modulation.

math.AP↗

arXiv2025

Soliton Amplification in the Korteweg-de Vries Equation by Multiplicative Forcing

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation with small multiplicative forcing. Forcing breaks the conservative structure of the KdV equation, leading to substantial changes in energy over long times. We show that, for small forcing, the inserted energy is almost fully absorbed by the soliton, resulting in a drastically changed amplitude and velocity. We decompose the solution to the forced equation into a modulated soliton and an infinite dimensional perturbation. Assuming slow exponential decay of the forcing, we show that the perturbation decays at the same exponential rate in a weighted Sobolev norm centered around the soliton.

math.AP↗

arXiv2024

Well-posedness of a parametrically forced nonlinear Schrödinger equation driven by translation-invariant noise

We prove well-posedness in $H^σ(\mathbb{R})$ for any $σ\in [0,\infty)$ of a parametrically forced nonlinear Schrödinger equation (PFNLS) in one dimension driven by multiplicative Stratonovich noise which has spatially homogeneous statistics. The noise is white in time and correlated in space. We first construct local mild solutions via a fixed-point argument. We then formulate a blow-up criterion by showing that the equation has persistence of integrability and regularity as long as the $L^2(\mathbb{R})$-norm of the solution remains finite. Afterwards we derive a pathwise estimate on the $L^2(\mathbb{R})$-norm using a mild Itô formula. Our results also apply to the standard cubic NLS equation driven by multiplicative translation-invariant Stratonovich noise.

math.AP↗

arXiv2024

Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation

We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space $H^{s}$ for any $s \in [0,\infty)$, demonstrating persistence of regularity.

math.DS↗

arXiv2024

Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise

This paper studies the behavior of solitons in the Korteweg-de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.

math.AP↗