Related papers: White noise for KdV and mKdV on the circle
We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDE of KdV-type, including generalized KdV and Benjamin-Ono equations. In this investigation, we consider the…
It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly…
After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this PhD thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness…
This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal…
In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with…
The stability of periodic traveling wave solutions to dispersive PDEs with respect to `arbitrary' perturbations is still widely open. The focus is put here on stability with respect to perturbations of the same period as the wave, for…
We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV…
The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic…
We will present exact solutions for three variations of stochastic Korteweg de Vries-Burgers (KdV-Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include…
In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with…
We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions…
We characterize the local smoothness and the asymptotic growth rate of the L\'evy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for…
We study the large deviations principle for locally periodic stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to…
We consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and semi-supervised metrics…
The effect of transport noise on a 2D fluid may depend on the space-scale of the noise. We investigate numerically the dissipation properties of very small-scale transport noise. As a test problem we consider the Kelvin-Helmholtz…
It is possible to construct L\'evy white noises as generalized random processes in the sense of Gel'fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two…
We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting,…
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the $I$-method and spectral analysis following Pego and Weinstein, we are…
We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation. We show that these waves are transversely unstable with…
In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov…