Related papers: Almost sharp wave kinetic theory of multidimension…
Starting from the stochastic Zakharov-Kuznetsov equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be…
Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the…
Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this…
We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves).…
We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schr\"odinger equation with stochastic forcing. The kynetic equation is obtained as a result of a double limiting…
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar…
Existence of non-resonant solutions of time-periodic type are established for the Kuznetsov equation with a periodic forcing term. The equation is considered in a three-dimensional whole-space, half-space and bounded domain, and with both…
We consider here the local existence of strong solutions for the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-pi /2, pi /2)^d, d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution…
We discuss a spectral property for the virial operator of the 2D Zakharov-Kuznetsov (ZK) equation. This is a crucial ingredient to establish blow-up or asymptotic stability of solitary waves in higher-dimensional problems. This model in 3D…
Direct numerical simulation of three-dimensional acoustic turbulence has been performed for both weak and strong regimes. Within the weak turbulence, we demonstrate the existence of the Zakharov-Sagdeev spectrum $\propto k^{-3/2}$ not only…
The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be…
This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact…
This paper is a continuation of our previous study arXiv:2507.01288 on the scattering problem for the Zakharov-Kuznetsov equation (ZK). When the space dimension is three, we construct a global solution to (ZK) which scatters to a given free…
The main purpose here is the study of dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Dispersive blow-up refers to point singularities due to the focusing of short or long waves. We will construct initial data such that…
In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg-de Vries (KdV)…
We study the dynamics of the collision of two solitary waves for the Zakharov-Kuznetsov equation in dimension $2$ and $3$. We describe the evolution of the solution behaving as a sum of $2$-solitary waves of nearly equal speeds at time…
We present a systematic study of the dynamical scaling process leading to the establishment of the Kolmogorov--Zakharov (KZ) spectrum in weak 3-wave turbulence. In the finite capacity case, in which the transient spectrum reaches infinite…
In this paper, we study the nonlinear dispersive waves including the rarefaction and dispersive shock waves in the discrete modified KdV equation through the numerical simulations of the dispersive Riemann problems. In particular, we…
We consider the Zakharov-Kuznestov (ZK) equation posed in a limited domain (0,1)_{x}\times(-\pi /2, \pi /2)^d, d=1,2 supplemented with suitable boundary conditions. We prove that there exists a solution u \in \mathcal C ([0, T]; H^1(\dom))…
We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schr\"odinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main…