3D Zakharov-Kuznetsov Equation Research Articles

Resonant decompositions and global well-posedness for 2D Zakharov-Kuznetsov equation in Sobolev spaces of negative indices

The Cauchy problem for Zakharov-Kuznetsov equation on R2 is shown to be global well-posed for the initial data in Hs provided s>−113. As conservation laws are invalid in Sobolev spaces below L2, we construct an almost conserved quantity by multilinear correction term following the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to the KdV equation, the main difficulty is to handle the resonant interactions due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.

Numerical study of soliton stability, resolution and interactions in the 3D Zakharov–Kuznetsov equation

We present a detailed numerical study of solutions to the Zakharov–Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg–de Vries equation, though, not completely integrable. This equation is L2-subcritical, and thus, solutions exist globally, for example, in the H1 energy space.We first study stability of solitons with various perturbations in sizes and symmetry, and show asymptotic stability and formation of radiation, confirming the asymptotic stability result in Farah et al. (0000) for a larger class of initial data. We then investigate the solution behavior for different localizations and rates of decay including exponential and algebraic decays, and give positive confirmation toward the soliton resolution conjecture in this equation. Finally, we investigate soliton interactions in various settings and show that there are both a quasi-elastic interaction and a strong interaction when two solitons merge into one, in all cases always emitting radiation in the conic-type region of the negative x-direction.

Symmetry analysis and conservation laws of a further modified 3D Zakharov-Kuznetsov equation

We study a further modified (3 + 1)-dimensional(3D) Zakharov-Kuznetsov equation. The classical Lie symmetry method will be employed to obtain solutions of a further modified 3D Zakharov-Kuznetsov equation. We will also establish the multiplier method to derive conservation laws of the underlying equation. Moreover, a brief physical interpretation of the derived conserved quantities is conferred.

Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

This paper is concerned with the Cauchy problem of the 2D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space Hs(R2) for s>−1/4, and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in L2(R2).

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

Abstract The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

Decay of regular solutions for the critical 2D Zakharov–Kuznetsov equation posed on rectangles

An initial-boundary value problem for the critical generalized 2D Zakharov–Kuznetsov equation posed on rectangles is considered. The existence, uniqueness, and exponential decay rate of global regular solutions for small initial data are established.

Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term −∂x2u. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in Hs(R2) for s>1/2, and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in Hs(R2) for s≥0. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in Hs(R2) for s>−1/2.

Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$

The Cauchy problem of the 3DZakharov-Kuznetsov equation$$u_{t}+\partial_{\tilde{x}_{*,1}}\Delta u +(u^2)_{\tilde{x}_{*,1}}=0, (x,t)\in \mathbb{R}^3 \times\mathbb{R}, \ x=(\tilde{x}_{*,1},\tilde{x}_{*,2},\tilde{x}_{*,3});$$is considered. It is shown that it is globally well-posed in energy space $H^1(\mathbb{R}^3)$.It answer an open problem: Is it globally well-posed in energy space $H^1 (\mathbb{R}^3)$ for 3D Z-K equtation [10,12,13]? Moreover, in 4-D and more higher dimension, it is shown that it is locally well-posed in $H^1(\mathbb{R}^n)$ with $n\geq 4$. The method in this paper combine the linear property of the equation (dispersive property) with nonlinear property of the equation (energy inequality). We mainly extend the spaces $\mathbf{F}^s$ and $\mathbf{N}^s$ in one dimension [4] to higher dimension.

Global Regular Solutions to One Problem of Saut-Temam for the 3D Zakharov–Kuznetsov Equation

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on an unbounded domain is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the $H^2$-norm for small initial data are proven.

Global regular solutions for the 3D Zakharov-Kuznetsov equation posed on unbounded domains

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on unbounded domains is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the H2-norm for small initial data are proven.

Conservation laws and exact solutions for a 2D Zakharov–Kuznetsov equation

This paper aims to compute conservation laws for the 2D Zakharov–Kuznetsov equation using Noether’s approach through an interesting method of increasing the order of the 2D Zakharov–Kuznetsov equation. Moreover, exact solutions for the 2D Zakharov–Kuznetsov equation are obtained using the Kudryashov method and Jacobi elliptic function method.

Local Existence of Strong Solutions to the 3D Zakharov-Kuznetsov Equation in a Bounded Domain

We consider here the local existence of strong solutions for the Zakharov-Kuznetsov (ZK) equation posed in a limited domain \(\mathcal{M}=(0,1)_{x}\times(-\pi/2, \pi/2)^{d}\), d=1,2. We prove that in space dimensions 2 and 3, there exists a strong solution on a short time interval, whose length only depends on the given data. We use the parabolic regularization of the ZK equation as in Saut et al. (J. Math. Phys. 53(11):115612, 2012) to derive the global and local bounds independent of ϵ for various norms of the solution. In particular, we derive the local bound of the nonlinear term by a singular perturbation argument. Then we can pass to the limit and hence deduce the local existence of strong solutions.

Exponential decay of the [formula omitted]-norm for the 2D Zakharov–Kuznetsov equation on a half-strip

We formulate on a half-strip an initial boundary value problem for the 2D Zakharov–Kuznetsov equation. The existence of a unique weak solution was proven as well as exponential decay of the H1-norm for small initial data.

Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip

We formulate on a half-strip an initial boundary value problem for the Zakharov–Kuznetsov equation. Existence and uniqueness of a regular solution as well as the exponential decay rate of small solutions as t→∞ are proven.

The Cauchy problem for the 3D Zakharov-Kuznetsov equation

We prove that the Cauchy problem for the three-dimensional Zakharov-Kuznetsov equation is locally well-posed for data in $H^s(\R^3)$, s > $\frac{9}{8}$.