Two-dimensional Zakharov-Kuznetsov Equation Research Articles

On Inner Regularity of Solutions of Two-Dimensional Zakharov–Kuznetsov Equation

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov–Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a half-strip. In this problem, different types of initial conditions (e.g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of domains: a plane, a half-plane, and a strip.

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

<p style='text-indent:20px;'>In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [<xref ref-type="bibr" rid="b3">3</xref>] here extended for the high-order Kadomtsev-Petviashvili. A similar result is also obtained for the two-dimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the <inline-formula><tex-math id="M1">\begin{document}$ y $\end{document}</tex-math></inline-formula>-derivative of the norm.</p>

Structure-preserving reduced-order modeling of Korteweg–de Vries equation

Computationally efficient, structure-preserving reduced-order methods are developed for the Korteweg–de Vries (KdV) equations in Hamiltonian form. The semi-discretization in space by finite differences is based on the Hamiltonian structure. The resulting skew-gradient system of ordinary differential equations (ODEs) is integrated with the linearly implicit Kahan’s method, which preserves the Hamiltonian approximately. We have shown, using proper orthogonal decomposition (POD), the Hamiltonian structure of the full-order model (FOM) is preserved by the reduced-order model (ROM). The reduced model has the same linear–quadratic structure as the FOM. The quadratic nonlinear terms of the KdV equations are evaluated efficiently by the use of tensorial framework, clearly separating the offline–online cost of the FOMs and ROMs. The accuracy of the reduced solutions, preservation of the conserved quantities, and computational speed-up gained by ROMs are demonstrated for the one-dimensional single and coupled KdV equations, and two-dimensional Zakharov–Kuznetsov equation with soliton solutions.

Loomis-Whitney-type inequalities and low regularity well-posedness of the periodic Zakharov-Kuznetsov equation

Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in the fully periodic case with initial data in Sobolev spaces Hs, s>1, is proved. Frequency dependent time localization is utilized to control the derivative nonlinearity. The new ingredient to improve on previous results is a nonlinear Loomis-Whitney-type inequality.

Global Well-posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in Hs(ℝ2) for all $${\textstyle{5 \over 7}}$$ < s < 1 via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, forced Zakharov-Kuznetsov equation is also established in Hs(ℝ2) for $${\textstyle{{10} \over {11}}}$$ < s < 1.

Well-Posedness for the Two-Dimensional Zakharov-Kuznetsov Equation

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $ B^\frac{1}{2}_{2,1}(\mathbb{R}^2)$ is established by using atomic spaces.

On Inner Regularity of Solutions of Two-Dimensional Zakharov-Kuznetsov Equation

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov-Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a semistrip. In this problem, different types of initial conditions (e. g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of areas: a plane, a half-plane, and a strip.

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

Initial–boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.

On the impossibility of solitary Rossby waves in meridionally unbounded domains

Evolution of weakly nonlinear and slowly varying Rossby waves in planetary atmospheres and oceans is considered within the quasi-geostrophic equation on unbounded domains. When the mean flow profile has a jump in the ambient potential vorticity, localized eigenmodes are trapped by the mean flow with a non-resonant speed of propagation. We discuss amplitude equations for these modes. Whereas the linear problem is suggestive of a two-dimensional Zakharov-Kuznetsov equation, we found that the dynamics of Rossby waves are effectively linear and confined to zonal waveguides of the mean flow. This eliminates even the ubiquitous Korteweg-de Vries equations as the underlying model for spatially localized coherent structures in these geophysical flows.

A New Technique for Simulation the Zakharov–Kuznetsov Equation

In this article, a new technique is proposed to simulated two-dimensional Zakharov–Kuznetsov equation with the initial condition. The idea of this technique is based on Taylors' series in its derivation. Two test problems are presented to illustrate the performance of the new scheme. Analytical approximate solutions that we obtain are compared with variational iteration method (VIM) and homotopy analysis method (HAM). The results show that the new scheme is efficient and better than the other methods in accuracy and convergence.

Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation

Initial-boundary value problems in a bounded rectangle with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global well-posedness in the classes of weak and regular solution are established. As applications of the developed technique results on boundary controllability and long-time decay of weak solutions are also obtained.

Initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.

Solitary wave solutions for variant two-dimensional Zakharov-Kuznetsov equation

Solitary wave solutions for variant two-dimensional Zakharov-Kuznetsov equation

A solution of the two-dimensional Zakharov-Kuznetsov equation using lattice-Boltzmann and He's semi-inverse method

A solution of the two-dimensional Zakharov-Kuznetsov equation using lattice-Boltzmann and He's semi-inverse method

A discontinuous Galerkin finite element method for the Zakharov-Kuznetsov equation

In this paper, we develop and analyze a discontinuous Galerkin (DG) method for the two-dimensional nonlinear Zakharov-Kuznetsov (ZK) equation. The DG method could be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability and an error estimate are discussed carefully. Finally, a numerical example for the nonlinear problem is given to show that the scheme attains the optimal $(k+1)$ th order of accuracy for piecewise $Q^{k}$ polynomials of degree k when $k\geq2$ .

Stabilization of Regular Solutions for the Zakharov-Kuznetsov Equation Posed on Bounded Rectangles and on a Strip

AbstractInitial–boundary-value problems for the two-dimensional Zakharov–Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. An exponential decay rate of regular solutions for the original nonlinear problems is proved.

Explicit multi-symplectic method for the Zakharov—Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov—Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.

A note on the 2D generalized Zakharov–Kuznetsov equation: Local, global, and scattering results

We consider the generalized two-dimensional Zakharov–Kuznetsov equation ut+∂xΔu+∂x(uk+1)=0, where k⩾3 is an integer number. For k⩾8 we prove local well-posedness in the L2-based Sobolev spaces Hs(R2), where s is greater than the critical scaling index sk=1−2/k. For k⩾3 we also establish a sharp criteria to obtain global H1(R2) solutions. A nonlinear scattering result in H1(R2) is also established assuming the initial data is small and belongs to a suitable Lebesgue space.

Well-Posedness for the ZK Equation in a Cylinder and on the Background of a KdV Soliton

We establish the well-posedness of two Cauchy problems for the two-dimensional Zakharov–Kuznetsov equation motivated by the study of transverse stability properties of the N-soliton φ N of the Korteweg–de Vries equation. They differ by the functional setting: Sobolev spaces H s (ℝ × 𝕋), for the first one, φ N + H 1(ℝ2) for the second one. In the latter case, the solution is shown to be global.

Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations

The time evolution of multi-symplectic PDEs with periodic boundary conditions in one and two space dimensions is considered. We introduce the idea of a multi-symplectic Fourier transformfor systems on periodic domains leading to a semi-discretization on Fourier space, and to the concept of multi-symplecticity on Fourier space. The spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it — a discrete wave model. A by-product of the semi-discretization is that it leads automatically to a system of Hamiltonian ODEs in time when truncated. We show that the one-dimensional shallow water equations and the two-dimensional Zakharov–Kuznetsov equation are multi-symplectic and derive spectral discretizations for these systems and present numerical experiments.