Constructing solitary pattern solutions of the nonlinear dispersive Zakharov–Kuznetsov equation

In this paper, the nonlinear dispersive Zakharov–Kuznetsov equation was solved by using the sine–cosine method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions were found.     

A note on the unique continuation property for Zakharov–Kuznetsov equation

We prove that if a sufficiently smooth solution to the initial value problem associated with the Zakharov–Kuznetsov equation$u_t+(u_{\mathit{xx}}+u_{\mathit{yy}}{)}_x+{\mathit{uu}}_x=0,(x,y)\in {\mathbb{R}}^2,t\in \mathbb{R}$is supported compactly in a nontrivial time interval then it vanishes identically.     

On the nonlinear self-adjointness of the Zakharov–Kuznetsov equation

A new general theorem, which does not require the existence of Lagrangians, allows to compute conservation laws for an arbitrary differential equation. This theorem is based on the concept of self-adjoint equations for nonlinear equations. In this paper we show that the Zakharov–Kuznetsov equation is self-adjoint and nonlinearly self-adjoint. This property is used to compute conservation laws corresponding to the symmetries of the equation. In particular the property of the Zakharov–Kuznetsov equation to be self-adjoint and nonlinearly self-adjoint allows us to get more conservation laws.     

The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov equation ${?}_{t}u+{?}_{x_1}\mathrm{\Delta }u={?}_{x_1}(u^{m+1})$ on three and higher dimensions. We mainly study the local well-posedness and the small data global well-posedness in the modulation space $M_{2,1}^0({\mathbb{R}}^n)$ for $m\ge 4$ and $n\ge 3$. We also investigate the quartic case, i.e., $m=3$.     

Solitary wave solution of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity

In this paper, we obtain an exact 1-soliton solution of the Zakharov–Kuznetsov equation, with power law nonlinearity, by the solitary wave ansatz method. A couple of conserved quantities of this equation are also calculated by using this 1-soliton solution.     

Numerical computation of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves

The main aim of the present work is to propose a new and simple algorithm for fractional Zakharov–Kuznetsov equations by using homotopy perturbation transform method (HPTM). The Zakharov–Kuznetsov equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions. The homotopy perturbation transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. HPTM is not limited to the small parameter, such as in the classical perturbation method. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.     

Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions

We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.     

Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetzov equation \(\partial _t u + \partial _x \Delta u = \partial _x ( u^{m+1} )\) on two or three space dimensions. We mainly study the two dimensional case and give the local well-posedness and the small data global well-posedness in the modulation space \(M_{2,1}(\mathbb {R}^2)\) for \(m \ge 4\). Moreover, for the quartic case (namely, \(m = 3\)), the local well-posedness in \( M_{2,1}^{1/4}(\mathbb {R}^2)\) is given. The well-posedness on three dimensions is also considered.     

Exponential decay for the linear Zakharov–Kuznetsov equation without critical domain restrictions

Initial–boundary-value problems for the linear Zakharov–Kuznetsov equation posed on bounded rectangles are considered. The spectral properties of a stationary operator are studied in order to show that the evolution problem posed on a bounded rectangle has no critical restrictions on its size. The exponential decay of regular solutions is established.     

Single peak solitary wave solutions for the Fornberg–Whitham equation

The qualitative theory of differential equations is applied to the Fornberg–Whitham equation. Smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation under inhomogeneous boundary condition are obtained. The conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the Fornberg–Whitham equation. The results presented in this article extend and improve the previous results.     

1-soliton solution of the Zakharov–Kuznetsov equation with dual-power law nonlinearity

In this paper, the Zakharov–Kuznetsov equation, with dual-power law nonlinearity is solved by using the solitary wave ansatze and 1-soliton solution is obtained. Using this soliton solution, a couple of conserved quantities, of this equation, are calculated.     

1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation

This paper obtains the solitary wave solution of the generalized Zakharov–Kuznetsov modified equal width equation. The solitary wave ansatz method is used to carry out the integration of this equation. A couple of conserved quantities are calculated. The domain restriction is identified for the power law nonlinearity parameter.     

Single peak solitary wave solutions for the generalized Camassa–Holm equation

This Letter presents all possible smooth, peaked and cusped solitary wave solutions for the generalized Camassa–Holm equation under the inhomogeneous boundary condition.The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the generalized Camassa–Holm equation.     

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.     

Numerical solutions of two-dimensional Burgers’ equations by lattice Boltzmann method

In this paper, the two-dimensional Burgers’ equations with two variables are solved numerically by the lattice Boltzmann method. The lattice Bhatnagar–Gross–Krook model we used can recover the macroscopic equation with the second order accuracy. Numerical solutions for various values of Reynolds number, computational domain, initial and boundary conditions are calculated and validated against exact solutions or other published results. It is concluded that the proposed method performs well.     

Application of the homotopy analysis method to the Poisson–Boltzmann equation for semiconductor devices

This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive an approximate solution to the nonlinear Poisson–Boltzmann equation for semiconductor devices. Specifically, this paper presents an analytic solution to potential distribution in a DG-MOSFET (Double Gate-Metal Oxide Semiconductor Field Effect Transistor). The DG-MOSFET represents one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry.     

New peakon,solitary wave and periodic wave solutions for the modified Camassa–Holm equation

In this paper, an independent variable transformation is introduced to solve the modified Camassa–Holm equation using the bifurcation theory and the method of phase portrait analysis. Some peakons, solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained.