Quantum Zakharov System Research Articles

Scattering for quantum Zakharov system in two space dimensions

In this paper, we study long time behavior of solution to the quantum Zakharov system in two dimensions. We construct a small global solution to the quantum Zakharov system which scatters to a given free solution by using space-time resonance method developed by Gustafson-Nakanishi-Tsai [Commun. Contemp. Math. 11 (2009), pp. 657–707] and Germain-Masmoudi-Shatah [Int. Math. Res. Not. IMRN 3 (2009), 414–432; J. Math. Pures Appl. (9) 97 (2012), pp. 505–543] etc.

On the convergence of the solution for a reduced model of the vectorial quantum Zakharov system

In this paper, we study the limit behavior of the smooth solution for a reduced vectorial quantum Zakharov system which describes the interaction between the quantum Langmuir waves and quantum ion-acoustic waves in the plasmas. We first give the local existence and uniqueness of the solution to the quantum Zakharov system. Then we derive the uniform bounds of solution with appropriate initial data, and prove that the solution of the quantum Zakharov system converges to the solution of the nonlinear Schrödinger system.

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Global Existence of the Solution for a Reduced Model of the Vectorial Quantum Zakharov System

Global Existence of the Solution for a Reduced Model of the Vectorial Quantum Zakharov System

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Multiplicity of solutions for the stationary quantum Zakharov system with combined nonlinearities

In this article, we prove the existence and multiplicity of nontrivial solutions for the stationary quantum Zakharov system with a sign-changing potential and a general nonlinearity involving a combination of a superlinear f ( x , u ) source term and a sublinear term ( K ( x ) | u | q − 2 u , K ∈ L 2 2 − q ( R 3 ) , where 1 < q < 2 ) .

The local discontinuous Galerkin method for the nonlinear quantum Zakharov system

In this paper, we design an energy conserving local discontinuous Galerkin (LDG) method to solve the quantum Zakharov system (QZS). We first give theoretical proof for the conservation of mass and energy and then focus our discussion on the priori estimates for the semi-discrete approximation of the QZS. With the proof for the L2 boundedness of the numerical solutions and the discrete Poincaré inequalities, we obtain the optimal error estimates. Next, we develop a decoupled and implicit linear fully-discrete scheme using the semi-implicit spectral deferred correction (SISDC) method for temporal discretization. Another strength of our approach is the arbitrary high accuracy which depends only on the degree of the approximating piecewise polynomials. Finally, several numerical examples are reported to validate it and the conservation properties. Especially, the second-order convergence rate of the QZS to its limiting model in the semi-classical limit is achieved as expected.

MULTILINEAR WEIGHTED ESTIMATES AND QUANTUM ZAKHAROV SYSTEM

We consider the well-posedness theory of the compact case of one-dimensional quantum Zakharov system with the periodic boundary condition. The global well-posedness for sufficiently regular data is shown. The semi-classical limit as is obtained on a compact time interval whereas the quantum perturbation proves to be singular on an infinite time interval.

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

In this paper, we propose and analyze a new fourth-order compact finite difference scheme for solving the quantum Zakharov system (QZS). The new scheme perfectly inherits the mass and energy conservation possessed by the QZS, while the energy preserved by the existing schemes expressed by two-level's solution at each time step. By using the energy method to analyze an equivalent form of the difference scheme, without any requirement on the grid ratio, we establish rigorously the optimal error estimate of the proposed scheme in the energy norm. The accuracy is shown to be fourth order in space and second order in time. Numerical examples are presented to verify the theoretical results and show the effectiveness of the proposed scheme.

Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

&lt;p style='text-indent:20px;'&gt;In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.&lt;/p&gt;

Local well-posedness for the quantum Zakharov system in three and higher dimensions

We study the Cauchy problem associated with a quantum Zakharov-type system in three and higher spatial dimensions.Taking the quantum parameter to unit and developing Fourier restriction norm arguments, we establish local well-posedness property for wider range than the one known for the Zakharov system.

A Conservative Linearly-Implicit Compact Difference Scheme for the Quantum Zakharov System

This paper is devoted to developing and analysing a highly accurate conservative method for solving the quantum Zakharov system. The scheme is based on a linearly-implicit compact finite difference discretization and conserve the mass as well as energy in discrete level. Detailed numerical analysis is presented which shows the method is fourth-order accurate in space and second-order accurate in time. Several numerical examples are reported to confirm the conservation properties and high accuracy of the proposed scheme. Finally the compact scheme is applied to study the convergence rate of the quantum Zakharov system to its limiting model in the semi-classical limit.

Scattering for 3D quantum Zakharov system in $L^2$

Scattering for 3D quantum Zakharov system in $L^2$

Asymptotic preserving trigonometric integrators for the quantum Zakharov system

We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime $$\vartheta = 1$$ as well as in the classical regime $$\vartheta \rightarrow 0$$ without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit $$\vartheta \rightarrow 0$$ uniformly in the time discretization parameter. Numerical experiments underline the favorable error behavior of the new schemes with first- and second-order time convergence uniformly in $$\vartheta $$ , first-order asymptotic convergence in $$\vartheta $$ and long time structure preservation properties.

Stationary quantum Zakharov systems involving a higher competing perturbation

We consider the stationary quantum Zakharov system with a higher competing perturbation $$\displaylines{ \Delta ^2u-\Delta u+\lambda V(x)u=K(x)u\phi -\mu | u|^{p-2}u \quad \text{in }\mathbb{R}^3, \cr -\Delta \phi +\phi =K(x)u^2 \quad \text{in }\mathbb{R}^3, }$$ where \(\lambda &gt;0\), \(\mu&gt;0\), \(p&gt;4\) and functions \(V\) and \(K\) are both nonnegative. Such problem can not be studied via the common arguments in variational methods, since Palais-Smale sequences may not be bounded. Using a constraint approach proposed by us recently, we prove the existence, multiplicity and concentration of nontrivial solutions for the above problem. For more information see https://ejde.math.txstate.edu/Volumes/2020/06/abstr.html

Local well-posedness for the quantum Zakharov system

Local well-posedness for the quantum Zakharov system

Semi-classical Limit for the Quantum Zakharov System

In this paper, we prove the semi-classical limit for the quantum Zakharov system, that is, the quantum Zakharov system converges to the classical Zakharov system as the quantum parameter goes to zero, including a convergence rate. We improve the results of Guo-Zhang-Guo [11].

Global well-posedness and scattering for the quantum Zakharov system in 𝐿²

We study the Cauchy problem for the quantum Zakharov system in the class of square-integrable functions on the Euclidean space of general dimensions. The local well-posedness is proven for dimensions up to eight, together with global existence for dimensions up to five, as well as scattering for small initial data in dimensions greater than three.

Global well-posedness and scattering for the quantum Zakharov system in $L^2$

Global well-posedness and scattering for the quantum Zakharov system in $L^2$

The stationary quantum Zakharov system perturbed by a local nonlinearity

We study a class of stationary quantum Zakharov systems with a perturbation as follows Δ2u−Δu+λV(x)u=uϕ+μf(x)|u|p−2uin R3,−Δϕ+ϕ=u2,in R3,where λ>0, μ∈R, p>1, and both f(x) and V(x) are nonnegative functions. By using the Nehari manifold, we prove the nonexistence, existence and multiplicity of nontrivial solutions, depending on the parameters λ,μ and p.