Abstract:
We investigate the existence of envelope soliton solutions in collisionless quantum plasmas, using the quantum-corrected Zakharov equations in the kinetic case, which describes the interaction between high frequency Langmuir waves and low frequency plasma density variations. We show the role played by quantum effects in the nonlinearity/dispersion balance leading to the formation of soliton solutions of the quantum-corrected nonlinear Schrodinger (QNLS) equation.

Abstract:
Quantum Zakharov equations are obtained to describe the nonlinear interaction between quantum Langmuir waves and quantum ion-acoustic waves. These quantum Zakharov equations are applied to two model cases, namely the four-wave interaction and the decay instability. In the case of the four-wave instability, sufficiently large quantum effects tend to suppress the instability. For the decay instability, the quantum Zakharov equations lead to results similar to those of the classical decay instability except for quantum correction terms in the dispersion relations. Some considerations regarding the nonlinear aspects of the quantum Zakharov equations are also offered.

Abstract:
A numerical analysis of the self-interaction induced by a relativistic electron/positron beam in the presence of an intense external longitudinal magnetic field in plasmas is carried out. Within the context of the Plasma Wake Field theory in the overdense regime, the transverse beam-plasma dynamics is described by a quantumlike Zakharov system of equations in the long beam limit provided by the Thermal Wave Model. In the limiting case of beam spot size much larger than the plasma wavelength, the Zakharov system is reduced to a 2D Gross-Pitaevskii-type equation, where the trap potential well is due to the external magnetic field. Vortices, "beam halos" and nonlinear coherent states (2D solitons) are predicted.

Abstract:
This paper continues the study of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for well-prepared initial data. We apply this result to the Euler-Maxwell equations describing laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler-Maxwell equations in terms of WKB approximate solutions which leading terms are solutions of the Zakharov equations. Because of transparency properties of the Euler-Maxwell equations, this study is led in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of Joly, M\'etivier and Rauch, recent results of Lannes on norms of pseudodifferential operators, and a semiclassical, paradifferential calculus.

Abstract:
A generalized -expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics. 1. Introduction In recent years, with the development of symbolic computation packages like Maple and Mathematica, searching for solutions of nonlinear differential equations directly has become more and more attractive [1–7]. This is because of the availability of computers symbolic system, which allows us to perform some complicated and tedious algebraic calculation and help us find new exact solutions of nonlinear differential equations. In 2008, Wang et al. [8] introduced a new direct method called the -expansion method to look for travelling wave solutions of nonlinear evolution equations (NLEEs). The method is based on the homogeneous balance principle and linear ordinary differential equation (LODE) theory. It is assumed that the traveling wave solutions can be expressed by a polynomial in , and that satisfies a second-order LODE . The degree of the polynomial can be determined by the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NPDEs. The coefficients of the polynomial can be obtained by solving a set of algebraic equations. Many literatures have shown that the -expansion method is very effective, and many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized -expansion method [9], the modified -expansion method [10], the extended -expansion method [11], the improved -expansion method [12], and the -expansion method [13] have been proposed. As we know, when using the direct method, the choice of an appropriate auxiliary LODE is of great importance. In this paper, by introducing a new auxiliary LODE of different literature [8], we propose the generalized -expansion method, which can be used to obtain travelling wave solutions of NLEEs. In our contribution, we will seek exact solutions of the Zakharov equations [14]: which are one of the classical models on governing the dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves, where is the perturbed number density of the ion (in the low-frequency response), is the slow variation amplitude of the electric field intensity, is the thermal transportation velocity of the

Abstract:
In this paper we study an initial boundary value problem for Zakharov's equations, describing the space propagation of a laser beam entering in a plasma. We prove a strong instability result and prove that the mathematical problem is ill-posed in Sobolev spaces. We also show that it is well posed in spaces of analytic functions. Several consequences for the physical consistency of the model are discussed.

Abstract:
We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinski\v{\i}-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinski\v{\i}-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over $I\!\!\!\,R^+\times I\!\!\!\,R^+$. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model.

Abstract:
We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic blow-up wave solutions and solitary wave solutions.

Abstract:
In this paper, we obtain several types of exact traveling wave solutions of the generalized Zakharov equations and the complex coupled KdV equations by using improved Riccati equations method. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method can also be applied to solve more nonlinear partial differential equations. Key Words: Improved Riccati equations method; Generalized Zakharov equations; Complex coupled KdV equations; Solitary wave solutions; Periodic wave solutions