Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction

Standing wave solutions of coupled nonlinear Hartree equations with nonlocal interaction are considered. Such systems arises from mathematical models in Bose–Einstein condensates theory and nonlinear optics. The existence and non-existence of positive ground state solutions are proved under optimal conditions on parameters, and various qualitative properties of ground state solutions are shown. The uniqueness of the positive solution or the positive ground state solution are also obtained in some cases.     

Solitons in coupled nonlinear Schrödinger equations with variable coefficients

We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright–Dark and Bright–Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.     

Applications of some transformations for several variable-coefficient nonlinear evolution equations from plasma physics,arterial mechanics,nonlinear optics and Bose–Einstein condensates

Variable-coefficient nonlinear evolution equations have occurred in such fields as plasma physics, arterial mechanics, nonlinear optics and Bose–Einstein condensates. This paper is devoted to giving some transformations to convert the original nonlinear evolution equations, e.g., the variable-coefficient nonlinear Schrödinger, generalized Gardner and variable-coefficient Sawada–Kotera equations to simpler ones or even constant-coefficient ones. Based on some constraints, we simplify the original equations and derive the associated chirp solitons, Lax pairs, and Bäcklund transformations from the original equations by means of the aforementioned transformations.     

Renormalization group symmetries in problems of nonlinear geometrical optics

We construct renormalization group symmetries in the geometrical optics approximation for the boundary value problem of the system of equations describing the propagation of strong radiation in a nonlinear medium. Using the renormalization group symmetries, new exact and approximate analytic solutions to the equations of nonlinear geometrical optics are obtained. Explicit analytic expressions are presented that characterize the spatial evolution of a laser beam having an arbitrary dependence on intensity at the nonlinear medium boundary. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 369–388, June, 1997.     

On plane wave solutions of non-symmetric field equations of unified theories of Einstein,Bonnor and Schrödinger

Summary Various attempts have been made to obtain wave solutions of nonsymmetric unified field equations of Einstein, Bonnor and Schrödinver in different space-times. Lal and Ali, Lal and Khare, Lal and Singh have obtained wave solutions of these field equations in different space-times under certain restrictions. In this paper plane wave solutions of both the weak and strong, nonsymmetric unified field equations of Einstein, Bonnor and Schrödinger have been investigated in a space which is more general than Pandey and Rao plane wave metric. It has been shown that the plane wave solutions of Einstein and Bonnor's equations exist but Schrödinger's field equations have no solution in this space-time.     

Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields

The isovector fields (infinitesimal generators of Lie groups) of Einstein vacuum equations for stationary axially symmetric rotating fields, in conventional form, that is a coupled system of nonlinear partial differential equations (PDEs) of second order are derived using the geometric prolongation technique. Some symmetry transformations and similarity (exact) solutions of Einstein vacuum equations are obtained.     

On some particular solutions of the Einstein gravitational equations

Particular wave and non-wave solutions of Einstein's gravitational equations outside the mass region were obtained. The criterion for wave solutions of the Einstein equations was established.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 341–352, November, 1995.     

AN ASYMPTOTIC SOLUTION OF THE NONLINEAR REDUCED WAVE EQUATION

1 IntroductionIn the theory of nonlinear optics[1], one is led to consider a nonlinear reduced wave equationwhere k is the wave number and n = n'(IuI') is a function of intensity Of the field and is calledthe index of refraction.We consider the case when the optical beam propagates in a quadratic index nledia sothat n'(luI') can be written aswhere no and n1 are constants.With IuI = O(k--'), d > 0, and following the discussion of this equation in two dimensionalspace (see [2]), we writeSubst…     

Multidimensional nonlinear wave equations with multivalued solutions

We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.     

Remarques sur l'équation de Schrödinger non linéaire avec potentiel harmoniqueRemarks on the nonlinear Schrödinger equation with harmonic potential

Bose–Einstein condensation is usually modeled by nonlinear Schrödinger equations with harmonic potential. We study the Cauchy problem for these equations, in particular the wave collapse phenomenon. For this, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation. We state wave collapse criteria, allowing a range of positive values for the energy. To cite this article: R. Carles, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 763–766.     

Approximate conservation laws of nonlinear perturbed heat and wave equations

We construct approximate conservation laws for non-variational nonlinear perturbed (1+1) heat and wave equations by utilizing the partial Lagrangian approach. These perturbed nonlinear heat and wave equations arise in a number of important applications which are reviewed. Approximate symmetries of these have been obtained in the literature. Approximate partial Noether operators associated with a partial Lagrangian of the underlying perturbed heat and wave equations are derived herein. These approximate partial Noether operators are then used via the approximate version of the partial Noether theorem in the construction of approximate conservation laws of the underlying perturbed heat and wave equations.     

薛定谔方程的整体几何光学近似

整体几何光学方法是一种新的求解高频线性波动方程初值问题的渐进近似理论.该理论最初是对WKB初值数据问题提出来的.在本文中,我们将采用不同的方法,对这一方法予以重新推导,使得该理论同样适用于初值为扩展WKB函数的情形.特别地,我们将建立的理论用于薛定谔方程传播子的半经典近似上来.结果表明,整体几何光学方法提供的波场近似恰好是Kay提出的半相空间公式的一个实例.作为副产品,我们指出Van Vleck近似中起到关键作用的Maslov指标可以通过一个简单的代数关系式来确定.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.     

Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

Seven common errors in finding exact solutions of nonlinear differential equations

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.     

Diffusion d'impulsions non linéaires radiales focalisantes dans R1+3

We study the validity of geometric optics for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is critical, linear geometric optics is valid before the focal point, as well as after. The matching between these two régimes is described by a scattering operator, which enlarges the support of the waves.     

Absorption d'impulsions non linéaires radiales focalisantes dans R1+3

We study the validity of geometric optics for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur. When the equation is dissipative, pulses are absorbed. When the equation is accretive, the family of pulses becomes unbounded.