Nonlinear Geometric Optics Research Articles

Resonant Leading Term Geometric Optics Expansions with Boundary Layers for Quasilinear Hyperbolic Boundary Problems

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2] by Coulombel et al. studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost periodic profiles as in [2].

Nonlinear geometric optics for contact discontinuities in three dimensional compressible isentropic steady flows

In this paper, the stability of supersonic contact discontinuities in the three-dimensional compressible isentropic steady Euler flows is investigated by using the nonlinear geometric optics. We construct the asymptotic expansions of highly oscillatory contact discontinuities when a planar contact discontinuity is perturbed by a small amplitude high frequency oscillatory incident wave, and deduce there exists a large amplification of amplitudes in the reflected and refracted oscillatory waves when the high frequency oscillatory wave strikes the contact discontinuity front at three critical angles. Moreover, we obtain that the leading profiles of highly oscillatory waves are described by an initial boundary value problem of Burgers-transport equations, and the leading profile of contact discontinuity front satisfies an initial value problem of a Hamilton-Jacobi equation, respectively. The amplification phenomenon shows that this supersonic contact discontinuity is only weakly stable in the sense of Wang and Yu [“Stability of contact discontinuities in three-dimensional compressible steady flows,” J. Differ. Equ. 255, 1278–1356 (2013)].

Gaussian beam evolution in nonlinear inhomogeneous plasma

AbstractThe method of nonlinear complex geometrical optics (NCGO) is proposed in this paper for description of the evolution of a spatially narrow Gaussian beam (GB) in an inhomogeneous nonlinear plasma. NCGO method deals with first-order ordinary differential equations for the complex curvature of the wave front and for GB amplitude and for second-order ordinary differential equation for GB width. Thus, NCGO simplifies the description of GB diffraction and self-focusing effects as compared to the known methods of plasma physics and this way it can be assumed to be attractive and comprehensive approach in problems of plasma heating by electromagnetic waves. Moreover, we demonstrate in this paper some regularity for nonlinear inhomogeneous plasma in the framework of which central ray of a GB is not subjected to nonlinear refraction within NCGO method boundary applicability. On the contrary, the beam width, wave front curvature, and GB amplitude are modified by diffraction and self-focusing processes. General properties of the beam propagation are illustrated with results of numerical modeling for two particular cases: GB diffraction and self-focusing along curvilinear trajectory with torsion in axially symmetric plasma column and GB reflection from nonlinear inhomogeneous plasma layer. We prove in this paper that NCGO is new effective method of plasma physics, which can be applied for improvement of ray tracing techniques and plasma diagnostics.

Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem

The Calderón problem is the mathematical formulation of the inverse problem in Electrical Impedance Tomography and asks for the uniqueness and reconstruction of an electrical conductivity distribution in a bounded domain from the knowledge of the Dirichlet-to-Neumann map associated to the generalized Laplace equation. The 3D problem was solved in theory in late 1980s using complex geometrical optics solutions and a scattering transform. Several approximations to the reconstruction method have been suggested and implemented numerically in the literature, but here, for the first time, a complete computer implementation of the full nonlinear algorithm is given. First a boundary integral equation is solved by a Nyström method for the traces of the complex geometrical optics solutions, second the scattering transform is computed and inverted using fast Fourier transform, and finally a boundary value problem is solved for the conductivity distribution. To test the performance of the algorithm highly accurate data is required, and to this end a boundary element method is developed and implemented for the forward problem. The numerical reconstruction algorithm is tested on simulated data and compared to the simpler approximations. In addition, convergence of the numerical solution towards the exact solution of the boundary integral equation is proved.

Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws

We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate this approach more clearly, we focus first on the Cauchy problem for the hyperbolic systems with compact support initial data of small bounded variation and establish that the L 1-estimate between the entropy solution and the geometric optics expansion function is bounded by O(ϵ2), independent of the time variable. This implies that the simpler geometric optics expansion functions can be employed to study the behavior of general entropy solutions to hyperbolic systems of conservation laws. Finally, we extend the results to the case with non-compact support initial data of bounded variation.

Nonlinear geometric optics for reflecting uniformly stable pulses

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting pulses is close to the uniquely determined exact solution for small wavelengths ε. Pulses reflecting off fixed noncharacteristic boundaries are considered under the assumption that the underlying boundary problem is uniformly spectrally stable in the sense of Kreiss. There are two respects in which these results make rigorous the formal treatment of pulses in Majda and Artola (1988) [16], and Hunter, Majda and Rosales (1986) [10]. First, we give a rigorous construction of leading pulse profiles in problems where pulses traveling with many distinct group velocities are, unavoidably, present; and second, we provide a rigorous error analysis which yields a rate of convergence of approximate to exact solutions as ε→0. Unlike wavetrains, interacting pulses do not produce resonances that affect leading order profiles. However, our error analysis shows the importance of estimating pulse interactions in the construction and estimation of correctors. Our results apply to a general class of systems that includes quasilinear problems like the compressible Euler equations; moreover, the same methods yield a stability result for uniformly stable Euler shocks perturbed by highly oscillatory pulses.

On the demodulation of short radio waves propagating through the lower ionosphere

The relationships for the amplitude of electron temperature oscillations, as well as for the phase and modulation amplitude of high-power short radio waves propagating through the lower ionosphere, are obtained on the basis of nonlinear geometrical optics.

Passage of short radio waves through the lower ionosphere

The principles of nonlinear geometric optics are used to formulate a method for estimating the amplitude of short radio waves during their passage through the lower ionosphere.

Singular pseudodifferential calculus for wavetrains and pulses

We develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderon-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on $L^2$, whose norm is measured with respect to some small parameter. Our main improvement with respect to an earlier work by Williams consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend previous results on the existence of highly oscillatory solutions to nonlinear hyperbolic problems. The results are also used in a companion work to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than any other one considered before. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in geometric optics.

Energy cascades for NLS on the torus

We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system.

Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so-called extended nonlinear coupled mode equations, or xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic, and numerical methods, the existence and properties of spatially localized structures of the xNLCME system for the case where the refractive index profile consists of a periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a spectral band edge. In this case, stationary xNLCME is well approximated by an infinite system of coupled, stationary, nonlinear Schrödinger (NLS) equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS$^\epsilon$, which interpolates between infinitely many decoupled NLS equations ($\epsilon=0$) and xNLS ($\epsilon=1$). Using bifurcation methods we show existence of solutions for a range of $\epsilon\in(-\epsilon_0,\epsilon_0)$ and, by a numerical continuation method, establish the continuation of certain branches all the way to $\epsilon=1$. Finally, we perform time-dependent simulations of a truncated xNLCME and find the small-amplitude near–band edge gap solitons to be robust to both numerical errors and the NLS approximation.

Diffractive Geometric Optics for Bloch Wave Packets

We study, for times of order $${1/\varepsilon}$$ , solutions of wave equations which are $${\fancyscript{O}(\varepsilon^2)}$$ modulations of an $${\varepsilon}$$ periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order $${\varepsilon}$$ . We construct accurate approximate solutions of the three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of the small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales $${t=\fancyscript{O}(1)}$$ . A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H 1 asymptotics when the envelopes are only H 1.

Pulse splitting in the anomalous group-velocity-dispersion regime

We investigate experimentally the role that the initial temporal profile of ultrashort laser pulses has on the self-focusing dynamics in the anomalous group-velocity dispersion (GVD) regime. We observe that pulse-splitting occurs for super-Gaussian pulses, but not for Gaussian pulses. The splitting does not occur for either pulse shape when the GVD is near-zero. These observations agree with predictions based on the nonlinear Schrödinger equation, and can be understood intuitively using the method of nonlinear geometrical optics.

Light propagation in media with a highly nonlinear response: An analytical study

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics [L.L. Tatarinova, M.E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78 (2008) 021806(R)(1—4)]. Numerous particular cases of media exhibiting high-order nonlinear refractive indices are considered. We obtain analytical expressions for determining the self-focusing position and a new exact expression for calculating the filament intensity. The constructed solutions allowed us to revise a so-called self-focusing scaling law, i.e., the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.

Resonant Leading Order Geometric Optics Expansions for Quasilinear Hyperbolic Fixed and Free Boundary Problems

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths (ε). Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type [21] was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption. Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular (because they involve coefficients of order ) fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form.

Rigorous weakly nonlinear geometric optics for surface waves

This paper is concerned with surface waves, solutions of hyperbolic nonlinear boundary value problems. We construct BKW solutions in the weakly nonlinear regime with infinite expansion in powers of e. We rigorously justify this expansion, constructing exact solutions, which admit the asymptotic expansions. We also show that the solution is not necessarily localized at the order O(e∞) in the interior, even if the data are.

A deterministic model for the propagation of turbulent oscillations

The deterministic point of view on turbulent fluid motion is to consider the Cauchy problem for equations of Navier–Stokes type associated with large Reynolds numbers and with singular initial data. Although the corresponding mathematical study has a lot progressed, it remains limited by fundamental difficulties related to the presence of instabilities. Precisely, the purpose of this article is to show on a realistic two-dimensional model that, up to some extent, such instabilities can be managed. This is achieved in the framework of a supercritical nonlinear geometric optics. The aim is to provide a theory allowing to take into account the interaction of a large amplitude monophase oscillating wave with waves oscillating at smaller frequencies in the other direction. The effect is that very complicated phenomena can occur in the inertial range, including for instance the production of new scales.

Predicting the filamentation of high-power beams and pulses without numerical integration: A nonlinear geometrical optics method

Predicting the filamentation of high-power beams and pulses without numerical integration: A nonlinear geometrical optics method

Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics

We demonstrate that laser beam collapse in highly nonlinear media can be described, for a large number of experimental conditions, by the geometrical optics approximation within high accuracy. Taking into account this fact we succeed in constructing analytical solutions of the eikonal equation, which are exact on the beam axis and provide (i) a first-principles determination of the self-focusing position, thus replacing the widely used empirical Marburger formula, (ii) a mathematical condition for obtaining the filament intensity, (iii) a benchmark solution for numerical simulations, and (iv) a tool for the experimental determination of the high-order nonlinear susceptibility. Successful comparison with experiment is presented.

Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type

A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q) T , where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σα (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.