Normal numbers and selection rules

Given anormal number x=0,x ₁ x ₂ ··· to base 2 and aselection rule S ?{0, 1}*=∪ _n=0 ^/t8 {0, 1} ⁿ , we define a subsequencex,=0, \(\chi _{t_1 } \chi _{t_2 } \) ·· where {t ₁<t ₂<···}={i; x ₁ x ₂···x _i?1 εS}.x _s is called aproper subsequence ofx if lim_i/∞ ti/S is said topreserve normality if for any normal numberx such thatx _s is a proper subsequence ofx, x _s is also a normal number. We prove that ifS/～ _s is a finite set, where ～ _s is an equivalence relation on {0, 1}* such that ξ ～ _s η if and only if {ζ; ξζ εS}={ζ; ηζ εS}, thenS preserves normality. This is a generalization of the known result in finite automata case, where {0, 1}*/～ _s is a finite set (Agafonov [1]).     

Harmonic generation in an electromagnetic wave

A plane electromagnetic wave is propagated in a non-linear non-dissipative isotropic dielectric. A method is given for the calculation of the change in the amplitudes and phases of harmonics of all orders, as the wave progresses, for any specified dependence of the dielectric constant on the magnitude of the electric field and for any initial wave form. The calculations are valid provided the distance of travel is sufficiently small that shocks are not formed. The formulae for the amplitudes and phases of the various harmonics take a particularly simple form when the initial wave-form is sinusoidal and the non-linearity of the dielectric is not too great.The results are compared with those obtained by the usual methods of calculation which assume that only harmonics of very low order are generated. The discrepancy in the amplitudes of these low order harmonics, calculated by the two methods is, for most practical purposes, insignificant.

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Résumé Nous discutons la propagation d'une onde électromagnétique plane dans un milieu diélectrique, non-linéaire, non-dissipatif et isotrope. Nous donnons une méthode pour le calcul du changement des amplitudes et des phases des harmoniques d'ordres variés. Les calculs sont valides sous la condition que la distance parcourue par l'onde n'est pas assez grande pour le développement d'un choc. Nous faisons la comparaison entre nos résultats et ceux qui sont obtenus par les méthodes usuelles dans lesquelles on fait l'hypothèse que seulement les harmoniques de petits ordres sont produites. Les différences entre les amplitudes des harmoniques de petits ordres, calculées par les deux méthodes, ne sont pas importantes du point de vue pratique. D'autre part, notre méthode est valable pour le calcul des amplitudes et des phases des harmoniques de tous les ordres pour une onde d'une forme quelconque.

     

Functional integration and variational equations of nonlinear optics

A variational principle for the derivation of nonlinear optics equations is proposed on the basis of a functional representation of the matrix element of the evolution operator for the optical system. The equations of a single-mode field in a medium consisting of resonant and nonresonant two-level atoms are derived as an example. The competition between self-induced transparency, nonlinear scattering, and self-focusing is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 170–175, 1990.     

The modulation equations of nonlinear geometric optics

The purpose of this paper is to present an introduction to the central aspects of the modulation equations of nonlinear geometric optics. That means regularity properties and existence in the large.     

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.     

Geometrical optics and post shock behavior for nonlinear conservation laws

In this paper we present an approach to the study of nonlinear waves and shocks associated with signaling problems for hyperbolic systems of conservation laws. Our approach employs a nonlinear phase variable, treats problems for which the flux function (and hence the matrix coefficients in the quasilinear system of partial differential equations) has explicit spatial dependence, and provides the post shock representation for the solution.     

On generation of nonlinear operator semigroups and nonlinear evolution operators

Generation of nonlinear operator semigroups and nonlinear evolution operators are proved by a different method, based on the theory of difference equations.     

Validity of nonlinear geometric optics with times growing logarithmically

The profiles (a.k.a. amplitudes) which enter in the approximate solutions of nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperbolic systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The approximations of nonlinear geometric optics typically have percentage error which tends to zero uniformly on bounded time intervals as the wavelength tends to zero. Under suitable hypotheses when the amplitude is uniformly bounded in space and time we show that the percentage error tends to zero uniformly on time intervals which grow logarithmically. The proof relies in an essential way on the fact that one has a corrector to the leading term of geometric optics.

     

Comparison of selection rules for ordinal optimization

The evaluation of performance of a design for complex discrete event systems through simulation is usually very time consuming. Optimizing the system performance becomes even more computationally infeasible. Ordinal optimization (OO) is a technique introduced to attack this difficulty in system design by looking at “order” in performances among designs instead of “value” and providing a probability guarantee for a good enough solution instead of the best for sure. The selection rule, known as the rule to decide which subset of designs to select as the OO solution, is a key step in applying the OO method. Pairwise elimination and round robin comparison are two selection rule examples. Many other selection rules are also frequently used in the ordinal optimization literature. To compare selection rules, we first identify some general facts about selection rules. Then we use regression functions to quantify the efficiency of a group of selection rules, including some frequently used rules. A procedure to predict good selection rules is proposed and verified by simulation and by examples. Selection rules that work well most of the time are recommended.     

Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear optics

By using coupled amplitude-phase formulation, we construct the quartic anharmonic oscillator equation from the coupled higher order nonlinear Schrödinger equation. For arbitrary physical parameter values, we discuss the construction of new cnoidal wave solutions and the exact solutions of both bright (GVD < 0) and dark (GVD < 0) solitary waves. In addition, we investigate the pairs of dark–dark and bright–bright solitary waves.     

A nonlinear hyperbolic problem arising from a question of nonlinear optics,Part II

Summary A modified analytical and numerical method is presented for a preliminary study of the problem of Graffi-Cesari in Nonlinear Optics [1, 12]. This problem concerns electromagnetic laser wave propagation in a nonlinear (quartz) crystal slab embedded between two linear media, and is a suitably simplified version of the experimental situation of Franken, Ward and co-workers (1961). A new version of an existence and uniqueness theorem is proved in a functional class which is chosen following Cesari's works on quasilinear hyperbolic systems in the Schauder canonic form, under a set of restrictions upon the admissible slab widtha of the crystal. The present method consists essentially in centering the relevant functional ball at the linear solution, and yields numerical values fora of the order of 2 mm, with a good agreement with values of quartz crystal widths used in experiments.

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Sommario Si presenta una trattazione analitica e numerica modificata, relativa al problema di Graffi-Cesari in Ottica Non Lineare, studiato in [1] e [12]. Si dimostra un teorema di esistenza, unicità e dipendenza continua (dall' onda laser incidente) per il campo elettromagnetico entro la lamina di cristallo, seguendo, come in [1], lo schema teorico elaborato da L. Cesari in [8–12]. Il presente metodo fornisce valori numerici per lo spessore ammissibile della lamina di cristallo (di quarzo) dell' ordine di 2 mm, in buon accordo con valori dello spessore delle lamine (di quarzo) usate negli esperimenti.

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Research supported by the GNFM of the CNR.     

Mathematical models of intraresonant frequency transformations in nonlinear optics and their numerical realization

A system of m₀ + 2m₁ quasilinear equations of Schrödinger type is studied in a cylindrical region; homogeneous boundary conditions are imposed on the lateral surface of the cylinder. We present mathematical models which describe the intraresonant frequency transformation (interaction of the waves) in nonlinear optics. An iteration method for their approximation is developed, and questions of its correctness are investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 61–67, 1990.     

A nonlinear hyperbolic problem arising from a question of nonlinear optics, Part II

A modified analytical and numerical method is presented for a preliminary study of the problem of Graffi-Cesari in Nonlinear Optics [1, 12]. This problem concerns electromagnetic laser wave propagation in a nonlinear (quartz) crystal slab embedded between two linear media, and is a suitably simplified version of the experimental situation of Franken, Ward and co-workers (1961). A new version of an existence and uniqueness theorem is proved in a functional class which is chosen following Cesari's works on quasilinear hyperbolic systems in the Schauder canonic form, under a set of restrictions upon the admissible slab widtha of the crystal. The present method consists essentially in centering the relevant functional ball at the linear solution, and yields numerical values fora of the order of 2 mm, with a good agreement with values of quartz crystal widths used in experiments.     

A nonlinear hyperbolic problem arising from a question of nonlinear optics,part I

Summary Analytical and numerical results are given for the problem of Graffi, recently proposed by D. Graffi in order to give a rigorous mathematical theory of the experiments carried out by Franken, Ward and coworkers (1961) in Nonlinear Optics, following the theoretical scheme developed by L. Cesari in some recent papers concerning boundary value problems for quasilinear hyperbolic systems in the Schauder canonic forms.Numerical results refer to the estimate by defect of the maximal slab widtha of the nonlinear medium (a quartz crystal slab), under which the relevant existence and uniqueness theorem holds, in a suitable functional class which is chosen following Cesari's works and is well-suited for applications to Mathematical Physics.

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Riassunto Si presentano risultati analitici e numerici relativi al problema di Graffi, recentemente proposto da D. Graffi per una teoria matematica degli esperimenti di Franken, Ward et al. (1961) in Ottica Non Lineare, seguendo lo schema teorico elaborato da L. Cesari in alcuni recentissimi lavori su certi problemi ai limiti per sistemi iperbolici quasi lineari nella forma canonica di Schauder.I risultati numerici si riferiscono alla stima per difetto del massimo spessore del mezzo non lineare (lamina di cristallo di quarzo) per cui restano validi teoremi di esistenza, unicità e dipendenza continua dall'onda (laser) incidente, in una opportuna classe funzionale, scelta seguendo i lavori citati di Cesari.

     

Free boundary problems and biological systems with selection rules

Archiv der Mathematik - We prove the local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida (Phys...     

Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide

The propagation of modulated light in a 2d nonlinear photonic waveguide is investigated in the framework of diffractive optics. It is shown that the dynamics obeys a nonlinear Schr?dinger equation at leading order. We compute the first and second corrector and show that the latter may describe some dispersive radiation through the structure. We prove the validity of the approximation in the interval of existence of the leading term.     

关于一个非线性几何光学中的非严格双曲守恒律系统的研究

研究了一个产生于非线性几何光学中的非严格双曲守恒律系统.该系统具有强非线性流函数项,且狄拉克激波可能同时出现在解的两个状态变量中.通过未知函数的一个变换,该系统的非线性流函数项得到弱化,从而其黎曼问题被完全解决.