Semilinear Schroedinger equations by Thierry Cazenave

By Thierry Cazenave

The nonlinear Schrödinger equation has got loads of recognition from mathematicians, relatively as a result of its functions to nonlinear optics. it's also an outstanding version dispersive equation, because it is frequently technically less complicated than different dispersive equations, resembling the wave or the Korteweg-de Vries equation. From the mathematical perspective, Schrödinger's equation is a fragile challenge, owning a mix of the homes of parabolic and elliptic equations. necessary instruments in learning the nonlinear Schrödinger equation are power and Strichartz's estimates.

This publication provides quite a few mathematical points of the nonlinear Schrödinger equation. It reports either difficulties of neighborhood nature (local life of suggestions, forte, regularity, smoothing influence) and difficulties of worldwide nature (finite-time blowup, worldwide life, asymptotic habit of solutions). In precept, the tools provided practice to a wide type of dispersive semilinear equations. the 1st bankruptcy remembers uncomplicated notions of practical research (Fourier remodel, Sobolev areas, etc.). in a different way, the e-book is usually self-contained.

By Thierry Cazenave

The nonlinear Schrödinger equation has got loads of recognition from mathematicians, relatively as a result of its functions to nonlinear optics. it's also an outstanding version dispersive equation, because it is frequently technically less complicated than different dispersive equations, resembling the wave or the Korteweg-de Vries equation. From the mathematical perspective, Schrödinger's equation is a fragile challenge, owning a mix of the homes of parabolic and elliptic equations. necessary instruments in learning the nonlinear Schrödinger equation are power and Strichartz's estimates.

This publication provides quite a few mathematical points of the nonlinear Schrödinger equation. It reports either difficulties of neighborhood nature (local life of suggestions, forte, regularity, smoothing influence) and difficulties of worldwide nature (finite-time blowup, worldwide life, asymptotic habit of solutions). In precept, the tools provided practice to a wide type of dispersive semilinear equations. the 1st bankruptcy remembers uncomplicated notions of practical research (Fourier remodel, Sobolev areas, etc.). in a different way, the e-book is usually self-contained.

Similar mathematics books

Multiparameter Eigenvalue Problems and Expansion Theorems

This e-book offers a self-contained remedy of 2 of the most difficulties of multiparameter spectral thought: the lifestyles of eigenvalues and the growth in sequence of eigenfunctions. the consequences are first bought in summary Hilbert areas after which utilized to vital operators and differential operators.

Séminaire Bourbaki, Vol. 1, 1948-1951, Exp. 1-49

Desk of Contents

* 1 Henri Cartan Les travaux de Koszul, I (Lie algebra cohomology)
* 2 Claude Chabauty Le théorème de Minkowski-Hlawka (Minkowski-Hlawka theorem)
* three Claude Chevalley L'hypothèse de Riemann pour les corps de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function)
* four Roger Godement Groupe complexe unimodulaire, I : Les représentations unitaires irréductibles du groupe complexe unimodulaire, d'après Gelfand et Neumark (representation thought of the advanced unique linear group)
* five Léo Kaloujnine Sur los angeles constitution de p-groupes de Sylow des groupes symétriques finis et de quelques généralisations infinies de ces groupes (Sylow theorems, symmetric teams, endless crew theory)
* 6. Pierre Samuel los angeles théorie des correspondances birationnelles selon Zariski (birational geometry)
* 7 Jean Braconnier Sur les suites de composition d'un groupe et los angeles travel des groupes d'automorphismes d'un groupe fini, d'après H. Wielandt (finite groups)
* eight Henri Cartan, Les travaux de Koszul, II (see 1)
* nine Claude Chevalley, L'hypothèse de Riemann pour les groupes de fonctions algébriques de caractéristique p, II,, d'après Weil (see 3)
* 10 Luc Gauthier, Théorie des correspondances birationnelles selon Zariski (see 6)
* eleven Laurent Schwartz, Sur un mémoire de Petrowsky : "Über das Cauchysche challenge für ein process linearer partieller Differentialgleichungen im gebiete nichtanalytischen Funktionen" (partial differential equations)
* 12 Henri Cartan, Les travaux de Koszul, III (see 1)
* thirteen Roger Godement, Groupe complexe unimodulaire, II : los angeles transformation de Fourier dans le groupe complexe unimodulaire à deux variables, d'après Gelfand et Neumark (see 4)
* 14 Marc Krasner, Les travaux récents de R. Brauer en théorie des groupes (finite groups)
* 15 Laurent Schwartz, Sur un deuxième mémoire de Petrowsky : "Über das Cauchysche challenge für procedure von partiellen Differentialgleichungen" (see 11)
* sixteen André Weil Théorèmes fondamentaux de l. a. théorie des fonctions thêta, d'après des mémoires de Poincaré et Frobenius (theta functions)
* 17 André Blanchard, Groupes algébriques et équations différentielles linéaires, d'après E. Kolchin (differential Galois theory)
* 18 Jean Dieudonné, Géométrie des espaces algébriques homogènes, d'après W. L. Chow (algebraic geometry)
* 19 Roger Godement, Sommes keeps d'espaces de Hilbert, I (functional research, direct integrals)
* 20 Charles Pisot, Démonstration élémentaire du théorème des nombres premiers, d'après Selberg et Erdös (prime quantity theorem)
* 21 Georges Reeb, Propriétés des trajectoires de certains systèmes dynamiques (dynamical systems)
* 22 Pierre Samuel, Anneaux locaux ; advent à los angeles géométrie algébrique (local rings)
* 23 Marie-Hélène Schwartz, Compte-rendu de travaux de M. Heins sur diverses majorations de los angeles croissance des fonctions analytiques et sous-harmoniques (complex research, subharmonic functions)
* 24 Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable (connections on fiber bundles)
* 25 Roger Godement, Sommes keeps d'espaces de Hilbert, II (see 19)
* 26 Laurent Schwartz, Sur un mémoire de okay. Kodaira : "Harmonic fields in riemannian manifolds (generalized power theory)", I (Hodge theory)
* 27 Jean-Pierre Serre, Extensions de groupes localement compacts, d'après Iwasawa et Gleason (locally compact groups)
* 28 René Thom, Les géodésiques dans les variétés à courbure négative, d'après Hopf (geodesics)
* 29 Armand Borel, Groupes localement compacts, d'après Iwasawa et Gleason (see 27)
* 30 Jacques Dixmier, Facteurs : category, measurement, hint (von Neumann algebras)
* 31 Jean-Louis Koszul, Algèbres de Jordan (Jordan algebras)
* 32 Laurent Schwartz, Sur un mémoire de ok. Kodaira : "Harmonic fields in riemannian manifolds (generalized capability theory)", II (see 26)
* 33 Armand Borel, Sous-groupes compacts maximaux des groupes de Lie, d'après Cartan, Iwasawa et Mostow (maximal compact subgroups)
* 34 Henri Cartan, Espaces fibrés analytiques complexes (analytic geometry, fiber bundles)
* 35 Charles Ehresmann, Sur les variétés presque complexes (almost-complex manifolds)
* 36 Samuel Eilenberg, Exposition des théories de Morse et Lusternick-Schnirelmann (Morse conception, Lyusternik-Schnirelmann category)
* 37 Luc Gauthier, Quelques variétés usuelles en géométrie algébrique (algebraic geometry)
* 38 Jean-Louis Koszul, Cohomologie des espaces fibrés différentiables et connexions (Chern-Weil theory)
* 39 Jean Delsarte, Nombre de options des équations polynomiales sur un corps fini, d'après A. Weil (Weil conjectures)
* forty Jacques Dixmier, Anneaux d'opérateurs et représentations des groupes (operator algebras, illustration theory)
* forty-one Roger Godement, Théorie des caractères dans les groupes unimodulaires (unimodular groups)
* forty two Pierre Samuel, Théorie du corps de sessions neighborhood selon G. P. Hochschild (local category box theory)
* forty three Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory)
* forty four Jean-Pierre Serre, Groupes d'homotopie (homotopy groups)
* forty five Armand Borel, Cohomologie des espaces homogènes (cohomology of homogeneous areas of Lie groups)
* forty six Samuel Eilenberg, Foncteurs de modules et leurs satellites, d'après Cartan et Eilenberg (homological algebra)
* forty seven Marc Krasner, Généralisations non-abéliennes de los angeles théorie locale des corps de sessions (local fields)
* forty eight Jean Leray, los angeles résolution des problèmes de Cauchy et de Dirichlet au moyen du calcul symbolique et des projections orthogonales et obliques (Dirichlet difficulties and Cauchy difficulties for partial differential equations, symbolic calculus)
* forty nine Pierre Samuel, Sections hyperplanes des variétés normales, d'après A. Seidenberg (algebraic geometry, hyperplane sections, general kind)

Additional resources for Semilinear Schroedinger equations

Sample text

Now, using any integers you like, find the longest possible sequence which does not contain any increasing or decreasing sequences of five numbers. 96 Fencing! Joe Appleyard wanted to build a fence to protect his orchard. The fence was to be built 90 m down one side of a valley and 78 m up the other side. The slope of the valley sides are shown, together with the heights of the valley sides above the valley's 54 m bottom. 5 m high. How many panels will be needed? I 30 m 97 Triangular Nim Place 15 coins (or counters) to form a triangular array as shown.

A player loses when, having recorded a number, the other player identifies two subsets of the numbers so far recorded which have the same total. Suppose, for example, the players have recorded 19 2 27 39 5 11 after six turns, with A playing first, and B having just recorded 11. If A is wide awake, he will spot that 19 + 27 = 2 + 39 + 5 and be the winner. Interestingly, this game should never go beyond ten moves for it can be shown that, given any ten numbers less than 100, there will always be two subsets of them which have the same total.

Chief Mustafa's pride and joy were his eleven fine white oxen. After his death, his principal wife made it known that her late husband wanted the oxen shared between his three eldest sons, Yusuf, Raheem and Ibrahim, so that they have \, \ and \ respectively. Not wanting to end up with having to dissect any of the beautiful beasts, they consulted the village oracle. She soon put them out of their misery by adding her one and only ox to the eleven and then giving six to Yusuf, three to Raheem, two to Ibrahim, and finally taking her own back!