Scattering of solutions to the fourthorder nonlinear. It is also a good model dispersive equation, since it is often technically simpler than other. Semilinear schrodinger equations courant lecture notes in mathematics. We discuss when a minimizing sequence with respect to e.
Particularly useful tools in studying the nonlinear schrodinger equation. Equations schrodinger equation with a cubic nonlinearity 1. Thierry cazenave universite pierre et marie curie 10 semilinear schrodinger equations courant institute of mathematical sciences new york university new york, new york american mathematical society providence, rhode island. An introduction to semilinear evolution equations, oxford lecture series in mathematics and its applications, oxford university press, oxford, 1998. The standard treatment of the subject is presented in the books of cazenave 4.
The nonlinear schrodinger equation with a quadratic potential. Oxford lecture series in mathematics and its applications. We will next prove a result of cazenave and weissler 5, which states that scattering holds in a. Deng y 1989 the existence and nodal character of the solutions in rn for semilinear elliptic equation involving critical sobolev exponent acta math. Nonlinear schrodinger equations, morawetz estimates, scattering. We will indicate possible extensions and references. Sharp thresholds of global existence and blowup for a. Communications in partial differential equations, vol. Request pdf a relaxation scheme for the nonlinear schrodinger equation in this paper, we present a new numerical scheme for the. For the rest of the paper, we focus on the study of finitetime radial blowup solutions, and prove a result on the concentration of the l 3 norm at. In the fifth chapter we return to the semilinear equations nls and nlw.
Consider the semilinear schrodinger equation nls in arbitrary dimensions. Request pdf on jan 1, 2003, thierry cazenave and others published semilinear schrodinger equation find, read and cite all the research you need on researchgate. Semilinear schrodinger equations courant lecture notes in. Asymptotic properties of standing waves for maxwell. Strang splitting for a semilinear schrodinger equation. Weissler example, in dimension n 1, if u0x xin a neighborhood of 0, then the resulting solution wt,x of 1. Semilinear schrodinger equations pdf free download epdf. Cazenave, t semilinear schrodinger equations, courant lecture. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or kortewegde vries equation.
This book presents in a selfcontained form the typical basic properties of solutions to semilinear evolutionary partial. Inhomogeneous equations and abstract semilinear problems 50 4. Local and global analysis cbms regional conference series in mathematics, amer. An introduction to nonlinear schrodinger equations thierry cazenave. This volume represents a broad survey of current research in the fields of nonlinear analysis and nonlinear differential equations. New york uni versity, courant institute of mathematical sciences, ams. In this paper we study the cauchy problem for the semilinear heat and schr odinger equations, with the nonlinear term fu juju. A note on berestyckicazenaves classical instability.
Cazenave, semilinear schrodinger equations, courant lecture. Journal of mathematical analysis and applications, vol. Semilinear schrodinger equations thierry cazenave download. Moreover, we investigate the dynamical properties of blowup solutions, including \l2\concentration, blowup. Semilinear schrodinger equations, courant lecture notes in mathematics, 10. New york university, courant institute of mathematical sciences, new york. For dispersive equations, this type of space was rst employed by cazenave weissler math z. Inhomogeneous strichartz estimates for the schrodinger equation. All of the results were obtained under a frame of the variational method or a frame of potential well theory.
Schrodinger schrodinger equation with a cubic nonlinearity. The nonlinear schrodinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. Cazenave 2003 for local wellposedness in such regularity classes and for related results. Semilinear schrodinger equations thierry cazenave publication year. The nonlinear schrodinger equation has received a great deal of attention from mathematicians, in particular because of its applications to nonlinear optics.
Semilinear schrodinger equations with and without potentials, and with various nonlinearities arise as models for diverse physical phenomena, including bose einstein condensates and as a description of the envelope dynamics of a general. Thierry cazenave and alain haraux translated by yvan martel. Critical exponent for the semilinear wave equation with timedependent damping. Remarks on the semilinear schrodinger equation request pdf. Fixed point methods for the study of semilinear evolution. Unconditional uniqueness results for the nonlinear. This equation occurs in various chapters of physics, including nonlinear optics, superconductivity. Cazenave, semilinear schrodinger equations courant lecture notes 10, amer. Contributions to nonlinear analysis a tribute to d. Carles, semiclassical analysis for nonlinear schr odinger equations, world scienti c 2008 c sulem and pl.
Global existence of small analytic solutions to schrodinger equations with quadratic nonlinearity. Semilinear schrodinger equations courant lecture notes by thierry cazenave and a great selection of related books, art and collectibles available now at. From the mathematical point of view, schrodingers equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. Find all the books, read about the author, and more. Nonlinear dispersive equations ucla department of mathematics. Page personnelle thierry cazenave sorbonneuniversite. An introduction to semilinear elliptic equations thierry cazenave laboratoire jacqueslouis lions umr cnrs 7598 b. A relaxation scheme for the nonlinear schrodinger equation.
For the sake of simplicity, we will mostly consider two model equations. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the kortewegde vries equation. As their applications, we combine the classical morawetz estimate and the. Liouville theorems and classification results for a. Ponce, introduction to nonlinear dispersive equations, springer 3 terence tao, nonlinear dispersive equations. Scattering theory for the defocusing fourth order nls with.