Is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data? For the linear wave equation, for instance, the answer is no. However, in [10] T. Kato found a local smoothing property of the Korteweg-de Vries equation: the solution of the initial value problem is, locally, one derivative smoother than the initial datum. Kato's proof uses, in a curcial way, the algebraic properties of the symbol for the Korteweg-de Vries equation and the fact that the underlying spatial dimension is one. Actually, judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect, one would be inclined to believe this was a special property of the K-dV equation. This is not, however, the case. In this paper, we attempt to describe a general local smoothing effect for dispersive equations and systems. The smoothing effect is due to the dispersive nature of the linear part of the equation. All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations are included. We study, thus, equations and systems of the form

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Local smoothing properties of dispersive equations

We consider a class of nonlinear Schr6dinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t ~ + oo) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result of asymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different for t ~ + ov from that+for t --. - or. Moreover the solution acquires an extra constant asymptotic phase e '~-.

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Multichannel nonlinear scattering for nonintegrable equations

We consider the scattering problem for the nonlinear Schrodinger equation in 1+1 dimensions:
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where ∂ = ∂/∂x,λ∈R∖{0},μ∈R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL2(R) or in the Sobolev spaceH1(R)., The modified wave operators are introduced in order to control the long range nonlinearity λ|u|2u.

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Long range scattering for nonlinear Schrödinger equations in one space dimension

It is shown that the initial value problem for the nonlinear Schrodinger equations ∂ t u = i Δ u + P ( u , ∇ x u , u ¯ , ∇ x u ¯ ) , t ∈ ℝ , x ∈ ℝ n , where Ρ (.) is a polynomial having no constant or linear terms, is locally well posed for a class of “small” data u 0 . The main ingredients in the proof are new estimates describing the smoothing effect of Kato type for the group { e i t Δ } − ∞ ∞ . This method extends to systems and other dispersive models.

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Small solutions to nonlinear Schrödinger equations

In the theory of water waves, the 2D generalisation of the usual cubic 1D Schrodinger equation turns out to be a family of systems: the Davey-Stewartson systems. For special values of the parameters characterising these systems, one obtains systems of the inverse scattering type. The authors' work addresses the very general case and their methods belong to the more standard theory of nonlinear partial differential equations. Well-posedness of the Cauchy problem and also finite-time blow-up are studied.

http://iopscience.iop.org/article/10.1088/0951-7715/3/2/010/pdf

On the initial value problem for the Davey-Stewartson systems

On solutions of the initial value problem for the nonlinear Schrödinger equations

t>0, xe~", has a regularizing property that the solutions become smooth instantly for t > 0 even if their initial data are not smooth, for example, in L ~ (N"). Our aim of this paper is to show that the NLS equation (1.1)-(1.2) has a similar regularizing effect. More precisely all solutions of finite energy to (1.1)- (1.2) are smooth for tq=0 provided the initial functions in Hi(N) (or in

On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension

The Cauchy problem for the coupled Maxwell–Schrodinger equations in Rd in the Lorentz gauge is considered. The viscosity method is used to establish local existence. In one and two space dimensions, global solutions are obtained.

The Cauchy problem for the coupled Maxwell–Schrödinger equations

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/98758/1/0541-2.pdf

Kuniaki Nakamitsu

Papers

On solutions of the initial value problem for the nonlinear Schrödinger equations

On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension

The Cauchy problem for the coupled Maxwell–Schrödinger equations

THE CAUCHY PROBLEM FOR THE COUPLED MAXWELL-SCHRODINGER EQUATIONS(Nonlinear Evolution Equations:Theory and Applications)