In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L 2 -unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.

Abstract. Fröhlich's polaron Hamiltonian describes an electron coupled to the quantized phonon field of an ionic crystal. We show that in the strong coupling limit the dynamics of the polaron is approximated by an effective non-linear partial differential equation due to Landau and Pekar, in which the phonon field is treated as a classical field.

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In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.

In this paper we study the blowup problem of nonlinear heat equations. Our result show that for a certain family of initial conditions the solution will blowup in finite time, the blowup parameters satisfy some dynamics which are asymptotic stable, moreover we provide the remainder estimates. Compare to the previous works our approach is analogous to one used in bifurcation theory and our techniques can be regarded as a time-dependent version of the Lyapunov-Schmidt decomposition. * Supported by NSERC under Grant NA7901.This result was further improved in several papers (see e.g. [22,21,25,14,31,45,15,16,17,5,34,35,36]). A blowup solution satisfying the bound ( 5) is said to be of type I. This bound was proven under various by an interpolation result. Note that the first four eigenvectors of L 0 are e − αx 2 4 , xe − αx 2 4 , (αx 2 − 1)e − αx 2 4 and (αx 3 − 3x)e − αx 2 4 with the eigenvalues −2α, −α, 0 and α. Thus for the case n = 2, using that the integral kernel of e −rL0 is positive and therefore e −rL0 g ∞ ≤ f −1 g ∞ e −rL0 f ∞ for any f > 0 and using that e −rL0 e − α 4 z 2 = e 2αr e − α 4 z 2 and e −rL0 (αz 2 − 1)e − α 4 z 2 = (αz 2 − 1)e − α 4 z 2 , we find thatThis implies (85). To prove (86) we note that U 0 (x, y) is, by definition, the integral kernel of the operator e − α 4 z 2 e −rL0 e α 4 z 2 . Thus, taking g(x) = x n e − α 4 x 2 in (85) yields (86).A version of the following lemma is proved in [5].Lemma 12. For any function g and positive constants σ and r we have z −3 e αz 2 4 U α (σ + r, σ)P α g ∞ [e 2αr r(1 + r)β 1/2 (σ) + e −αr ] z −3 e αz 2 4 g ∞ .