Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.     

Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

For the critical focusing wave equation ${\square u = u^5 \, {\rm on} \, \mathbb{R}^{3+1}}$ in the radial case, we establish the role of the “center stable” manifold ${\Sigma}$ constructed in Krieger and Schlag (Am J Math 129(3):843–913, 2007) near the ground state (W, 0) as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń et al. (Nonlinearity 17(6):2187–2201, 2004). The underlying topology is stronger than the energy norm.     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ ₀ liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity.     

Stability and Asymptotic Stability for Subcritical gKdV Equations

 We prove in this paper the stability and asymptotic stability in H ¹ of a decoupled sum of N solitons for the subcritical generalized KdV equations The proof of the stability result is based on energy arguments and monotonicity of the local L ² norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16]. Received: 8 October 2001 / Accepted: 2 July 2002 Published online: 14 October 2002     

Critical Density Points for Threshold Voter Models on Homogeneous Trees

In this paper we study the threshold voter models on homogeneous trees. We give the definition of the ‘critical density’ of the models and obtain an estimation of it. Furthermore, we study the exponential rate at which the process converges to an absorbed state δ ₀ in the subcritical case. At the end of the paper, we propose two conjectures about the phase transition phenomenons of our models.     

Two Unconditional Stable Schemes for Simulation of Heat Equation on Manifold Using DEC

To predict the heat diffusion in a given region over time, it is often necessary to find the numerical solution for heat equation. However, the computational domain of classical numerical methods are limited to flat spacetime. With the techniques of discrete differential calculus, we propose two unconditional stable numerical schemes for simulation heat equation on space manifoldand time. The analysis of their stability and error is accomplished by the use of maximum principle.     

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation

We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in hyperbolic coordinates to the study of an ODE. Similar arguments extend to higher dimensions and other long range type nonlinear problems. Mathematics Subject Classifications (2000): 35L15, 74J30, 76B15 ★ Part of this work was done while H.L. was a Member of the Institute for Advanced Study, Princeton, supported by the NSF grant DMS-0111298 to the Institute. H.L. was also partially supported by the NSF Grant DMS-0200226. † Also a member of the Institute of Advanced Study, Princeton. Supported in part by NSF grant DMS-0100490.     

Critical Behavior of the Generalized Srnoluchovski's Coagulation Equation

The critical behavior of the gelation of polymers is studied by means of the generalized Srnoluchovski's coagulation equation. The exact solution of the kinetic equation with a factorial coagulation rate R(i₁, i₂, . . . i_n) = s_i1 × s_i2 × . . . × s_in and s_k = A × k + B is derived for the monodisperse initial condition c_m(O) = δ_{m, 1}. It is shown that a gelation transition takes place within a finite time t_c and the gelation time t_c ii characterized by the parameters A and B.     

Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space

The critical and super-critical dissipative quasi-geostrophic equations are investigated in . We prove local existence of a unique regular solution for arbitrary initial data in H ^2-2α which corresponds to the scaling invariant space of the equation. We also consider the behavior of the solution near t = 0 in the Sobolev space.     

Quantum Invariants for Solving the Time-Dependent Schrödinger Equation in One Dimension

Extending the work of Lewis and Leach on classical invariants for solving the classical equation of motion in one-dimensional system, the quantum invariants in polynomial form of momentum are obtained. The involved Hamiltonian is time-dependent and quadratic in momentum.     

Global Regularity for the Maxwell-Klein-Gordon Equation with Small Critical Sobolev Norm in High Dimensions

We show that in dimensions n 6 one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation [17, 7, 14, 12] but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.Acknowledgements The authors are deeply indebted to Sergiu Klainerman, without whose encouragement and insight this project would not even have been initiated. We have tremendously benefited from numerous discussions with him, in particular on the issue of the Cronstrom gauge. We would also like to thank Joachim Krieger and the anonymous referee for their helpful comments and suggestions.I.R. is a Clay Prize Fellow and supported in part by the NSF grant DMS-01007791T.T. is a Clay Prize Fellow and supported in part by a grant from the Packard Foundation     

TYZ Expansion for the Kepler Manifold

The main goal of the paper is to address the issue of the existence of Kempf’s distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non-compact manifold. Motivated by the recent results for compact manifolds we construct Kempf’s distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of n − 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter ( standing for Planck’s constant and , ). Moreover, the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We show that our estimates are sharp by analyzing the nonharmonic behaviour of T _m for . The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces. The first author was supported in part by the project PRIN (Cofin) n. 2006019457 with M.I.U.R., Italy. The second author was supported in part by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

Instability Threshold for a Calutron (Isotope Separator) with Only One Isotope Species

A calculation of the instability threshold for an isotope separator with only one isotope species yields a threshold comparable to that for several species. The beam interacts with a neutralizing cloud of electrons. As a side issue, a scaling law is derived that predicts that the beam particle density scales as the square of the magnetic field.     

The Critical Attractive Random Polymer in Dimension One

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e ^– for every self-intersection and e ^/(2d) for every self-contact to the probability of an n-step simple random walk on ^d , where , >0 are parameters. It is known that for d=1 and > the chain collapses down to finitely many sites, while for d=1 and < it spreads out ballistically. Here we study for d=1 the critical case = corresponding to the collapse transition and show that the end-to-end distance runs on the scale _n = (log n)^–1/4. We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size _n times a constant. Moreover, we derive the asymptotics of the partition function.