arXiv subjects

Justin Holmer

Publications and source records attributed to Justin Holmer.

At least 37 records · Page 2Linked to original sources

arXiv2015

On the Klainerman-Machedon Conjecture of the Quantum BBGKY Hierarchy with Self-interaction

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3β-1}V(N^β\bullet).$ For interaction parameter $β\in(0,\frac23)$, we prove that, as $N\rightarrow \infty ,$ the limit points of the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman-Machedon in 2008. This allows for the application of the Klainerman-Machedon uniqueness theorem, and hence implies that the limit is uniquely determined as a tensor product of solutions to the Gross-Pitaevski equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović (2011) for $β\in (0,\frac14)$ and subsequently by X. Chen (2012) for $β\in (0,\frac27]$. We build upon the approach of X. Chen but apply frequency localized Klainerman-Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the potential part to extend the range to $β\in (0,\frac23)$. Overall, this provides an alternative approach to the mean-field program by Erdös-Schlein-Yau (2007), whose uniqueness proof is based upon Feynman diagram combinatorics.

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arXiv2014

Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy

We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3β-1}V(N^{β}\cdot )$ . By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the $N$-body wave function, for $β\in \left( 0,1\right] $, we derive the Gross-Pitaevskii hierarchy with $2$-body interaction. In particular, we establish that, in the $N\rightarrow \infty $ limit, all $k$-body scattering processes vanishes if $k\geqslant 3$ and thus provide a direct answer to a question raised by Erdös, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for $β\in \left( 0,1\right) $ and weakly for $β=1$. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three body estimate for which we have the estimates to achieve $β<1$, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon in [47] for $β\in \left( 0,1\right) $.

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arXiv2014

Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by $a^{3β-1}V(a^β\cdot)$ where $\int V\leqslant 0$ and $a$ matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D $N$-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty $. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.

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arXiv2013

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation

We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N^{β-1}V(N^β\cdot) where \int V\leqslant 0. We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the one dimensional focusing cubic NLS with a quadratic trap as the N->\infty limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.

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arXiv2012

A solution to the focusing 3d NLS that blows up on a contracting sphere

We rigorously construct radial $H^1$ solutions to the 3d cubic focusing NLS equation $i\partial_t ψ+ Δψ+ 2 |ψ|^2ψ=0$ that blow-up along a contracting sphere. With blow-up time set to $t=0$, the solutions concentrate on a sphere at radius $\sim t^{1/3}$ but focus towards this sphere at the faster rate $\sim t^{2/3}$. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov (1975) and independently later in Holmer-Roudenko (2007), where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the $L_x^3$ norm within the "wide" radius $\sim |\nabla u(t)|_{L_x^2}^{-1/2}$ but not within the "tight" radius $\sim |\nabla u(t)|_{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail in Chapter 7 of Sulem-Sulem (1998).

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arXiv2012

On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics

We consider the 3D quantum many-body dynamics describing a dilute bose gas with strong confining in one direction. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty$. We find that this diverging coefficient is counterbalanced by the limiting structure of the density matrices and establish the convergence of the BBGKY hierarchy. Moreover, we prove that the limit is fully described by a 2D cubic NLS and obtain the exact 3D to 2D coupling constant.

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arXiv2012

Low regularity bounds for mKdV

We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H^s when s<1/4. In spite of this, we establish that for -1/8 < s < 1/4, the solution satisfies global in time H^s(R) bounds which depend only on the time and on the H^s(R) norm of the initial data. This result is weaker than global well-posedness, as we have no control on differences of solutions. Our proof is modeled on recent work by Christ-Colliander-Tao and Koch-Tataru employing a version of Bourgain's Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.

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arXiv2011

Phase-driven interaction of widely separated nonlinear Schrödinger solitons

We show that, for the 1d cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces.

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arXiv2010

Dynamics of KdV solitons in the presence of a slowly varying potential

We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $\partial_t u = -\partial_x (\partial_x^2 u + 3u^2-bu)$, where $b(x,t) = b_0(hx,ht)$, $h\ll 1$ is a slowly varying, but not small, potential. We option an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale $δh^{-1}\log h^{-1}$, together with an estimate on the error of size $h^{1/2}$. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Martel-Merle (2005). The results are supported by numerics. The proof does not rely on the inverse scattering machinery and is expected to carry through for the $L^2$ subcritical gKdV-$p$ equation, $1<p<5$. The case of $p=3$, the modified Korteweg-de Vries (mKdV) equation, is structurally simpler and more precise results can be obtained by the method of Holmer-Zworski (2007).

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arXiv2010

Blow-up solutions on a sphere for the 3d quintic NLS in the energy space

We prove that if $u(t)$ is a log-log blow-up solution, of the type studied by Merle-Raphaël (2001-2005), to the $L^2$ critical focusing NLS equation $i\partial_t u +Δu + |u|^{4/d} u=0$ with initial data $u_0\in H^1(\mathbb{R}^d)$ in the cases $d=1, 2$, then $u(t)$ remains bounded in $H^1$ away from the blow-up point. This is obtained without assuming that the initial data $u_0$ has any regularity beyond $H^1(\mathbb{R}^d)$. As an application of the $d=1$ result, we construct an open subset of initial data in the radial energy space $H^1_{rad}(\mathbb{R}^3)$ with corresponding solutions that blow-up on a sphere at positive radius for the 3d quintic ($\dot H^1$-critical) focusing NLS equation $i\partial_tu + Δu + |u|^4u=0$. This improves Raphaël-Szeftel (2009), where an open subset in $H^3_{rad}(\mathbb{R}^3)$ is obtained. The method of proof can be summarized as follows: on the whole space, high frequencies above the blow-up scale are controlled by the bilinear Strichartz estimates. On the other hand, outside the blow-up core, low frequencies are controlled by finite speed of propagation.

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arXiv2010

Effective dynamics of double solitons for perturbed mKdV

We show that an interacting double soliton solution to the perturbed mKdV equation is close in $H^2$ to a double soliton following an effective dynamics obtained as Hamilton's equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof.

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arXiv2010

A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle

We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + Δu + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + Δu + |u|^4 u=0, that blow-up on a circle.

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arXiv2010

Divergence of infinite-variance nonradial solutions to the 3d NLS equation

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + Δu + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and by $Q(x)$ the ground state solution to $-Q+ΔQ+|Q|^2Q=0$, we prove the following: if $M[u]E[u] \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then either $u(t)$ blows-up in finite positive time or $u(t)$ exists globally for all positive time and there exists a sequence of times $t_n\to +\infty$ such that $\|\nabla u(t_n)\|_{L^2} \to \infty$. Similar statements hold for negative time.

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arXiv2009

Blow-up criteria for the 3d cubic nonlinear Schrödinger equation

We consider solutions $u$ to the 3d nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^2u=0$. In particular, we are interested in finding criteria on the initial data $u_0$ that predict the asymptotic behavior of $u(t)$, e.g., whether $u(t)$ blows-up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for $H^1$ data) if $M[u]E[u]\leq M[Q]E[Q]$, where $M[u]$ and $E[u]$ denote the mass and energy of $u$, and $Q$ denotes the ground state solution to $-Q+ΔQ +|Q|^2Q=0$. Here, we prove a new sufficient condition for blow-up using an interpolation type inequality and the virial identity that is applicable to certain initial data satisfying $M[u]E[u]>M[Q]E[Q]$. Our condition is similar to one obtained by Lushnikov (1995) but our method allows for an adaptation to radial, infinite-variance initial data that can be stated conceptually: for real initial data, if a certain fraction of the mass is contained in the unit ball, then blow-up occurs. We also show analytically (if one takes the numerically computed value of $\|Q\|_{\dot H^{1/2}}$) that there exist Gaussian initial data $u_0$ with negative quadratic phase such that $\|u_0\|_{\dot H^{1/2}} < \|Q\|_{\dot H^{1/2}}$ but the solution $u(t)$ blows-up. We conclude with several numerically computed examples.

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arXiv2009

Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations

In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schrödinger equation $i\partial_t v +Δv =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schrödinger (NLS) equation $i\partial_t u + Δu -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\partial_t u + Δu -(|x|^{-γ}\star|u|^{2})u=0$ for interaction powers $1<γ<2$, such that $\|u(t)-v(t)\|_{H^1} \to 0$ as $t\to +\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\geq 3$, while for $p\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<γ<2$, and it was previously known that wave operators cannot exist for $0<γ\leq 1$, while no result was previously known in the range $1<γ<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008).

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arXiv2009

On the 2d Zakharov system with L^2 Schrödinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.

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arXiv2009

Fast soliton scattering by attractive delta impurities

We study the Gross-Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit.

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arXiv2008

Geometric structure of NLS evolution

We clarify the relation between the Hamiltonian and Lagrangian approaches to nonlinear evolution equations, focusing specifically on the nonlinear Schroedinger equation. In particular, we explain the least action principle and the Noether theorem in this context.

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