RESEARCH DOSSIER

GIDEON SIMPSON

Contents

1. Overview 12. Metastability & Materials Science 22.1. Parallel Replica Dynamics 52.2. Diffusive Molecular Dynamics 83. Nonlinear Wave Equations 123.1. Derivative Nonlinear Schrodinger Equations 123.2. Weak Turbulence 20References 23

1. Overview

My research program is in applied mathematics, numerical analysis, and scientific computing,and I am drawn to problems that include:

Multiscale Modeling: Many natural and engineered systems found in Earth science, optics,materials science, and other fields, exhibit behavior on multiple spatial and temporal scales.It may be unclear which of these scales are essential to the physical problem of interest.Consequently, multiscale modeling typically begins with a “primitive” mathematical modelfully resolving all scales. Numerically integrating this model may be very costly, and itmay be challenging to interpret the simulation output, but, if solved, the primitive modelwould yield a physically consistent answer. Given such a model, accepted as sound in botha physical and mathematical sense, we then seek to identify additional assumptions andapproximations leading to an alternative model that may be less computationally expensiveto simulate and easier to interpret. Some of the recurring research problems are thus:What is the primitive model? What are acceptable approximations? Can the error ofthe approximations be quantified and controlled? How do different models relate to oneanother?Representative publications include [30,69,80,83,84,87].

Nonlinear Wave Equations: Such equations, including the nonlinear Schrodinger equa-tion, appear in a variety of physical applications, including optics, fluid mechanics, andgeophysics. The nonlinearity of these equations is well-known to produce rich dynamics,including solitary waves and singularity formation. Some of the recurring research problemsinclude: Are the equations well-posed? Do they possess solitary wave solutions? Are thesolitary waves stable or unstable? Is there a finite time singularity, and, if so, what is therate of blowup? Provided these equations are well-posed, the predictions for solitary wavesand singularities explain observations from laboratory experiments, where available, andfrom numerical simulations.

Date: August 8, 2017.

1

Representative publications include [2, 18,23,56,57,62,63,82].Numerical Algorithms: Whether the underlying research problem is in multiscale model-

ing or nonlinear waves, its study benefits from direct numerical simulation. In multiscalemodeling, insight is gained from a direct comparison of a primitive model with a reducedmodel, while the regime of validity for an asymptotic prediction for singularity formationin a nonlinear wave equation may best be examined with simulation. Such explorationsoften require novel algorithms and software implementations. In turn, the convergence andefficiency of the algorithms may warrant investigation.Representative publications include [4, 10,21,22,34,46,68,73,74,79,81,85].

These three broad research areas often overlap with motivations in one leading to new researchproblems in another. For example, a reduced model of a multiscale problem may lead to an equationpossessing a solitary wave, demanding stability analysis. Alternatively, a solitary wave solution ora derived multiscale model may require some novel computation.

My approach is problem dependent, and I apply a combination of tools, including asymptoticmethods, numerical simulation, and rigorous analysis. Indeed, my work often intersects with com-putational physics, analysis of differential equations, and probability. This mixture of problems andapproaches dates to my graduate work, where I looked at a multiscale modeling problem in Earthscience (magma migration), along with the well-posedness and stability properties of a relatednonlinear wave equation, [82–84,86,88].

Here, I review some of my research on these topics. Section 2 reviews some projects on multiscalematerials modeled at the atomistic scale, including the parallel replica dynamics algorithm and thediffusive molecular dynamics model. Section 3 presents projects on nonlinear waves, includingsolitons and singularity formation for a class of derivative nonlinear Schrodinger equations andnumerical methods for weak turbulence.

For brevity, not all of my past and present projects are presented here. Some of these otherprojects include:

Gaussian Approximation via Relative Entropy: This includes collaborations with F.J.Pinski, A.M. Stuart, H. Weber, and D. Watkins, [73, 74,85]

Solitary Wave and Singularity Stability and Dynamics: This includes collaborationswith R. Cote, C. Munoz, D. Pilod, J.L. Maruzola, R. Asad, S.G. Raynor, and I. Zwiers,[6, 23,60,61,63,64,89]

Solitary Wave Computations: This includes collaborations with M. Spiegelman, D. Olson,S. Shukla, D. Spirn, and D. Ginsberg, [35,68,81]

Solitary Waves in Nonlinear Maxwell: This includes collaborations with M.I. Weinsteinand D.E. Pelinovksy, [69,87]

Nonlinear Waves in Magma Dynamics: This includes collaborations with M. Spiegel-man, M.I. Weinstein, P. Rosenau, D.M. Ambrose, J.D. Wright, and D.G. Yang, [3,82,86,88].

Multiscale Models for Magma Migration: This incldues collaborations with M. Spiegel-man and M.I. Weinstein, [83, 84].

2. Metastability & Materials Science

Projects in this section were supported in part by the grant:

Title: Theory and Computation for Mesoscopic Materials ModelingAgency: US Department of Energy (DOE), Subcontract from University of MinnesotaRole: Co-PI; Lead PI – M. Luskin, School of Mathematics, University of MinnesotaDuration: 08/01/2014 – 08/14/2018Award Amount Total/ Drexel: $549,513 / $88,715

2

0.0 0.2 0.4V(x)

1.5

1.0

0.5

0.0

0.5

1.0

1.5

x

0 2000 4000 6000 8000 10000t

X t

Figure 1. A trajectory evolving under (2.2) with V (x) = 14(x2 − 1)2. At the value

of β = 20, a significant fraction of the trajectory is spent in the basins of attractionof V (x) associated with x = ±1, with rare transitions between the basins.

Molecular dynamics (MD) treats materials at the atomistic scale at finite temperature. Thisbegins with the introduction of a potential, V (x), x = (x1, x2, . . . , xn) ∈ R3n, d = 3n, for a systemwith n atoms in R3. A simple example consists solely of pair interactions for an elemental material:

(2.1) V (x) =∑

1≤i<j≤nφ(|xi − xj |)

At inverse temperature β, the evolution of the system can be modeled by stochastic differentialequations, such as the overdamped Langevin equation:

(2.2) dx(t) = −∇V (x(t))dt+√

2β−1dw(t),

with w(t) an Rd valued Brownian motion. This has the dynamics of a noisy gradient descent; the−∇V term drives the system towards local minima of V , while the additive noise continuouslyperturbs it. Figure 1 shows the solution of (2.2) for a double well potential V (x) = 1

4(x2 −1)2, revealing the phenomenon of metastability. At sufficiently low temperature, the trajectorytends to spend much of its time “vibrating” about local minima, undergoing infrequent, but rapid,transitions among them. These are the so called metastable states of the system.

A physical problem is diagrammed in Figure 2, where a vacancy diffuses through a crystallinelattice in 2D. As the point defect moves, the system migrates between basins of attraction of thepotential energy. At sufficiently low temperatures with high energy barriers these minima willbe metastable. The migration of defects, of one type or another, is the fundamental mechanismby which macroscopic material deformation occurs. Thus, it is essential to be able to model thisevolution for predictive simulation.

In real materials, the vibrational timescale is on the order of 10−15 s, while the physicallyinteresting time scale of defect motion could be on the order of 10−6 s. Such time scale separationsare found in metals, carbon nanotubes, and proteins, [72,97]. This time scale separation is onerous,even with the availability of parallel computers. For example, suppose a direct numerical integrationof (2.2) resolved the 10−15 s time scale by taking time steps of ∆t ∼ 10−16 s, and assume that asingle rare event has a characteristic time scale of 10−6 s. This requires 1010 time steps. Given thata single evaluation of ∇V may take 10−6 s of wall clock time per atom, integrating such a systemto observe a first escape from a metastable region would take about 2.5 hours of wall clock time

3

Line Defect

Point Defect(Vacancy)V (x)

State A

State B

State C

State A

State B

State C

State A

State B

State C

Figure 2. A diagram for the migration of a vacancy through from State A to StateB and then State C. These states reflect the entire configuration of the crystal, andnot just the defect position.

per atom, [71]. Thus, to simulate a “small” system of one hundred atoms would require about tendays of simulation.

This time scale problem has motivated a number of algorithms to reach physically relevant timescales, either by accelerating the occurence of transitions or by averaging out the vibrational timescale. At the heart of all of the algorithms that circumvent direct integration is the observationthat there is a separation between the time it takes to “relax” to a local equilibrium associatedwith a metastable state, and the time it takes for the trajectory to exit said metastable state. Thisis captured in Figure 1, where, rapidly, the trajectory becomes trapped in a basin and, after sometime, it escapes to the other basin.

In general, let Ω ⊂ Rd be a basin of attraction associated with V , in which x(t) is expectedto remain for a long time. Denote by µt the marginal distribution of the trajectory, starting atx0 ∈ Ω, conditioned on having not left Ω:

(2.3) µt(•) = P (x(t) ∈ • | x(0) = x0, x(s) ∈ Ω, s ≤ t) , µt(Ω) = 1.

Alternatively, if we define T to be the first exit time, when x(t) first reaches ∂Ω, then

(2.4) µt(•) = P (x(t) ∈ • | x(0) = x0, T > t) .

The distribution of trajectories that persist in Ω for a “very” long time is given by ν, the quasista-tionary distribution (QSD) with ν = limt→∞ µt, in the weak sense. So for t large, x(t) ∼ µt ≈ ν.Note, however, if at time t, x(t) ∼ ν, it still leaves Ω in finite time, almost surely (a.s.).

The time scale on which µt converges to ν is thus the relaxation time scale, which is expectedto be much shorter than the first exit time scale. For (2.2), these time scales can be quantified byintroducing the generator, L,

(2.5) L = −∇V · ∇+ β−1∇2.

Equipping L with homogeneous Dirichlet boundary conditions on Ω, let 0 < λ1 < λ2 denote thefirst two eigenvalues of −L:

Characteristic Relaxation Time Scale = (λ2 − λ1)−1

Characteristic Escape Time Scale = (λ1)−1

Thus, for (2.2), metastability corresponds to (λ2 − λ1)−1 (λ1)−1.4

2.1. Parallel Replica Dynamics. A.F. Voter (Los Alamos National Laboratory) and collabora-tors have proposed several methods for overcoming the time scale challenge which have come tobe known as accelerated MD (AMD), including Parallel Replica Dynamics (ParRep), [72,97]. Theidea of all of these algorithms is to trade accuracy with respect to the trajectory within each statein exchange for a faster computation of the transitions amongst them. Turning back to Figure 2,it would be satisfactory to know which state, A, B or C, x(t) was in along with the time betweentransitions. Analogously, in Figure 1, it would be sufficient to know if the trajectory were in theupper or lower basin, and the time between transitions. The output of an AMD simulation is thusa coarse grained trajectory of the sequence of visited states and the times spent in each. My ownwork has focused on the ParRep algorithm, though the insight gained from it is informative for theother AMD methods.

ParRep exploits the property of the QSD that if x(0) ∼ ν, then the first exit time is exponentiallydistributed with parameter λ1,

(2.6) P(T > t) = e−λ1t.

The algorithm then runs N i.i.d. copies of (2.2), x(1)(t),x(2)(t), . . .x(N)(t), with x(k)(0) ∼ ν, andterminates when the first of these exits Ω. The computational speedup comes from the simplecalculation

(2.7) P(

min1≤k≤N

T (k) > t

)= e−Nλ1t, N min

1≤k≤NT (k) ∼ T (1),

where T (k) is the first exit time of the k-th replica. Thus, a first exit amongst the ensemble appearsmore rapidly, by a factor of N . The simulation clock is advanced according to the rule

(2.8) T p = N min1≤k≤N

T (k).

Aspects of ParRep I have considered include the role of N , sampling strategies for ν, and the impactof discretization.

2.1.1. Impact of Number of Replicas. Following up on the analysis in [49], I worked with M. Luskin(Minnesota) to establish:

Theorem 2.1 (Convergence of ParRep, [79]). Let (x(T ), T ) denote the hitting point and first exittime of (2.2) from the metastable region Ω, and let (xp(T p), T p) be the values generated fromParRep determined by numerical parameters εcorr > 0 and εphase > 0, tolerances associated with the“decorrelation” and “dephasing” steps of ParRep. Then the error over a single transition is:

(2.9) |P(x(T ) ∈ • | T > t)− P(xp(T p) ∈ • | T p > t)| . εcorr +N2εphase(1 + εphase)N−1

Parameters εcorr and εphase are related to the quality of the QSD sampling ν; reducing themreduces the error, but also reduces the efficiency. Their roles in the algorithm are discussed in[49,79,96]. The result of this theorem can be interpreted as the local error of the algorithm, havingmade a single escape from a metastable region; in a real simulation, the algorithm will be appliedin successive metastable regions.

An important aspect of this result was the apparent error growth in the upper bound with thenumber of replicas, N . While the difference in probabilities is bounded from above by unity, anumerical experiment in [79] indicated that though the upper bound in (2.9) might not be sharp,there is indeed growth in the error with N . Practitioners can guard against this problem by alteringthe values of εcorr and εphase as N changes.

5

0 20 40 60 80 100t

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

P(T

>t)

SerialCorrectedContinuous

Figure 3. Distribution of ParRep first passage times in an overdamped Langevinproblem discretized according to (2.10). The “Continuous” curve corresponds tothe naive use of (2.8) in the discrete in time problem. This leads to a nonphysicalstaircase. Using (2.12) instead, we obtain a physically consistent prediction; this isthe “Corrected” curve. “Serial” corresponds to direct integration of (2.2) withoutusing ParRep. Here, N∆t = 10. From [5].

2.1.2. Impact of Time Step. A novel, previously unnoticed, challenge appeared when I began exam-ining the impact of the finite time step discretization on ParRep. Supposing that (2.2) is integratedusing the Euler-Maruyama scheme, x∆t

n ≈ x(n∆t),

(2.10) x∆tn+1 = x∆t

n −∇V (x∆tn )∆t+

√2β−1∆tξn+1, ξn+1 ∼ N(0, Id×d).

The first exit time, τ ∈ N, is now the first n such that x∆tn /∈ Ω. Note that the first exit time of this

discrete process is denoted τ , in contrast to T , for the continuous in time problem (2.2); a physicaltime is obtained by multiplying τ by ∆t.

The problem occurs when applying ParRep with such a discretization. The naive implementationwould apply (2.8) with T p = N∆t · τ (k) and obtain exit times in the set

(2.11) N ·∆t,N · 2∆t,N · 3∆t, . . . .Consequently, when N∆t becomes large, the statistics of the first exit time will be nonphysical.This is captured in Figure 3 by the “staircase.” Indeed, the first exit time, τ , is geometricallydistributed in contrast to T , which is exponentially distributed; (2.7) does not hold for geometricrandom variables (r.v.). Thus, an error is introduced by virtue of discretization when using (2.8).

In [5], working with D. Aristoff (Colorado State) and T. Lelievre (ENPC/CERMICS), I formu-lated ParRep for discrete in time problems and developed the geometric r.v. analog of (2.8):

Lemma 2.1 (Acceleration Rule for Geometric Random Variables, [5]). Let τk, k = 1, . . . N , bei.i.d. geometric r.v. with parameter p. Let

M = minm ≥ 1 | there exists k s.t. τk ≤ m(2.12a)

K = argmin1≤k≤N

τk ≤M(2.12b)

τp = (N − 1)(M − 1) +K − 1 + τK(2.12c)

Then τp is equivalent in law to τ1.6

Multiplying τp by ∆t gives a physical time scale for a first exit when this is applied to ParRep.Instead of computing exit times in (2.11), they are in

(2.13) ∆t, 2∆t, 3∆t, . . . .This yields the “Corrected” curve in Figure 3. There is still an error associated with ∆t, but it isno longer amplified by N .

Lemma 2.1 is but one result in [5], which formulates the ParRep algorithm, generically, fordiscrete in time problems that suffer from metastability. Additionally, the ∆t correction of Lemma2.1 is a very minor alteration to existing ParRep software, as it only requires the tracking of theindex K, the lowest index replica that makes the first exit.

2.1.3. QSD Sampling. ParRep, along with other algorithms that address metastability, depends onsampling ν. This motivates an exploration of QSD sampling algorithms. An elementary approachis rejection sampling. Since the total variation distance between µt and ν is

(2.14) dTV(µt, ν) ≤ Ce−(λ2−λ1)t,

for large enough t?, µt? ≈ ν. Next, we integrate (2.2) till time t?. If x(t) ∈ Ω for all t ≤ t?, x(t?),is accepted as an approximate sample of ν. Otherwise, it is rejected, and the process is restarted.Denoting the probability of reaching time t? without exiting by p = P(T > t?), the expected numberof runs to obtain the N samples needed for ParRep is N/p. This value of t? and (2.14) determinesεcorr and εphase in Theorem 2.1.

With A. Binder (graduate student at Minnesota) and T. Lelievre (ENPC/CERMICS), I examinedan alternative sampling strategy based on a branching-interacting particle system:

Algorithm 2.1 (Sampling the QSD for Parallel Replica Dynamics, [10]). Initialize N replicasof (2.2), with initial positions in Ω. When one of the replicas exits, resample its position fromthe N − 1 survivors. Simultaneously, use convergence diagnostics to terminate this branching-interacting particle process when it has reached stationarity. The N replicas then approximatelysample ν. This is illustrated in Figure 4.

Such a sampling strategy has been explored before, but not in the context of ParRep; oneapplication is estimation of λ1, [50, 51, 76]. The first novelty of our work is that since ParReprequires an ensemble of N samples to run, this algorithm exploits the ensemble inherent to thealgorithm. In contrast to rejection sampling, nothing is discarded. A second novelty is the couplingof the algorithm to the Gelman-Rubin convergence diagnostic, which requires simultaneous copies ofthe problem to determine stationarity, [33]. Thus, we arrive at an algorithm that can, in principle,be run asynchronously, that is not more costly than naive rejection sampling, and introducesconvergence diagnostics to test for when to terminate integration. The “cost” of this algorithm isthat it introduces some amount of correlation amongst the replicas, though this can be mitigated;see [10] for some discussion of this.

This algorithm does not make explicit use of (2.14), as it remains an open problem on how tocompute λ2. Indeed, the use of the convergence diagnostic was to dynamically estimate t? becauseneither the pre-exponential constant nor the spectral gap in bound (2.14) are explicit or are easyto estimate in general.

In Algorithm 2.1, if Euler-Maruyama is used, the discretization introduces a O(∆t) bias, [58];for any bounded smooth function,

(2.15) |E[f(x(tn))]− E[f(x∆tn )]| ≤ Kn∆t.

Furthermore, numerical stability issues may restrict the use of large time steps, reducing efficiency.Working with N. Bou-Rabee (Rutgers), we are in the process of reducing the bias and allowing

7

X1t

X2t

X3t

(a)

X1t

X2t

X3t

(b)

X1t

X2t

X3t

(c)

X1t

X2t

X3t

X1t

(d)

Figure 4. The branching-interacting particle system used to sample a QSD, [10].Initially, each copy propagates independently, as in (a). Eventually, one of thereplicas reaches the boundary, as in (b). The replica that has reached the boundaryis then killed, as in (c), and resampled from the surviving trajectories, as in (d).

for larger values of ∆t. This project has taken two tracks. First, we have applied the Metropolisintegrator (MALA), [11,52],

(2.16) x∆tn+1 =

xpn+1 = x∆t

n −∇V (x∆tn )∆t+

√2β−1∆tξn ζn < r(x∆t

n ,xpn+1), ζn ∼ U(0, 1)

x∆tn otherwise.

In the above expressions, xpn+1 is the proposed move, and r is Metropolis acceptance probability.

This ensures detailed balance with respect to the invariant measure e−βV (x) (not the QSD). How-ever, preliminary numerical experiments show that larger time steps are viable with the Metropolisstep, improving the performance.

This approach omits an important detail, determining when a numerical trajectory has left themetastable region so as to sample the QSD. In the above schemes, all that is known is that theexit occurs in the interval [n∆t, (n + 1)∆t]. Motivated by work in [12], instead of solving (2.2)by discretizing in time, we instead discretize the generator, L, to compute the transition rates toneighboring points in a discretization of the domain, Ω. This builds the boundary ∂Ω into thealgorithm and computes exact exit times. An investigation of this approach is currently underway,along with benchmark problems. This would be of value to a variety of algorithms, includingParRep, as many of them rely on the ability to sample from the QSD.

2.2. Diffusive Molecular Dynamics. Diffusive Molecular Dynamics (DMD), is another methodthat takes advantage of the time scale seperation in atomistic systems, [53, 77]. The key idea isto extend the classical MD model with additional degrees of freedom that distinguish atoms fromvacancies, or two different species from one another, and to allow these degrees of freedom to evolvealong with the positions, [26,27,53,77]. In the works that have appeared in the materials literaturethus far, DMD appears to be an attractive approach that gives both qualitative and quantitativepredictions for material evolution and equilibrium behavior. DMD also bears some resemblance tothe recent work in [95].

2.2.1. Vacancy Problem and Relative Entropy. In [80], I presented a formulation of DMD by startingwith n “atomic sites”, instead of n atoms. Each site has position xi in the computational domain,Ω ⊂ R3. At each of these sites, there is a value, ai ∈ 0, 1, indicating if it is an atom or a vacancy;see Figure 5. On this state space, (Ω× 0, 1)n, a DMD pair potential takes the form

(2.17) V (x,a) =∑

1≤i<j≤naiajφ(|xi − xj |),

8

xi

(a) Classical MD

xj , cjxi, ci

(b) DMD

Figure 5. DMD adds a probability of occupancy degree of freedom to each of theatomic sites, in contras to classical MD, where there are only atoms and vacancies.

which only includes contributions when both sites i and j are occupied by atoms. Given a set ofdesired mean concentrations ci = E[ai], a Boltzmann ensemble is given by

(2.18) ν(dx,a) = Z−1∏i

|Ω|ai−1e−βV (x,a)+βµ·adx.

with partition function

(2.19) Z =∑

a∈0,1n

∏i

|Ω|ai−1

∫Ωne−βV (x,a)+βµ·adx.

The µ = (µ1, µ2, . . . , µN ), are chemical potentials enforcing ci = E[ai]. Thus, the free energy F isgiven by

(2.20) F = −β−1 logZ + µ · c.The term

∏i |Ω|ai−1 is introduced in (2.19) as we adopt the perspective that if a site corresponds

to a vacancy, its position within the computational domain is indistinguishable. This guaranteesthat the problem is well defined in the large volume, fixed vacancy density, limit.

Having computed the free energy, practitioners make a modelling assumption to evolve theconcentrations in time, [26,27,53,77]. One such example is the deterministic flow

(2.21)dcidt

=∑j∈N(i)

kij

(∂F

∂cj− ∂F

∂ci

).

Here, the kij is a mobility term, governing the rate of transfer between sites i and j, and N(i) isthe set of sites neighboring i.

As n will be large, direct summation and quadrature of Z is impractical, suffering from the“curse of dimensionality.” Instead, an approximate distribution is formulated with potential

(2.22) V (x,a; k,X) =n∑i=1

1

2aiki|xi −Xi|2.

where k and X are free parameters. They are selected such that the free energy difference betweenthe ensembles associated with V and V is minimized. Since the coordinates in (2.22) are separablewith Gaussian densities, the approximate free energy can be computed from integrals over Ω ⊂ R3,as opposed to integrals over Ωn, [53]. This yields an approximate free energy, F , that is substitutedinto (2.21).

9

With M. Luskin and D. Srolovitz (Penn), I reformulated the problem of solving for k and X

in (2.22) as a relative entropy minimization problem between ν and ν corresponding to V and V .Recall that relative entropy, [28,59], measures the distance between measures, ν and µ:

(2.23) R(µ||ν) =

Eµ[log dµ

dν

]µ ν,

∞ otherwise.

We found that DMD requires a careful definition of the state space in order to ensure the measuresare well defined, that they are absolutely continuous, and that the large volume, fixed vacancydensity limit exists. This motivated, for instance, the adoption of the finite volume Ω, and afore-mentioned |Ω|ai−1 term in (2.18). Such structure had been ambiguous in the earlier materialsscience literature, [26, 27, 53, 77]. Having properly formulated these aspects of the problem, weobtain:

Theorem 2.2 (Relative Entropy Minimization for DMD, [80]). Subject to assumptions on V , theproblem of obtaining

(2.24) (X,k) ∈ argminR(ν||ν)

is well posed in the sense that minimizing sequences with respect to relative entropy subsequentiallyconverge. Furthermore, the induced distributions, along the subsequence, converge in total variation.

This result is essential to running DMD because, as noted, some approximation of the free energymust be used due to the curse of dimensionality. Additionally, by framing it in the context of arelative entropy approximation of one distribution by another, there is the possibility of looking atapproximate potentials that are richer than the Einstein model, (2.22).

2.2.2. Binary Alloy Models and Evolution Equations. While the results in [80] gave some mathe-matical structure to DMD, an outstanding question remains as to how the system evolves in time– is there any justification for the evolution given by (2.21)? Working with Luskin (Minnesota),P. Plechac (Delaware), and B.A. Farmer (postdoc at Drexel & Minnesota), we consider the relatedbinary alloy problem. In place of a, we let σ ∈ ±1n, and instead of (2.17), the potential is

(2.25) V (x,σ) =∑

1≤i<j≤nφσi,σj (|xi − xj |).

The “spin” σi indicates now which of two species of atom is present at site i, as in an aluminum-magnesium alloy. The values of σi and σj now determine the type of pair potential.

The goal is now to construct a stochastic process that, after making approximations including ascale separation and a mean field approximation, leads to a deterministic flow in the spirit of (2.21).

To that end, we first identify two stochastic processes, each with e−βV (x,σ) as its invariant measure,subject to the net mass constraint

∑σi = M . The joint process will then also have e−βV (x,σ) as

its invariant measure.The first process is the diffusion, (2.2), with V = V (x,σ) and σ held constant. Holding x

constant, the second process is a spin exchange model, akin to an Ising model, where the reactionrates of evolving σ to some other σ′ are assumed to satisfy detailed balance:

(2.26) r(σ → σ′; x)e−βV (x,σ) = r(σ′ → σ; x)e−βV (x,σ′),

also preserving∑σi =

∑σ′i. Letting Lx denote the generator of the diffusion an Lσ denote the

generator of the spin exchange, there exists a joint stochastic process with generator

(2.27) Lε = ε−1Lx + Lσ,

with ε > 0 reflecting a time scale separation between the two processes.10

The selection of ε is motivated as follows. Given a particular spin, σ, the diffusion begins in somebasin of V (x,σ). As this basin is likely metastable, the system will persist in it for a relativelylong time. The coupling to the spin exchange process in (2.27) allows for a computationally cheaptransition out of one basin and into another. Thus, ε roughly captures ratio of the time it takes forthe system to relax to a local minimum with respect to the time it takes for the system to evolveinto some other conformation.

The scale separation is now exploited by examining the evolution equation

(2.28)dv

dt= v = Lεv, v = v(x,σ, t),

where v is some observable. Making a standard Fredholm expansion, v = v0 + εv1 + ε2v2 + . . ., weconclude that v0 is x independent and the leading order model takes the form

(2.29) v0 = Eµ(•|σ)[Lσ]v0

This corresponds to a spin exchange process with reaction rates obtained by averaging against aQSD, µ(dx | σ(t)). Making use of the scale separated master equation, we obtain

(2.30)d

dtE[σi] =

∑j∈N(i)

E[(σj − σi)r(σ → σij)].

Some details have been omitted from this presentation for clarity. Choosing tanh type reactionrates for (2.26) and making a mean field approximation in (2.30),

(2.31) si =1

2τ−1

∑j∈N(i)

(sj − si)− (1− sisj) tanh

(βV (sij)− V (s)

sj − si

), x ∼ µ(dx | s(t))

Here, si = E[σi], and sij corresponds to s with the i-th and j-th coordinates exchanged. This modelcaptures the scale separation of the problem and overcomes the metastability in the system. Thefast vibrations have been averaged out, and the atomic positions, x(t), are instantaneously sampledfrom µ(dx | s(t)). Meanwhile, the composition, s(t), slowly evolves under the deterministic flow.

In contrast, model (2.21) leads to equations of the form(2.32)

si = β−1kc

∑j∈N(i)

tanh−1(sj)− tanh−1(si)− βV (sij)− V (s)

sj − si+ βJij(sj − si), x ∼ µ(dx | s(t)),

where kc > 0 is a mobility constant and µ is a Gaussian distribution obtained by relative entropyapproximation. Both µ and µ act to sample a mode of the energy landscape e−βV (x,s) at fixed s,and they will agree in the low temperature limit. While (2.31) differs from (2.32), the terms thatappear are qualitatively similar, and a very similar disagreement was noted by Penrose in [70].There, he compared two deterministic approximations of the Ising model on a fixed lattice. As isthe case in our work, the mean field approximation of the master equation leads to an equationlike (2.31), while a gradient flow leads to an equation like (2.32).

Thus, the work in [30] succeeds in identifying a series of approximations and assumptions thatlead to DMD type equations. Though it is not fully rigorous, I contend that our derivation leadsto better constrained equations, as they are a consequence of the scale separation, the choice ofreaction rates, and the mean field approximation, amongst other ingredients. Ongoing projectsin DMD include developing suitable problems to throughly test these assumptions and make thetheory more rigorous.

11

3. Nonlinear Wave Equations

Projects in this section were supported in part by the grant:

Title: Computational and Analytical Challenges in Nonlinear Dispersive Wave EquationsAgency: National Science Foundation (NSF)Role: PIDuration: 09/01/2014 – 07/31/2018Initial Award Amount: $146,118Supplementary Award Amount: $5,000 for REU support

A canonical nonlinear dispersive wave equation is the nonlinear Schrodinger equation (NLS),

(3.1) iψt +∇2ψ + f(|ψ|2)ψ = 0, ψ : Rd+1 → C.

This is an ubiquitous leading order approximation to phenomena in many fields, including fluidmechanics, plasma physics, and optics. The interpretation of ψ depends on the application; inphotonics, |ψ|2 is the beam’s intensity. Though it has spawned several surveys, [17,31,92,93], NLSand related equations remain of interest because of their ubiquity and outstanding challenges.

The interplay of dispersion and nonlinearity in (3.1) leads to rich behaviour depending on fand the dimension d, including solitary wave solutions and finite time singularities. Solitary wavesolutions can be obtained by making the ansatz ψ = eiλtRλ(x), where Rλ solves the stationaryproblem

(3.2) − λRλ +∇2Rλ + f(|Rλ|2)Rλ = 0, Rλ : Rd → C.

Such solutions play an intrinsic role in the dynamics, as they may be stable attractors to the flow.A second feature of NLS is its potential to form singularities in finite time for many choices of fand d. This corresponds to the existence of a finite t? > 0, such that

(3.3) limtt?‖ψ(·, t)‖Hs = +∞,

and, often, a blowup rate can be inferred. For example, in the celebrated L2-critical case (wheref(s) = sσ and σd = 2), one has the log− log blowup, [31,48,65,66,93],

(3.4) ‖ψ(·, t)‖H1

tt?∼√

log | log(t? − t)|t? − t

.

3.1. Derivative Nonlinear Schrodinger Equations. A variation of (3.1) that I have spent timeinvestigating is the derivative nonlinear Schrodinger (DNLS),

(3.5) iΨt + Ψxx + i(|Ψ|2Ψ)x = 0, Ψ : R1+1 → C.

DNLS first appeared as a long wavelength approximation of Alfven waves from plasma physics, [93].This is in contrast to NLS, which usually arises from a slowly varying envelope approximation of aprimitive model. Making the gauge transformation,

(3.6) ψ(x, t) = Ψ(x, t) exp

i

2

∫ x

−∞|Ψ(y, t)|2dy

,

(3.5) becomes

(3.7) iψt + ψxx + i|ψ|2ψx = 0, ψ : R1+1 → C.

For sufficiently smooth data such that the gauge transform is well-defined, the two forms areequivalent. This latter form has appeared as a model for ultrashort optical pulses, [67]. My ownwork has focused on DNLS in form (3.7).

12

−5 0 5 10 0

0.1

0.2

0.3

0.4

0

5

10

t

x

|ψ|

Figure 6. Evolution of the solution to DNLS, (3.7), with Gaussian initial data.Note the emergence of rank ordered solitary waves and no suggestion of a finitetime singularity. From [56].

The cubic DNLS equation is integrable and L2-critical in the sense of scaling; given a solution ψthe L2 norm is invariant under the scaling

(3.8) ψλ(x, t) = λ12ψ(λ1x, λ2t).

Given that (3.7) is L2-critical, there had been speculation that it would form a finite time singularity.This suspicion was motivated by the behavior of other L2-critical equations, including, but notlimited to, focusing L2-critical NLS,

(3.9) iψt +∇2ψ + |ψ| 4dψ = 0, ψ : Rd+1 → C,and quintic generalized Korteweg-de Vries equation (gKdV),

(3.10) ut + u4ux + uxxx = 0, u : R1+1 → R,all of which were known to develop finite time singularities.

However, my numerical explorations of (3.7) suggested that there was no singularity, and theinitial condition would instead resolve into stable rank ordered solitary waves; see Figure 6 from [56],a collaboration with X. Liu (graduate student at Toronto) and C. Sulem (Toronto). Very recently,using integrable systems techniques, it has been proven that the solutions are global in time fordata with solitons, [45,54,55,78]. This sets DNLS apart from (3.9) and other L2-critical problems.Separate efforts, not relying on integrable systems techniques, obtained small data, global in time,existence for progressively larger data, [42, 43, 100, 101]. The most recent of these non-integrableefforts obtained global H1 solutions provided ‖ψ0‖2L2 < 4π .

Prior to these recent results using integrability, in a series of collaborations, I considered thegeneralization of (3.7),

(3.11) iψt + ψxx + i|ψ|2σψx = 0, ψ : R1+1 → C,deemed the generalized DNLS equation (gDNLS). gDNLS is L2-supercritical for σ > 1, and itreadily develops singularities in simulations. The exploration of an associated supercritical problemwas motivated by an analogous study of (3.9). In the cubic focusing case with radial data, NLS is

(3.12) iψt + ψrr +d− 1

rψr + |ψ|2ψ = 0, ψ : (0,∞)× R→ C.

As d 2, the problem descends from the supercritical to critical regime. In the same way, westudy (3.11), letting σ 1.

13

Note that (3.11) formally obeys the Hamiltonian flow

(3.13) ψt = −iδHδψ

, H[ψ] =

∫|ψx|2 +

1

(σ + 1)2ψσ+1Dxψ

σ+1, Dx =1

i∂x.

The flow also conserves mass and momentum,

(3.14) M[ψ] =

∫|ψ|2, P[ψ] = −

∫ψDxψ.

3.1.1. Well-Posedness. In collaboration with D.M. Ambrose (Drexel), I looked at the well-posednessof (3.11) on the torus, with solutions in Hs(T), [1]. For the cubic problem, a variety of results areavailable, with the earliest using the energy method to establish local in time well-posedness inH1(R) and higher regularity spaces, [32, 42, 43, 94]. More recent results, relying on harmonic anal-ysis techniques, demonstrated well-posedness in the lower regularity spaces Hs(R) and Hs(T) with

s < 1. For the non-cubic case, a low regularity result in H12 (R) was obtained in [40], but only for

σ ≥ 52 . In work published after [1], the case of general σ > 0 has been explored, [25,41].

We sought mild solutions of (3.11), σ ≥ 1, using the energy method on the torus. Our mainresults were

Theorem 3.1 (Existence in H1(T), [1]). Let σ ≥ 1 be given. Let u0 ∈ H1(T). There existsT > 0 and u ∈ L∞(0, T ;H1(T)) such that u is a mild solution of (3.11). As a mapping from[0, T ] → H1(T), u(t) is weakly continuous in time. Furthermore, for s such that 0 ≤ s < 1,u ∈ C(0, T ;Hs(T)).

and

Theorem 3.2 (Higher Regularity Well-Posedness, [1]). Let σ ≥ 1 be given. Let u0 ∈ H2(T).There exists T > 0 and a unique u ∈ C(0, T ;H2(T)) that is a mild solution of (3.11). The solutiondepends continuously upon the initial conditions in the Hs(T) norm for 0 ≤ s < 2.

The motivation for working on the torus is that it allows for easy application of the Aubins-Lionslemma to obtain the necessary compactness for subsequential convergence in L2([0, T ];L2). Indeed,denoting the mollified sequence of solutions by uε, we obtain a uniformly bounded sequenceuε ∈ L2([0, T ];H1), with ∂tuε ∈ L2([0, T ];H−1) also uniformly bounded. As H1 is compactlyembedded in L2, which is continuously embedded in H−1, the lemma is immediately applicable.Additionally, many numerical simulations of (3.11) have been conducted on periodic domains.

An interesting aspect of our proof is that we use a “conservative” mollifier, which is to saythat the mollifier Jε merely zeros out wave numbers with modulus in excess of ε−1. The mollifiedequation is then

(3.15) ψε,t = −Jε(|Jεψε|2σJεψε,x

)+ iJεψε,xx, ψε : T× R→ C,

which has, as an invariant, the mollified Hamiltonian (contrast with (3.13)) :

(3.16) Hε[f ] =

∫ 2π

0|fx|2 +

1

(σ + 1)2(Jεf)σ+1Dx(Jεf)σ+1 dx.

Were we to use a dissipative regularization, we would not have this as a conserved quantity. Theenergy Eε is then defined as

(3.17) Eε[ψε] ≡ Hε[ψε] + 12‖ψε‖2L2 + c‖Jεψε‖4σ+2

L4σ+2 ≥ 12‖ψε‖2H1 ,

which, as indicated, controls H1. Using the conservation of Hε, we then obtain

(3.18)d

dtEε[ψε] . (1 + Eε[ψε])p, p > 1.

14

This prevents a finite time explosion in H1 and gives a uniform bound on the mollified sequence.To prove that the limiting solution is indeed a mild solution, a second mollification is required inthe mild form of the equation, first sending ε 0 and then δ 0:

(3.19) Jδuε = eiJε∂2xtJδu0 −

∫ t

0eiJε∂

2x(t−s)JδJε

(|Jεuε|2σJεuε,x

)ds.

For the H1 solutions, uniqueness and conservation of the true Hamiltonian, (3.13), escaped us withthese methods. In the recent work [41], the authors obtained uniqueness and conservation on Hfor the H1(R) problem when σ ≥ 1. Their result relied on dispersive estimates. For the problemposed in H2(T), where we obtained stronger results, ‖uε‖2H2 can be used as the energy. This higherregularity case also required less delicate mollification.

3.1.2. Soliton Stability. Cubic DNLS, (3.7), has long been known to possess solitary wave solutions,and in [19, 38], it was shown that they were stable. Working with Liu and Sulem, I examinedsolitary wave solutions for the case of general σ > 0 in (3.11), obtaining explicit representations,and determining their stability properties, [57].

In particular, for all σ > 0, there is a two parameter family of exact solutions of (3.11),

ψω,c(x, t) = ϕω,c(x− ct) exp i

ωt+

c

2(x− ct)− 1

2σ + 2

∫ x−ct

−∞ϕ2σω,c(y)dy

(3.20a)

ϕω,c(y) =

(σ + 1)(4ω − c2)

2√ω(cosh(σ

√4ω − c2y)− c

2√ω

)

12σ

(3.20b)

Thus, for 4ω > c2 ≥ 0, there are exponentially localized solutions that travel and oscillate in time.Such solitary wave solutions are said to be orbitally stable if, for all ε > 0, there exists a δ > 0

such that

(3.21) ‖ψ0 − ψω,c‖H1 < δ ⇒ supt≥0

inf(θ,y)∈T×R

‖ψ(·, t)− eiθψω,c(t, · − y)‖H1 < ε,

and the solution is global in time. The stability results that we obtained are:

Theorem 3.3 (Solitary Wave Stability, [57] ). Assume 4ω > c2 ≥ 0. Then

• For σ ≥ 2, the solitary wave solution is orbitally unstable for all (ω, c).• For σ ∈ (1, 2), there exists z0 = z0(σ) ∈ (−1, 1) such that:

(1) the solitary wave solution is orbitally stable if c < 2z0√ω.

(2) the solitary wave solution is orbitally unstable if c > 2z0√ω.

• For σ ∈ (0, 1], the solitary wave solutions are orbitally stable.

The numerically computed values of z0 correspond to the zeros of the function

(3.22) F (z;σ) = (σ−1)2

[∫ ∞0

(cosh(y)− z)− 1σ dy

]2

−[∫ ∞

0(cosh(y)− z)− 1

σ−1(z cosh(y)− 1)dy

]2

,

plotted in Figure 7. The proofs of these results use the methodology of Grillakis, Shatah, & Strauss,along with Weinstein, [36, 37, 98, 99]. In [57], I assumed the necessary existence of weak solutionsfor which the invariants are conserved for all σ > 0, so as to concentrate on obtaining the stabilityresults. This assumption was proven to hold for σ ≥ 1 in [41]. Challenges remain for σ ∈ (0, 1),as [41] did not address conservation of H or uniqueness of the solution in this case.

There are two related aspects of these results that are noteworthy. The first is that when σ ∈(1, 2), the equation is L2-supercritical, but stable solitary waves are present. Numerical simulationsof a solitary wave in the stable regime for σ = 1.5 were consistent with this prediction, showinga stable profile out to t = 100, [57]. This is entirely different from focusing NLS with a power

15

−1 −0.5 0 0.5 1−10

−5

0

5

10

z

F

σ = 1σ = 1.2σ = 1.4σ = 1.6σ = 1.8σ = 2

(a)

−1 −0.5 0 0.5 1−10

−8

−6

−4

−2

0

2

z

F

σ = 0.2σ = 0.4σ = 0.6σ = 0.8σ = 1

(b)

Figure 7. The functions F (z;σ) defined in (3.22). The zero crossings, z0(σ), de-termine the transition between stability and instability of solitary wave solutions togDNLS. From [57].

nonlinearity, where, in the supercritical regime, all solitary waves are unstable, and they will tendto generate singularities. A consequence of this is that, for σ ∈ (1, 2), there are open sets of datain H1, about the solitary waves, that have global in time solutions, yet fail to scatter; the solitarywaves persist.

3.1.3. Singularity Formation. Working with Liu & Sulem, and later with Y. Cher (graduate studentat Toronto) & Sulem, I investigated the tendency for singularity formation in the supercriticalregime of (3.11), with σ > 1, and the behavior as σ 1. Initial simulations of the quintic problem(σ = 2), with large enough data strongly suggested a finite time singularity. To better understandthis, we applied the dynamic rescaling method, [93], defining the new variables

(3.23) ψ(x, t) = λ−12σ (t)u(ξ, τ), ξ =

x− x0(t)

λ(t), τ =

∫ t

0

ds

λ2(s)

which leads to the equations

(3.24) iuτ + uξξ + i|u|2σuξ + ia(

12σu+ ξuξ

)− ibuξ = 0, a = −λdλ

dt, b = λ

dx0

dt

Closures are now introduced for λ and x0 so as to move the singularity in H1 to τ = +∞. In [56],we chose them to make ‖uξ‖L2 an invariant of (3.24) and to keep x0 near the peak of the wave:

(3.25) λ(t) = ‖uξ(·, 0)‖qL2‖ψx(·, t)‖−q

L2 , q−1 = 1 + 12

(1σ − 1

), x0(t) =

∫x|ψxx|2∫|ψxx|2

Integrating (3.24) with these closures, we found that u tended to a fixed profile, while a and bboth tended to positive constants, plotted in Figure 8. This allows us to infer blowup and obtaina profile at times in the original time scale, t, close to t?. Using (3.24), we see that if a → A > 0sufficiently rapidly, as in Figure 8, then we infer

(3.26) λ2(t) ∼ 2A(t? − t)⇒ ‖ψx‖L2 ∼ |t? − t|σ−14σ− 1

2 .

Another important observation was the tendency of u(ξ, τ)→ S(ξ)eiCτ , a time harmonic profile,with (S,A,B,C) solving the nonlinear eigenvalue problem

(3.27) Sξξ − CS + iA(

12σS + ξSξ

)− iBSξ + i|S|2σSξ = 0, lim

ξ→±∞S = 0.

16

0 0.2 0.4 0.6 0.8 110

−2

10−1

100

101

102

τ/τM

a(τ)aM

σ = 1.1, aM = 0.13, τM = 9σ = 1.3, aM = 1.09, τM = 4.6σ = 1.5, aM = 3.77, τM = 1.5σ = 1.7, aM = 0.54, τM = 6

(a)

0 0.2 0.4 0.6 0.8 1

100

101

τ/τM

b(τ)bM

σ = 1.1, bM = 3.18, τM = 9σ = 1.3, bM = 2.76, τM = 4.6σ = 1.5, bM = 2.66, τM = 1.5σ = 1.7, bM = 1.24, τM = 6

(b)

Figure 8. Estimates of the limiting values of a and b from (3.24) for different valuesof σ, rescaled by the value at the terminal time of integration, τM . In all cases, thereis a tendency towards a positive constant, implying singularity formation. From [56].

−20 −10 0 10 201.2

1.4

1.6

1.8

2

0

0.5

1

1.5

2

2.5

σ

ξ

|Q|

(a)

1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

σ

α

(b)

1 1.2 1.4 1.6 1.8 21.5

1.6

1.7

1.8

1.9

2

2.1

2.2

σ

β

(c)

Figure 9. Numerically computed solutions, (Q,α, β) of (3.28) for 1.08 ≤ σ ≤ 2.Note that α > 0 at all computed values, implying singularity formation, and itappears that α 0 as σ 1. From [56].

After a simple rescaling, (3.27) becomes

(3.28) Qξξ −Q+ iα(

12σQ+ ξQξ

)− iβQξ + i|Q|2σQξ = 0, lim

ξ→±∞Q = 0.

where α and β are universal (i.e., independent of u0). In the α = 0 limit, up to a simple rescaling,the solitary wave solution (3.20), solves (3.28).

Thus, under the assumption that u → S(ξ)eiCτ , if we numerically solve (3.28) at a value of σ,and observe that α > 0 (the singularity parameter), we can infer singularity formation. The resultsof these computations are shown in Figure 9, and from these computations, we infer the local (inthe neighborhood blowup point x?) blowup, after rescaling is

(3.29) ψ(x, t)tt?∼

(1

2α(t? − t)

) 14σ

Q

(x− x?√2α(t? − t)

+β

α

)ei(θ?+ 1

2αlog t?

t?−t

).

This first effort left some questions unresolved. The time independent computations (Figure 9)show a tendency for α 0 as σ 1, but the computations only reached σ = 1.08. For the timedependent problem, we only reached σ = 1.1. This prompted a more detailed investigation, with

17

ξ

-20 -15 -10 -5 0 5 10 15 20

|Q|

0

0.5

1

1.5

2

2.5

3σ = 1.044

σ = 1.06

σ = 1.1

(a)

σ − 1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

a

0

0.05

0.1

0.15

0.2

(b)

σ − 1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

ǫ

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(c)

Figure 10. Plots of profiles, α and ε = 2 − β at different values of σ. Thesecomputations were performed down to σ = 1.044. Note the increasing symmetry ofthe profiles with respect to the origin, and that α tends to zero smoothly. From [18].

a goal of studying the problem closer to the critical limit of σ = 1. Part of the motivation for thiswas, again, to compare against (3.12), where it was concluded that the parameter a(d), the analogof α(σ) in (3.24), tended to zero as d 2 in a catastrophic way, [93],

(3.30) d− 2 ∼ a−1e−πa .

Thus, d = d(a), tends to two as the singularity parameter a 0 faster than any polynomial, anda(d) has a vertical asymptote at d = 2, the L2-critical case. The function, a(d), “catastrophically”tends to zero as d 2. This plays an essential role in the prediction of the log− log blowup.

The subsequent investigation, with Cher & Sulem, [18], made use of more computationallypowerful numerical schemes along with additional asymptotic analysis to conclude that in theσ 1 limit

α∝(σ − 1)γa , γa ≈ 3.2,(3.31a)

ε = 2− β∝(σ − 1)γb , γb = 2,(3.31b)

with results shown in Figure 10. Thus, α 0 in a non-catastrophic way, which is very differentfrom (3.30). We also refined (3.29), predicting that the blowup profile tends to

(3.32) Q(ξ)σ1∼ L(ξ) exp

−i(αξ2

4− bξ

2+

1

4

∫ ξ

0|L(η)|2dη

),

where L =√

81+4ξ2

is the algebraic soliton solution of DNLS solving

(3.33) Lξξ − L3 + 316L

5 = 0.

Such algebraic soliton solutions can be obtained from (3.20) in the limit 4ω c2.These predictions were obtained by making a phase-amplitude decomposition of (3.28), after

removing the quadratic piece of the phase, combined with matched asymptotics. The exact powersin (3.31) were obtained using nonlinear least squares fittings. The main difference between thecomputations in Figure 9 and those in Figure 10 were that the former made use of Matlab’sbvp4c boundary value problem solver, while the latter made use of my own second order finitedifference scheme written in PETSc, [7–9], which allowed for computation of the problem on largedomains with large numbers of mesh points. Our code gave us much more control, which helped toachieve smaller values of σ. Both methods made use of Robin type boundary conditions obtained

18

σ − 10.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1A2

−

a1−1/σ(k + l)(1− 1/σ)

(a)

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

10-300

10-200

10-100

100

σ − 1

A+

ǫ3/4√

aexp

(

−

π

a+ 2

3

ǫ3/2

a

)

(b)

Figure 11. Plots of A± (from (3.34)) as σ 1. Note the massive asymmetry.σ ≥ 1.044. From [18].

from (3.28) by omitting the nonlinear term:

(3.34) Qξ→±∞≈ A±|ξ|−

12σ

(1± β

2ασ|ξ|

)e− iα

(log |ξ|± βα|ξ| )

The slow, |ξ|− 12σ , decay and the oscillatory e−

iα

log |ξ| pose challenges for computation. Large do-mains are needed for the solution to be small enough to be in this linear regime. At the same time,as α 0, the solution remains oscillatory in the tails, requiring adequate resolution. This taxedall of our numerical schemes.

There is also a massive asymmetry in the profile, with the constants behaving as,

A+≈4ε34α−

12 exp

−πα

+2

3

ε32

a

(3.35a)

A−≈(k + l)α1− 1σ

(1− 1

σ

)(3.35b)

and these approximations (along with the numerical data) are shown in Figure 11. A notablefeature is that as σ 1.04, A+ is approaching the limit of double precision, where the smallestnumber, ignoring subnormal numbers, is about 10−308. This suggests that there needs to be areconsideration of how to handle the boundary.

This slow decay of the profile also precluded the time dependent simulations of (3.24) fromexploring values of σ < 1.1. The reason for this is that dynamic rescaling solves the problem in theneighborhood of the singularity. As we computed the time dependent problem by psuedo-spectralmethods on the periodic domain [−ξmax, ξmax), it is necessary that the solution be, approximately,periodic. This would be satisfied if ξmax were large enough such that |u| ≈ 0 at ±ξmax.

The problem with periodicity is visible in the time independent computation in Figures 9 (a)and 10 (a). There, the modulus is still far from zero at the left edge of the visualized domain. Theactual domains on which these profiles were computed were much larger. Very similar problems wereencountered in [47], where the authors studied singularity formation in critical and supercriticalgKdV using similar numerical methods. It is a working hypothesis of mine that this asymmetry willbe generic for blowup solutions to equations which have the advective type nonlinearity, |u|2σux orupux.

One option is to avoid (3.34) by not rescaling space, and use a modest sized computationaldomain with Dirichlet boundary conditions together with an adaptive mesh redistribution strategy,[15,16,24,44,75]. In contrast to dynamic rescaling, which uniformly scales the position of the mesh

19

points about the singularity, adaptive mesh redistribution only focuses additional mesh points whereneeded. Indeed, indexing the nodes as

(3.36) xmin = x0 < x1 < x2 < . . . < xN < xN+1 = xmax

the goal would be to choose them such that

(3.37)

∫ xi+1

xi

√1 +

|ψx|2‖ψ‖2∞

dx = Constant in i.

Then, rather than try to solve (3.28), I would integrate

(3.38) iuτ + λ(τ)uxx + iλ(τ)|u|2σux = 0.

This bears some resemblance to (3.24), except that only time has been rescaled,

(3.39) τ =

∫ t

0

ds

λ2(s), λ(t) = ‖ux(·, 0)‖q

L2‖ux(·, t)‖−qL2

The purpose of rescaling time is that it allows for an integration out to τ = +∞ using uniformtime steps.

The recent inverse scattering results tell us there should be no singularity at σ = 1. I expect myfuture computations will reflect this as follows. For a given initial condition, as σ is reduced, thetime scale for norm growth will slow as σ 1. Since the norm growth is monotonic with α, it willalso vanish. This project should also better constrain (3.31).

3.2. Weak Turbulence. Weak turbulence is a phenomenon by which a system remains well-posedfor all time, but energy is continuously transferred to progressively higher frequencies. In has beenconjectured (see, for instance, [14, 29]), that the defocusing cubic NLS equation in 2D is weaklyturbulent,

(3.40) iut +∇2u− |u|2u = 0, u : T2 × R→ C.The equation is globally well-posed, with a priori bounds on H1 in terms of the data. Here, weakturbulence corresponds to

(3.41) lim supt→∞

‖u‖Hs = +∞, s > 1,

even though, at all finite times, the norms are finite. To understand weak turbulence, an associatedthe dynamical system was considered in [20],

(3.42) − ibj = −|bj |2bj + 2b2j−1bj + 2b2j+1bj , j = 1, . . . , N, b0 = bN+1 = 0,

and it was proven that there exists a particular solution that propagates mass from low to high j.As the relationship between (3.40) and (3.42) is that |bj |2 is roughly the spectral energy densityof u in a dyadic shell, this implied an energy cascade. In weakly turbulent systems, this would begeneric.

In [22], working with J. Collinander (UBC), J.L. Marzuola (UNC), and T. Oh (Edinburgh), Inumerically solved (3.42) using a high order Runge-Kutta integrator. Evidence for generic energytransfers was observed. But the Runge-Kutta method does not preserve the structure of theequation. Indeed, (3.42) is known to conserve `2 and a Hamiltonian:

(3.43) M =∑|bj |2, H =

∑12 |bj |4 − 2<(b2jb

2j−1),

with the symplectic structure ibj = ∂H/∂bj .As the interesting feature of (3.42) is its weakly turbulent behavior, it is important to recognize

that the energy transfer is quite slow. Indeed, the norms of (3.40) are predicted to grow, at most,at polynomial rates, requiring long simulations to observe the energy transfer, [13, 90, 91]. Thus,

20

we are faced with a computational challenge: large time steps in our integrator are desired toefficiently reach weakly turbulent time scales, but small time steps are needed to preserve accuracy.Additionally, as a Hamiltonian system, it would be appealing to have an integrator for (3.42) thatpreserved its geometric structure.

Working with A.D. Jones and W. Wilson (undergraduate students at Drexel), I explored thechoice of integrator, the impact of time step, and how they relate to the needs of different simula-tions, [46]. This included formulating conservative schemes, preserving one of the two invariants,and comparing them against RK4, along with projection based methods that preserved both in-variants, [39]. One of the conservative schemes for (3.42), was a modified midpoint method, withbj,n ≈ bj(tn),

− i

∆t(bj,n+1 − bj,n) = −|b|2j,n+1/2bj,n+1/2 + 2bj,n+1/2

(b2j+1,n+1/2 + b2j−1,n+1/2

).(3.44)

This exactly preserves M , up to root finding error. Another, similar, scheme exactly preserves theenergy, H. Since M conserving scheme gives a priori bounds on |bj,n| for all time, we could proveconvergence:

Theorem 3.4 (Convergence of Mass Preserving Integrators, [46]). For a second order M conservingscheme, there exist ∆t?, K > 0 and C > 0, such that for all ∆t ≤ ∆t? and all n, the error is

(3.45) ‖bn − b(tn)‖2 ≤ C(eKtn − 1)∆t2

Such a result escaped us for the H preserving methods as we did not have analogous a prioriestimates.

For a given initial value problem, integrated out to a modest tmax, smaller time steps with higherorder methods will generally give better results than larger time steps with lower order methods.But for weak turbulence, the quantities of interest are the ensemble averages of observables at verylarge times. Thus, it may be acceptable to use larger time steps, hoping that the errors “averageout.” We found this to hold in our numerical experiments, where we observed: (i) for ensembleaverages of random initial conditions, the the impact of ∆t was modest on conservative methods;(ii) at large time step RK4 performed inconsistently in comparison to the conservative methods.These results are reproduced in Figures 12 from [46]. The norm that is used to measure energytransfer is

(3.46) ‖b(t)‖2hs =

N∑j=1

2(s−1)j |bj(t)|2.

Samples from the statistical ensemble of initial conditions were generated with

(3.47) b(k)j (0) =

1

4j−1· expiθ(k)

j , j = 1, . . . , N, θ(k)j ∼ U(0, 2π).

The conclusion is thus that, for studying the statistical behavior of weak turbulence, it is possibleto use a large fixed time step with a conservative method.

21

0 200 400 600 800 1000t

100

101

102

103

104

105

〈‖b‖ h

4〉

Mass Preserving Integrator

∆t = 0.1

∆t = 0.05

∆t = 0.025

∆t = 0.0125

∆t = 0.00625

(a)

0 20000 40000 60000 80000 100000t

100

102

104

106

108

1010

1012

1014

1016

1018

1020

1022

〈‖b‖ h

4〉

∆t = 0.1

Mass Preserving IntegratorEnergy Preserving IntegratorTrapezoidal MethodRK4Implicit Midpoint

(b)

Figure 12. Figure (a) reveals a robustness for a mass conserving scheme (3.44)with respect to ∆t, for the statistical ensemble. Figure (b) shows the tendency forenergy transfer to occur with respect to (3.46). Notice that with this larger timestep, RK4 deviates from the rest of the integrators.

22

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