Edinburgh Meeting Notes 3

Galina Perelman: 2 soliton collision in NLS

$$ i \partial\_t \psi = - \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R $$ where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$

This family of equations has solitary wave solutions $$ e^{i \theta(x,t) \phi (x - b(t), E)} $$ where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, …

If I set $\epsilon^2 = E$ and write $\phi(y, \epsilon^2) = \epsilon \hat{\phi}(\epsilon, \epsilon).$ We have then that $\hat{\phi}(z, \epsilon) = \phi_0 (z) + O(\epsilon^2)$ where $\phi_0$ is the standard soliton for cubic NLS. A calculation shows that $$| \phi(\cdot, \epsilon^2) |_{H^1} = O(\epsilon^{1/2}).$$ Let’s collect the parameters $\sigma = (\beta, E, b, v) \in R^4.$

The question I’d like to address:


Perturbative regime: $$\epsilon^2 = E_1 \ll 1, E_0 \thicksim 1, v_1 \thicksim 1.$$

Collision Scenario:


She draws a picutre:


$s = s(\frac{v_1}{2}), r = r (\frac{v_1}{2})$ where $s(k), r(k)$ are the translation and reflection coefficients of the linearized operator corresponding to $w(\cdot, \sigma_0 (t))$. Here we have $|s|^2(k) + |r|^2 (k) =1$. The only trace of nonlinearity appears in the phase.





$$ i {\bf{f}}_t = L(E) {\bf{f}}. $$

Here ${\bf{f}}$ is a (column) vector $(f, \overline{f})$. $$ L(E)= (-\partial_y^2 + E) \sigma_3 + V(E). $$ Here $\sigma_3$ is the Pauli matrix and $V$ is a certain matrix involving $V_1 = F(\phi^2) + F’ (\phi^2) \phi^2$ and $V_2 = F’ (\phi^2) \phi^2$.

She draws a spectral plane. Essential spectrum along real line in region $|x| > E$ and some eigenvalues drawn as x’s inside the gap and one above and below the real line on the imaginary axis. 0 is an eigenvalue. We have two explicit eigenfunctions $\xi_0$ and $\xi_1$.


$$\sigma(L(E)) \subset R, {\mbox{dim}} M(E) = 4 \iff \frac{d}{dE} | \phi(E) |_2^2 > 0.$$

These conditions imply the orbital stability of $\Phi$.



(H1): $$\frac{d}{dE} | \phi(E) |2^2 |{E=E_0} > 0$$




Theorem: For $\epsilon^{-1-\delta} \leq t \leq \delta \epsilon^{-2} | \ln \epsilon |$

$$ \psi (t) = w (\cdot, \sigma(t)) + \psi_+ (t) + \psi_{-} (t) + h(t)$$

is expressed as an explicit phase times a function $S^{\pm}$ which solves cubic NLS emerging from data built using thre reflection, transmission coefficients and $\phi_0 (y)$. 3. error estimates in terms of $\epsilon.$

Edriss Titi: Loss of smoothness in 3d Euler Equations

(joint work with Claude Bardos)


  1. Background
    • Euler
    • Classical
    • Nonuniqueness: De Lellis - Sh…
  2. Shear flow
  3. Vortex sheets induced by 3d shear flows
    • Examples
    • Differences between 2d and 3d Kelvin-Helmholtz problems
    • Comments on numerics

Euler equations


Vorticity stretching term distinguishes 2d and 3d.

Classical Wellposedness:





Shear flows:

$$u(x,t) = (u_1 (x_2), 0, u_3 (x_1 - t u_1 (x_2))).$$


This example due to DiPerna-Majda (1987).







This family of solutions is compactly supported in space and time.

Other spaces and optimal spaces:


Weak limit of oscillating initial data:

DiPerna-Majda example…



Numerical investigation of blowup for the 3d Euler


Question: Does the soluton of the following PDE blow up? $$ \partial_t u - \nu \Delta u = |\nabla u |^4? $$



Benoit Grébert: Hamiltonian Interpolation for Approximation of PDEs.

(joint work [Grébert-Faou] with Erwan Faou)


Take a PDE with solution u. Consider a numerical approximation $u^n$ built with a symplectic integrator which approximates $u(nh)$. We build a hamiltonian $H_h$ such that $$u^n = \Phi_{Hh}^{nh}(u_0) + very ~small.$$




  1. Finite dimensional Context (ODE)
  2. PDE Context
  3. Ideas of the proof (time permitting)

Finite Dimensional Context

We go back to Moser’s theorem. A discrete symplectic map close to the identity can be approximated by a Hamiltonian flow. Consider an analytic symplectic map $$ R^{2n} \ni (p,q) \longmapsto \Psi(p,q) \in R^{2n} $$ with $\Psi = Id + O(\epsilon)$. Then $\exists~ H_\epsilon$ such that $$\Psi = \Phi_{H\epsilon}^\epsilon + O(e^{-\frac{1}{c\epsilon}}).$$ ([Moser 1968], [Benettin-Giorgilli 1994])

Numerical Context: Suppose I have a Hamiltonian ODE system $$ (\dot{p}, \dot{q}) = X_H (p,q) $$ and an associated numerical discrete-time-step symplectic integrator $$ (p_n, q_n)= \Psi_h^n (p_0, q_0). $$ We then have that $\Psi_{h} = \Phi_{Hh} + O(e^{-1/ch}).$ We obtain that $H_h (p_n, q_n) = H_h (p_0, q_0) + n e^{-1/ch}$. So, we are observing that the modified energy is essentially conserved for exponentially long times.

Backward Error Analysis

PDE Context


Expand $u$ to get $$ u = \sum \xi_j e^{ijx}, ~ {\overline{u}}= \sum \eta_j e^{-ijx}. $$ We can then write, for each $j \in Z^d$, $$ {\dot{\xi}} = -i \frac{\partial H}{\partial \eta} $$ $$ {\dot{\eta}} = i \frac{\partial H}{\partial \xi}. $$ For the cubic NLS case, we obtain $$ H = \sum |j|^2 \xi_j \eta_j + \sum^* \xi_{k_1} \xi_{k_2} \eta_{l_1} \eta_{l_2} $$ where $\sum^*$ is the sum over all the parameters subject to the constraint $k_1 + k_2 = l_1 + l_2$.


Splitting Method: $$\Phi_{P+H0 } \thicksim \Phi_p^h \circ \Phi_{H0}^h =^? \Phi_{Hh}^h.$$

First naive idea: Use the Baker-Campbell-Haussdorf formula. We can then expand as a Lie series… to write $$\Phi_p^h \circ \Phi_{H0}^h = e^{h\mathcal{L}p}e^{h\mathcal{L}H0} = e^{h\mathcal{Hh}} $$ with $H_h = H_0 + P + \frac{h}{2}{ P, H_0 } + \dots. $$






$$ | \Phi_p^h \circ \Phi_{A_0}^1 (\xi, \eta) - \Phi_{Hh}^h(\xi, \eta) |_{l^1} \leq h^{N+1} (cN)^N $$


So, assuming that the numerical trajectory is bounded in $l^1$ (as opposed to the stronger claim that it is bounded in $H^k$ for $k$ large) then $$ H_h (u^n) = H_h (u_0) + Cn h^{N+1}. $$

Of course, I have to explain: what is $N$? This is related to a regularization condition. We know that $N = \frac{r-2}{r_0 - 2}$ where $r_0$ is the degree of $P$ (so 4 for cubic NLS). The parameter $r$ is determined by the condition: $ \forall ~ j = 1, \dots, r$ and for any $j$-tuple of integers $(k_1, \dots, k_j) \in Z^d$, we have $$|\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj}| \leq 2 \pi.$$

CFL: $|\lambda_k | \leq C$.


For cubic NLS, we end up obtaining $$Hh = \frac{1}{h} A_0 + Z_1 + h Z_2 + \dots $$ where $$ Z_1 = \sum^* \frac{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})}}{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})} - 1}. $$ You can now see how the zero divisor issue emerges and is resolved.