# 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:

Question: As $t \rightarrow -\infty$, suppose that $\psi(t) = w(\cdot, \sigma_0 (t)) + w(\cdot, \sigma_1 (t)) + o_{H^1} (1)$. Because of the galilean invariance we can arrange so that $\sigma_0$ does not move and we assume that $v_1 > 0$. So, we can arrange this data to have completely decoupled solitons as $t \rightarrow - \infty$. The question is then to understand the soliton collision and also what happens afterwards.

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

Collision Scenario:

1. $w(\cdot, \sigma_0 (t))$ is ‘preserved’.
2. $w(\cdot, \sigma_1 (t))$ splits into two outgoing waves of the cubic NLS. The splitting is controlled by the linearized operator associated to the large soliton $w_{\sigma_0}$.
Collision: $|t| \lesssim \epsilon^{-1-\delta}, ~ \delta > 0$. pre-interaction: $t leq - \epsilon^{-1-\delta}$ post-interaction: $t leq - \epsilon^{-1-\delta}$

She draws a picutre:

Long wide soliton to the left of a big soliton at the origin before the collision. After the collision the small soliton splits into two waves, one moving left and one moving right. The big soliton at the origin is drawn not centered at the origin.

$s = s(\frac{v_1}{2}), r = r (\frac{v_1}{2})$ 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.

This phenomena has been observed before by Holmer-Mazuola-Zworski and earlier by physicists. H-M-Z conisdered the cubic NLS with an external delta potential. For small incoming solitons, they have observed the small soliton splitting caused by the Dirac function potential.

Hypotheses:

(H0): $F \in C^\infty, F(\xi) = - 2 \xi + O(\xi^2), \xi \rightarrow 0.$ $F(\xi ) \geq - C\xi^q, C>0, q<2, \xi \geq 1$. (GWP in $H^1$) $\exists !$ ground state.

Linearization around $w(x, \sigma(t)) = e^{i\theta} \phi(x - b(t), E)$. We substitute $\psi = w + f$ and expand to obtain the following equation for $f$:

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

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 $\xi_1$.

$M(E)$ is the generalzied null space of $L(E)$. We have the following equivalence:

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

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

$Lf = \lambda f, ~\lambda \geq E, \lambda = E + k^2, ~ k \in R$. If $k^2 + I \notin \sigma_p (L(E))$ then $\exists ~! ~ f(x,k) = s(k) e^{i k x} (1, 0)^t + O(e^{-\gamma x})$ as $x\rightarrow + \infty, ~ \gamma > 0$ and $f(x,k) = e^{ikx} (1,0)^t + r(k) e^{-ikx}(1,0)^t + O(e^{\gamma x}), x \rightarrow - \infty$.

$w(x,\sigma, t), ~ j=0,1$ normalized as before.

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

(H2): $\epsilon^2 = E_1$ sufficiently small

(H3): $M(E + \frac{v_1^2}{4}) \notin \sigma_p (L(E_0))$ (Nobody knows how to prove no embedded eignevalues.)

Proposition: $\exists ~! ~ \psi \in C(R, H^1)$ such that ….

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

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

1. $\sigma(t) = (\beta(t), E_0, b(t), v_0), ~V_0 = \epsilon \kappa$ where $\kappa$ is an explicit constant and $$|\beta(t) - \beta_0 (t)|, |b(t) - v_0 t| \leq C \epsilon^2 t.$$