Generalized KdV Equation
Solitons
Results
Strategy of Proof
1. Weinstein’s Stability Result
2. Increment of $E_N (t)$:

Upper bounds on the orbital instability of the generalized KdV equations

Generalized KdV Equation ↩

$$ u_t + u_{xxx} + (u^p )_x = 0,~p=2,3,4.$$

For $p \geq 5$, the soliton is unstable. In fact, for $p=5$, there is finite time blowup.
Problem: Prove blowup for some $p > 5$.

Solitons ↩

The gKdVp equations have traveling wave solutions of the form $$ Q_c (t,x) = \psi_c (x -ct)$$ where $c>0$ is the speed, and $\psi_c > 0$, $\psi_c \in H^1$ and satisfies $$ \psi_{c, xx} + \psi^p - c \psi_c = 0. $$ These solutions are also called solitons. There is an explicit formula for $\psi_c$. In this talk, we’ll only discuss the solitons with speed $c=1$.

Notation: $Q = Q_1$, soliton solution of gKdVp. We will write $\psi = \psi_1$ as the ground state solution of $\psi_{xx} + \psi^p - \psi = 0$.

The ground state curve: $\Sigma = { \psi(x - x_0): x_0 \in R}$.

There is this idea called the Soliton Resolution Conjecture which predicts that the long time behavior of all solutions should be described in terms of solitons. This circle of ideas spawns various questions.

Natural Questions:

Are the solutions stable in $H^s$? For which $s$? For which $p$?
- [W]: Solitons are orbitally stable in $H^1$. This means that solutions which begin close to $\Sigma$ in $H^1$ remain close to $\Sigma$ for all time.
- [MM1, MM2]: Solitons are asymptotically stable in $H^1$. This means that there is some error term which shrinks to zero in some appropriate sense.
- These results apply in cases $p=2,3,4$ in $H^1$.

Below $H^1$:

[MV] The KdV solitons ($p=2$) are orbitally and asymptotically stable in $L^2$.
[RS] Perturbations of the the KdV solitons ($p=2$) are polynomially-in-time stable for $0 \leq s \leq 1$.

Results ↩

Theorem (Pigott 2010):

Let $p = 2$ or $p=4$ and let $0 \leq s <1$. If $u_0 \in H^s$ with $dist_{H^s}(u_0, \Sigma) = \sigma \ll 1$ and $u(t)$ solves gKdVp with $u(0) = u_0$, then $dist_{H^s} (u(t), \Sigma) \lesssim t^{1-s + \epsilon} \sigma$ for all $t \ll \sigma^{-1/)1-s+\epsilon}$.
Let $p=3$ and let $1/4 \leq s < 1$. Then the same thing holds.

Remark: The $p=3$ (mKdV) is known to be ill-posed in $H^s$ for $s < 1/4$.

Comments:

This result agrees with [RS] for $p=2$ and extends the work to $p = 3,4$.
The CKSTT work on cubic NLS on R below $H^1$ where the same bound is obtained for the cubic NLS soliton on the line.

These works rely on the $I$-method and a suitable almost conservation law. My proof relies on these inputs as well but is perhaps a bit more streamlined.

Strategy of Proof ↩

Use the $I$-method to derive an almost conservation law which can be iterated in time. Before we do that, let’s reconsider Weinstein’s stability result

Weinstein’s Stability Result ↩

Conserved Quantities
- Mass: $M(u(t)) = \int |u(t,x)|^2 dx = M(u_0)$.
- Hamiltonian: $H(u(t)) = \int (u_x^2 (t) - \frac{2}{p+1}u^{p+1}) dx = H(u_0)$.

Lyapunov Functional: ${\cal{L}} (u(t)) = H(u(t)) + M(u(t))$.

The function $\mathcal{L} (u(t))$ is non-increasing in time. (Actually, it is conserved.)
$\mathcal{L}$ is translation invariant.
Elements of $\Sigma$ are local minimizers for $\cal{L}$.
Weinstein’s Lemma: If $\psi \in \Sigma$ and $v \in H^1$ and small in $H^1$ and $\langle v, (\psi^p)x \rangle = 0$ then $\cal{L}(\psi + v) = \mathcal{L} (\psi) \sim \| v \|_{H^1}^2.$

Problem: We are in $H^s$ with $s<1$ so we can’t use $\mathcal{L}$ directly.

Idea: Smooth out the flow with a regularizing operator. Let $ I_N: H^s \rightarrow H^1$ be defined as a Fourier multiplier operator ${\hat{If}} (\xi) = m(\xi) \hat{f}(\xi)$ where $m$ is the usual suspect with change of behaviour at frequency $N$.

Question from Robert McCann: Is this the $I$ operator for the $I$-method? Should I think of this as an approximate identity?

Answers: Yes, it is the same $I$ and it is indeed a smoothing operator that can be thought of a convolution.

Decomposition: Decompose our solution $$ u(t,x) = Q(t, x - x_0 (t)) + w(t,x) $$ where $t \rightarrow x_0 (t)$ is a translation parameter.

Almost Conserved Quantity: We want to build a functional $E_N (t)$ that satisfies some nice properties. We want it to:

$\| D^s w(t) \|^2 \lesssim E_N (t) - \mathcal{L}(Q).$
If $t_0 \in R$ and $0 \leq \tilde{\sigma} \ll 1$ is such that $|E_N(t) - \mathcal{L} (Q) | \lesssim \tilde{\sigma}^2$ then there is a $\delta > 0$ such that $$ \sup_{t \in [t_0, t_0 +\delta]} E_N (t) - E_N (t_0) \lesssim N^{-\alpha + \epsilon} {\tilde{\sigma}}^2 $$ where $\alpha > 0$.

Remark: From iterating item 2, we obtain the distance claim of the theorem but with exponent $\alpha^{-1} (1-s)$ so we want to prove this with $\alpha = 1$.

Lemma: If $w \in H^s$ with $H^s$ size much smaller than $N^{s-1}$ ($N$ taken to be sufficiently large) then $u = \psi + w$ where $\psi \in \Sigma, ~ \langle w, I_N (\psi^p)_x \rangle = 0$….

…..a bit bogged down rephrasing the lemma……what should have been said here?

We choose to define $E_N (t) = \mathcal{L} ( Q(t, x - x_0 (t)) + I_N w(t))$, with $x_0 (t)$ chosen to ensure the orthogonality condition.

Question: Are there any issues selecting the translation parameter at low regularity?

Lemma: (Modified LWP Spacetime Control in $X^{s,b}$) For $I_N w \in H^1$ of some size $\sigma$ there exists a $\delta >0$ (depending upon the initial modified $H^1$ size) such that we have good control of $I_N w \in X^{1, 1/2+}$ localized to a $\delta$-sized interval with upper bound linear in $\sigma$.

Increment of $E_N (t)$: ↩

The increment of $E_N$ over a time interval of size $\delta$ starting at time $t_0$ is expressed as a sum of multilinear spacetime expressions involving $w$. These expressions are studied using the multilinear analysis toolbox to get the needed decay of $N^{-\alpha}.$

…some conversation about the bilinear estimate and the fact that it is more involved than Hölder-type steps in the recovery of the derivative.

Brian had a handout with references on one side and some representative multilinear expressions on the other.

Table of Contents