Week 11 Lecture 2: Self-similar blowup - Enclosing $Q_\infty$
We continue our study of the paper Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations.
In last lecture we looked at enclosing $Q_0$, the solution the ODE satisfying the boundary conditions at zero. Today we will look at enclosing $Q_\infty$, the solution satisfying the boundary conditions at infinity.
Recall that $Q_\infty$ is a solution to the ODE
\[(1 - i\epsilon)\left(Q_{\infty}'' + \frac{d - 1}{\xi}Q_{\infty}'\right) + i\kappa\xi Q_{\infty}' + i \frac{\kappa}{\sigma}Q_{\infty} - \omega Q_{\infty} + (1 + i\delta)|Q_{\infty}|^{2\sigma}Q_{\infty} = 0\]
satisfying appropriate boundary conditions at infinity (which we will get to).
Linear equation
We expect $Q_\infty$ to go to zero at infinity. Hence the non-linearity, $(1 + i\delta)|Q_{\infty}|^{2\sigma}Q_{\infty}$, becomes negligible in the limit and it suffices to study the linear equation
\[(1 - i\epsilon)\left(Q_{\infty}'' + \frac{d - 1}{\xi}Q_{\infty}'\right) + i\kappa\xi Q_{\infty}' + i \frac{\kappa}{\sigma}Q_{\infty} - \omega Q_{\infty} = 0.\]
This is a second order linear ODE with (after multiplication by $\xi$) polynomial coefficients. This type of ODEs are well studied in the literature. If one makes the change of coordinates
\[a = \frac{1}{2}\left(\frac{1}{\sigma} + i \frac{\omega}{\kappa}\right),\quad b = \frac{d}{2},\quad c = \frac{-i \kappa}{2(1 - i\epsilon)},\quad z = c\xi^2,\]
then one can write the equation as
\[zw'' + (b - z)w' - aw = 0,\]
which is Kummer's equation. One of the solutions to Kummer's equation is the confluent hypergeometric function, $U(a, b, z)$, and the other solution is a scaled version given by $V(a, b, z) = e^z U(b - a, b, -z)$. Since we are dealing with a linear second order ODE, all solutions are given by linear combinations of these two.
To understand the asymptotic behavior of $Q_\infty$ we will have to understand the asymptotic behavior of $U$ and $V$. From the asymptotic expansion of $U$ one gets the asymptotic behavior
\[U(a, b, z) \approx z^{-a}.\]
For $V$ this gives us
\[V(a, b, z) \approx e^{z}(-z)^{-(b - a)}.\]
In terms of solutions to our original linear ODE we let
\[P(\xi) = U(a, b, c\xi^2)\]
and
\[E(\xi) = e^{c\xi^2}U(b - a, b, -c\xi^2).\]
The solutions are then given by linear combinations of $P$ and $E$.
To get a better feeling for the asymptotic behavior. Let us look at the case when $d = 3$, $\sigma = 1$, $\epsilon = 0$, $\omega = 1$ and $\kappa$ is positive. We then have
\[a = \frac{1}{2}\left(1 + i \frac{1}{\kappa}\right),\quad b = \frac{3}{2},\quad c = -i\frac{\kappa}{2}.\]
This gives us
\[P(\xi) \approx z^{-a} = (c\xi^2)^{-a} \sim \xi^{-2a} = \xi^{-1 - i\frac{1}{\kappa}}\]
and
\[E(\xi) \approx e^{z}(-z)^{-(b - a)} = e^{c\xi^2}(-c\xi^2)^{-(b - a)} \sim e^{c\xi^2}\xi^{-2(b - a)} = e^{-i\frac{\kappa}{2}\xi^2}\xi^{-2 + i\frac{1}{\kappa}}.\]
We can see that both $P$ and $E$ decay at infinity. However, $E$ has exponentially increasing oscillations, which in this case stops it from having finite energy. Since any solution to the linear ODE is given by a linear combination of $P$ and $E$, the only way for this solution to have finite energy is for the coefficient in front of $E$ to be zero. This gives us our first approximation to $Q_\infty$, namely
\[Q_\infty(\xi) \approx \gamma P(\xi)\]
for some $\gamma \in \mathbb{C}$. This is, of course, not an exact solution to the ODE, it neglects the non-linearity. The next step is therefore to add a correction to this approximation to make it an exact solution.
Solution to full ODE
We can get an equation for the solution to the full ODE, including the non-linear term, using the classical method of variation of parameters. For this one treats the non-linearity as the right-hand side in a non-homogeneous equation. Making the ansatz
\[Q_\infty(\xi) = c_1(\xi)P(\xi) + c_2(\xi)E(\xi)\]
one gets that $c_1$ and $c_2$ should satisfy
\[c_1' = (1 + i\delta)|Q_\infty(\xi)|^{2\sigma}Q_\infty(\xi) \frac{E(\xi)}{(1 - i\epsilon)W(\xi)} \text{ and } c_2' = -(1 + i\delta)|Q_\infty(\xi)|^{2\sigma}Q_\infty(\xi) \frac{P(\xi)}{(1 - i\epsilon)W(\xi)},\]
where $W$ is the Wronskian associated with $P$ and $E$. To simplify the notation we let
\[J_E(\xi) = (1 + i\delta)\frac{E(\xi)}{(1 - i\epsilon)W(\xi)} \text{ and } J_P(\xi) = (1 + i\delta)\frac{P(\xi)}{(1 - i\epsilon)W(\xi)}.\]
To satisfy the boundary conditions we want $c_1$ to have a non-zero limit at infinity and $c_2$ to go to zero. It is therefore natural to use the representations
\[\begin{align*} c_1(\xi) &= \gamma + \int_{\xi_1}^\xi J_E(\eta)|Q_\infty(\eta)|^{2\sigma}Q_\infty(\eta)\, d\eta,\\ c_2(\xi) &= \int_{\xi}^\infty J_P(\eta)|Q_\infty(\eta)|^{2\sigma}Q_\infty(\eta)\, d\eta. \end{align*}\]
We let $I_E(\xi)$ and $I_P(\xi)$ denote the integrals with $J_E$ and $J_P$ respectively. This gives us the following equation for $Q_\infty$
\[Q_\infty(\xi) = \gamma P(\xi) + P(\xi)I_E(\xi) + E(\xi)I_P(\xi).\]
Note that both $I_E$ and $I_P$ themselves depend on the solution $Q_\infty$, so this doesn't directly give us an expression for $Q_\infty$. Instead, we introduce the operator
\[T(Q_\infty) = \gamma P(\xi) + P(\xi)I_E(\xi) + E(\xi)I_P(\xi).\]
We are then looking for fixed points of this operator. The first step in computing enclosures of $Q_\infty$ is therefore to prove the existence of a fixed point for this operator, and give bounds for it.
Existence of a fixed point
To prove the existence of a fixed point for $T$ we need to be to control the asymptotic behavior of $P$, $E$, $J_P$, $J_E$, $I_P$ and $I_E$. For $P$ and $E$ we have the following lemma
Let $\xi_1 > 1$. Under certain assumptions on $a$, $b$, $c$ and $\xi_1$ we have the bounds
\[\begin{align*} |P(\xi)| &\leq C_{P}\xi^{-\frac{1}{\sigma}},\\ |E(\xi)| &\leq C_{E}e^{\operatorname{Re}(c)\xi^{2}}\xi^{\frac{1}{\sigma} - d},\\ |J_{P}(\xi)| &\leq C_{J_{P}}e^{-\operatorname{Re}(c)\xi^{2}}\xi^{-\frac{1}{\sigma} + d - 1},\\ |J_{E}(\xi)| &\leq C_{J_{E}}\xi^{\frac{1}{\sigma} - 1} \end{align*}\]
for all $\xi \geq \xi_1$. The constants are explicitly computable, depending only on $a$, $b$, $c$ and $\xi_1$.
The proof of this is based on the asymptotic expansion of $U$, with explicit bounds for the remainder term, see Fungrim. The same expansion is used in Flint for the evaluation of $U$.
To get bounds for $I_P$ and $I_E$ we first need to decide on in which space we are searching for a fixed point. Since we expect the asymptotic behavior of $Q_\infty$ to be determined by $P$, which decays as $\xi^{-\frac{1}{\sigma}}$, and we are working on the interval $[\xi_1, \infty)$ it would be natural to use the norm
\[\|u\| = \sup_{\xi \geq \xi_1} \xi^{\frac{1}{\sigma}}|u(\xi)|.\]
However, when taking derivatives in $\kappa$ we get extra logarithmic factors and it is therefore better to use the norm
\[\|u\|_v = \sup_{\xi \geq \xi_1} \xi^{\frac{1}{\sigma} - v}|u(\xi)|.\]
for some small $v > 0$.
We can then bound $I_P$ and $I_E$ in terms of this norm.
Assume that $\xi_1 > 1$, $v \geq 0$ and $\operatorname{Re}(c) \geq 0$ and that the following inequalities are satisfied
\[(2\sigma + 1)v - 2 < 0 \text{ and } (2\sigma + 1)v - \frac{2}{\sigma} + d - 2 < 0.\]
Then, for $\xi \geq \xi_1$ we have the following bounds
\[\begin{align*} |I_{P}(\xi)| &\leq C_{I_{P}}\|Q_{\infty}\|_{v}^{2\sigma + 1}e^{-\operatorname{Re}(c)\xi^{2}}\xi^{(2\sigma + 1)v - \frac{2}{\sigma} + d - 2},\\ |I_{E}(\xi)| &\leq C_{I_{E}}\|Q_{\infty}\|_{v}^{2\sigma + 1}\xi_{1}^{(2\sigma + 1)v - 2}, \end{align*}\]
where
\[C_{I_P} = \frac{C_{J_P}}{|(2\sigma + 1)v - \frac{2}{\sigma} + d - 2|} \text{ and } C_{I_E} = \frac{C_{J_E}}{|(2\sigma + 1)v - 2|}.\]
We give the proof for $I_{E}$, the proof for $I_{P}$ is similar. To begin with we note that
\[|I_{E}(\xi)| \leq \int_{\xi_1}^\infty |J_E(\eta)||Q_\infty(\eta)|^{2\sigma + 1}\, d\eta.\]
From the above lemma we have
\[|J_E(\eta)| \leq C_{J_{E}}\eta^{\frac{1}{\sigma} - 1}.\]
Furthermore, we have
\[|Q_\infty(\eta)| \leq \|Q_\infty\|_v \eta^{-\frac{1}{\sigma} + v},\]
giving us
\[|Q_\infty(\eta)|^{2\sigma + 1} \leq \|Q_\infty\|_v^{2\sigma + 1} \eta^{(2\sigma + 1)v - \frac{1}{\sigma} - 2}.\]
Inserted into the integral we get
\[|I_{E}(\xi)| \leq C_{J_E}\|Q_\infty\|_v^{2\sigma + 1}\int_{\xi_1}^\infty \eta^{(2\sigma + 1)v - 3}\, d\eta.\]
The integral can be bounded by
\[\int_{\xi_1}^\infty \eta^{(2\sigma + 1)v - 3}\, d\eta \leq \frac{1}{|(2\sigma + 1)v - 2|}\xi_1^{(2\sigma + 1)v - 2},\]
giving us the result.
From this it is straightforward to prove the following bound for $T$.
Under the assumptions of the previous lemma, we have
\[\|T(Q_\infty)\|_v \leq C_P|\gamma|\xi_1^{-v} + C_T\xi_1^{-2 + 2\sigma v}\|Q_\infty\|_v^{2\sigma + 1}\]
and
\[\|T(u) - T(v)\|_v \leq M_\sigma C_T \|u - v\|_v (\|u\|_v^{2\sigma} + \|v\|_v^{2\sigma}),\]
with $C_T = C_PC_{I_E} + C_EC_{I_P}$ and a constant $M_\sigma$ depending only on $\sigma$.
This gives us all the ingredients to set up a fixed point problem. The above estimates shows that $T$ is a contraction of the ball $B_\rho = \{u: \|u\|_v \leq \rho\}$ into itself if $\rho$ satisfies
\[\begin{align} C_P|\gamma|\xi_1^{-v} + C_T\xi_1^{-2 + 2\sigma v}\|Q_\infty\|_v^{2\sigma + 1} &\leq \rho,\\ 2M_\sigma C_T\xi_1^{-2 + 2\sigma v}\rho^{2\sigma} &< 1. \end{align}\]
It then follows from the Banach fixed point theorem that $T$ has a unique fixed point in the ball $B_\rho$. This gives us both the existence of a solution and also bounds on the $\|\cdot\|_v$ norm of the solution! However, to be able to compute good enclosures of $Q_\infty$, just having bounds on its norm is not enough!
Enclosures for fixed point
A bound for the norm of $Q_\infty$ is only the first step. Next we need to use this bound to get better enclosures.
From the fixed point equation we have
\[Q_\infty(\xi) = \gamma P(\xi) + P(\xi)I_E(\xi) + E(\xi)I_P(\xi).\]
In practice we are only interested in computing the value at $\xi = \xi_1$. We can note that $I_E(\xi_1) = 0$, hence
\[Q_\infty(\xi_1) = \gamma P(\xi_1) + E(\xi_1)I_P(\xi_1).\]
We can compute enclosures of $P$ and $E$ using the implementation of $U$ in Flint. For $I_P$ we can combine the above lemma with the bound for $\|Q_\infty\|_v$ to get a bound for $|I_P(\xi_1)|$.
With the above approach we get a somewhat decent enclosure of $Q_\infty(\xi_1)$. However, for the proof to go through we need better enclosures than that. These improved bounds are based on integration by parts in $I_P$, which gives fairly lengthy calculations.
In addition, we also need bounds for the derivatives, both with respect to $\xi$ and with respect to $\gamma$ and $\kappa$. In the end we have to compute bounds for
\[\begin{align*} Q'(\xi) &= \gamma P'(\xi) + P'(\xi)I_{E}(\xi) + P(\xi)I_{E}'(\xi) + E'(\xi)I_{P}(\xi) + E(\xi)I_{P}'(\xi),\\ Q_{\gamma}(\xi) &= P(\xi) + P(\xi)I_{E,\gamma}(\xi) + E(\xi)I_{P,\gamma}(\xi),\\ Q_{\gamma}'(\xi) &= P'(\xi) + P'(\xi)I_{E,\gamma}(\xi) + P(\xi)I_{E,\gamma}'(\xi) + E'(\xi)I_{P,\gamma}(\xi) + E(\xi)I_{P,\gamma}'(\xi),\\ Q_{\kappa}(\xi) &= \gamma P_{\kappa}(\xi) + P_{\kappa}(\xi)I_{E}(\xi) + P(\xi)I_{E,\kappa}(\xi)\\ &\quad+ E_{\kappa}(\xi)I_{P}(\xi) + E(\xi)I_{P,\kappa}(\xi),\\ Q_{\kappa}'(\xi) &= \gamma P_{\kappa}'(\xi)\\ &\quad+ P_{\kappa}'(\xi)I_{E}(\xi) + P_{\kappa}(\xi)I_{E}'(\xi) + P'(\xi)I_{E,\kappa}(\xi) + P(\xi)I_{E,\kappa}'(\xi)\\ &\quad+ E_{\kappa}'(\xi)I_{P}(\xi) + E'(\xi)I_{P,\kappa}(\xi) + E_{\kappa}(\xi)I_{P}'(\xi) + E(\xi)I_{P,\kappa}'(\xi). \end{align*}\]
Some of these are relatively easy, others require more work. That is however outside the scope of these notes.