# A Curious NLS Simulation
Consider the following NLS equation with gain

$$i\partial_t\psi=-\frac{1}{2}\partial_x^2\psi-|\psi|^2\psi-iV(t)\psi
$$
where $x\in\mathbb{R},\ t\in[0,T],$ with a 4 parameter family of NLS soliton initial conditions given by
$$
\psi(x,0)=2\lambda{\rm sech}\left(2\lambda(x-\xi)\right)e^{i(cx+\theta)}.
$$
The time-dependent potential is a negative semi-definite bump in time, for example, like the one in the following figure:
For reasons that will become apparent later on in this note, the way I am building $V(t)$ is through a four parameter family of sigmoids
$$
D(t)=A\mathrm{tanh}(w(t-\tau))+D_0
$$
so that
$$
V(t)=\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{ln}\left(D(t)\right)
$$
The above $V(t)$ corresponds to $A=-60,\ w=1/32,\ \tau=80$ and $D_0=-75.$ Choosing soliton parameters $\theta=\frac{\pi}{4}, \ c=.01,$ and $\xi=0,$ we observe the following from a numerical simulation:
This picture is reminiscent of pictures one would see when performing a semi-classical study of a Schrodinger Equation. For example, this picture is taken from page 180 of P.D. Miller's asymptotic analysis book where he analyzes the dynamics of a free particle Schrodinger wave function via the method of stationary phase.

However, nonlinear interactions seem to complicate the dynamics. Indeed, a similar problem was studied by Biondini and Oregero two years ago. In their problem, discussed here https://arxiv.org/abs/2005.12708,
there is no time-dependent scattering, but instead, periodic solutions are evolved according to an NLS, with Planck's constant $\hbar$ reinstated. That is, using scalings consistent with their work, they solve
$$
i\varepsilon\partial_t\psi=-\varepsilon^2\partial_x^2\psi-2|\psi|^2\psi
$$
Using the initial condition
$$
\psi(x,0)=e^{-\sin^2(x)}.
$$
and setting $\varepsilon=0.026,$ B. and O. observe the following:
A potential project is to study the scattering problem shown above first, via a combination of detailed numerical simulations and a WKB-type analysis or one facilitated by an inverse scattering formalism, in order to understand what connection this has with the work of B. and O. as well as how it connects to other work done on semi-classical limits of Schrodinger equations.

## Better Literature Review
After a brief discussion with P. Kevrekidis, I now understand that these types of dynamics are well-understood from work due to Tovbis and Bertola, here: https://arxiv.org/abs/1004.1828. The interaction of the soliton with the time-dependent potential causes a gradient catastrophe.

A little more digging on my end, and I found one of the earliest papers on the topic due to P.D. Miller and S. Kamvissis
http://www.math.lsa.umich.edu/~millerpd/docs/J18.pdf

These earlier works differ from the more recent work of Biondini and Oregero in their assumptions on initial conditions, namely that B. and O. initial conditions are not localized in space.

The situation at the top remains different still. It may be interesting to understand, and place into context, why the gradient catastrophe in the soliton scattering problem manifests a spatio-temporal pattern resembling that which referenced from papers on semi-classical NLS dynamics.

## Some Numerical Experiments
I realized the parameters $c$ and $\theta$ don't really matter at all. Setting $c$ just transports the whole situation along some characteristic direction in space-time (please ignore incorrect labeling).

So with $c=\theta=\xi=0,$ and for a fixed potential, we can vary $\lambda.$ Setting $\lambda=1/130$ we see similar dynamics just on a slower time-scale.
Setting $\lambda=1/70,$ things start to look different:
Setting $\lambda=1/50,$ we see some kind of periodic envelope emerge,
And at $\lambda=1/40,$ I see this:
I'm very unsure of what "this" is. Perhaps this is a trivial rescaling of the dynamics and this is slightly obfuscated by the way I am plotting things.

Here's more one more picture about the nearly-periodic behavior of the dynamics, but when $\lambda=1/90$ and on a longer time-scale:

## Where to go next?
From a brief discussion, P. Miller brings up a good point:
"...if you don't have V(t), but still take the sech initial condition, as you mentioned you can assume that c=0 without loss of generality in this case, in which case the square modulus of the solution is exactly periodic in time, with a period that is increasing in the limit $\varepsilon\to$ 0. So I think that might explain the recurrence you see in your simulations. "

An analytical result we would like to see is that for a given initial soliton condition, the *shape* of $D(t)$ doesn’t affect microstructure, that is, the oscillations in the support of the solution. Here are two supporting numerical examples:

This
gives rise to
while this
does almost exactly the same thing

There is a similar result for the KdV equation, due to Tappert and Zabusky:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.27.1774
that confirms the experimental results of Madsen and Mei: https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/transformation-of-a-solitary-wave-over-an-uneven-bottom/4D5A97D39CA331D3BFBEBC22AF885807

The idea there is that given only a KdV soliton initial condition and the depth of potential appearing in that context, one can predict the number of solitons that will fission off of the initial condition. That is, the shape of the potential matters very little. Roy Goodman and I confirmed this numerically some years ago. There is also something similar due to W. Choi for the Benjamin-Ono equation: https://royalsocietypublishing.org/doi/10.1098/rspa.1997.0094

Somehow, the analogous situation for the NLS is much more complex than KdV or Benjamin-Ono. (Pun not intended)

## Toward Understanding the Dynamics

### Removing the Gain
Panos K., Alex T., and I met mid-October to try to understand more about the simulations above. The first observation made is that the gain term essentially "kicks" the soliton initial condition into something we are calling a non-equilibrium state. It seems reasonable to attempt to fit a hyperbolic secant to the profile. One question is: when?

With all scalings the same as above and with $\lambda=1/90$, I chose $T=110$ time units by examining the imaginary part of $\psi(x,T).$ I'm sure there are other ways to have decided which $T$ to use, but this is what I did. Observe the following picture:


Notice that $T=105$ is a bit early, and at $T=115,$ the top looks a little flat. By $T=120,$ the wavefunction now has three critical points. I insist that if you look at the real part or absolute value of $\psi,$ you wouldn't be able to detect such a hint as to when to perform the fit as easily.

So the least-squares fit of the intensity profile, $|\psi|^2$, at $T=110$ with
$$\psi_{\rm fit}=C_1\mathrm{sech}(C_2x)
$$
looks like this
where $(C_1,C_2)$ is found to be $(0.2305,.0226).$ Hey, not bad.

The next question is, what do the dynamics look like? Here is what a soliton kicked by the gain looks like:


and here is the above fitted state with no gain


It's absolutely clear now that we can remove the gain from the problem and investigate this problem with hyperbolic secant initial conditions of the above form evolving according to the usual NLS equation. However, it seems like we are a bit "far" from a soliton to do anything pertrubatively since the fited wavefunction looks something like
$\psi=2\lambda\mathrm{sech}(2\lambda(1-\varepsilon)x),$ where $\varepsilon\approx 0.9$. Something we can consider small with regards to the setup is nowhere to be found. This might be a point of discussion for us.

# Investigating the Spectrum
Sorry for the confusing notation, but let's use $\lambda$ to denote an eigenvalue of the Zakharov-Shabat system Recall, the inverse scattering formalism associates a linear eigenvalue problem, called the Zakharov-Shabat system, with a potential given by an integrable nonlinear wave equation. For the homogeneous NLS, the Zakharov-Shabat eigenvalue problem is given by

$$
\partial_{\xi}\varphi=-i\lambda\varphi +\psi(\xi,\zeta)\phi,\\
$$

$$
\partial_{\xi}\phi=i\lambda \phi-\psi^{*}(\xi,\zeta)\varphi.
$$
To find $\lambda$, I used a spectral collocation method I found in Jianke Yang's book on nonlinear waves. Here are synchronized movies of the wavefunction's intensity and the spectrum of the ZS system below it (slightly out of sync).


What we see is that, when the soliton interacts with the gain, we go from one pair of imaginary eigenvalues to 11.