Project A12 • Dynamics of the Gross–Pitaevskii equation and related dispersive equations

The word soliton is a combination of the terms solitary wave and particle. It describes (localized) traveling waves that exhibit some particle-like behavior, such as stability under perturbations or interaction with other solitons.

If $c=0$, then $q_0(t,x)=\tanh(x)$ denotes the (time-independent) black soliton, with the (light) density $|q_0|^2=1-{\text{sech}}^2(x)$ vanishing at the origin, which heuristically corresponds to the black point in nonlinear optics. If $c\in (-1,0)$ or $(0,1)$, then the dark solitons $q_c$ travels to the right or to the left at the constant speed $2|c|<2$, with a kink in the density function $|q_c|^2$ at the point $x=2ct$, which heuristically corresponds to the dark point in nonlinear optics. There are no soliton solutions travelling faster than the speed $2$.

Due to \eqref{NZBC}, the usual functional framework such as Sobolev spaces are not suitable for the solutions of \eqref{GP} anymore. We consider the tailor-made solution spaces $X^s$ for \eqref{GP}:

• The module of the unit circle $\mathbb{S}^1$ is motivated by the $\mathbb{S}^1-$symmetry of \eqref{GP};
• The constant functions $1=-1=i=-i$ are the same element in $X^s$, and have zero energy ${E}(1)=0$;
• $X^s$ is not a vector space;
• The soliton solutions $q_c(t,\cdot)\in X^s$ since \begin{align*} -(|q_c(t,x)|^2-1)=(1-c^2)\text{sech}^2\bigl(\sqrt{1-c^2}(x-2ct)\bigr)=\partial_x q_c(t,x); \end{align*}
• The finite-energy function $q\in X^s$ satisfies the nonzero boundary condition \eqref{NZBC} in the sense that $|q|^2-1\in H^{s-1}(\mathbb{R})$;
• If $s=1$, $X^1$ denotes the (classical) energy space \begin{equation*} X^1=\bigl\{q\in H^1_{\textrm{loc}}(\mathbb{R}; \mathbb{C})\,|\, E(q)=\|( |q|^2-1, \partial_xq)\|_{ L^2(\mathbb{R})}^2 <\infty\bigr \}/{\mathbb{S}^1}. \end{equation*}

The distance function $d^s(q,p)$ measures the differences between $q$ and $p$ in a nonlinear way:

• Because the integral in (ds) contains an infimum over phase rotations $\lambda \in \mathbb{S}^1$, it is possible for two functions $q, p$ to have different asymptotic phases at infinity while having a finite distance $d^s(q, p)$. This makes $(X^s, d^s)$ a natural space for functions with \eqref{NZBC};
• The distance $d^s(q,1)$ is (locally) equivalent to the energy $E^s(q)=\|( |q|^2-1, \partial_xq)\|_{ H^{s-1}(\mathbb{R})}^2$;
• For $s > \frac12$ and two (nowhere vanishing) functions $q = \sqrt{\rho} e^{i \phi}, p = \sqrt{\eta} e^{i \psi} \in H_{\textrm{loc}}^s(\mathbb{R})$, $d^s(q,p)$ is (locally) equivalent to both \begin{align*} \sum_{k \in \mathbb{Z}} \inf_{|\lambda| = 1} \|\lambda q - p\|_{W^{s,2}(\{x \in \mathbb{R}: \,|x - k| \leq 1\})} \end{align*} and \begin{align*} \theta^s((\rho, v), (\eta, w)) = \|\rho - \eta\|_{H^s(\mathbb{R})} + \|v - w\|_{H^{s-1}(\mathbb{R})} \,, \end{align*} where $v,w$ denote the phase velocities $\partial_x\varphi, \partial_x \psi$ respectively;
• $1+\mathcal{S}(\mathbb{R})$ is dense in $(X^s, d^s)$;
• The set of $1$ and the soliton profiles $\{Q_c: c \in [-1,1] \}$ is a strong deformation retract of $(X^s, d^s)$, and hence $X^s$ is homotopy equivalent to a circle;

The physical concept of scattering refers to the deviation of particles (e.g. photons) from their natural trajectory as they interact with some localized non-uniformity (e.g. a material). The goal of inverse scattering is to obtain information about the non-uniformity by observing the way particles scatter off it. In our case, the non-uniformity is given by a potential $q$, and the evolution is given by rewriting the spectral problem $L \phi = \lambda \phi$ to an ODE system \begin{equation*} \phi_x = \begin{pmatrix} - i \lambda & q \\ \overline{q} & i \lambda \end{pmatrix} \phi \,. \end{equation*} In the trivial case $q=1$ (no non-uniformity) the ODE system is $\phi_x = \left( \begin{matrix} -i\lambda & 1 \\ 1 & i \lambda \end{matrix} \right) \phi $, which has two fundamental solutions $\phi_l=e^{-izx}\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$, $\phi_r=e^{izx}\begin{pmatrix} i\beta\\ -i\alpha\end{pmatrix}$ where $z=\sqrt{\lambda^2-1}$, $\alpha=\sqrt{-i(\lambda+z)}$, $\beta=\sqrt{i(\lambda-z)}$ (we always take the square root in the closed upper complex plane). In our analogy, $\phi_l$ and $\phi_r$ represent particles travelling towards $-\infty$ and $+\infty$ respectively. Here the spectrum of $L$ consists of the continuous spectrum $(-\infty,-1]\cup [1,\infty)$, where the reflection coefficient $R(\lambda)=0$ and the transmission coefficient $T^{-1}(\lambda)=0$. This intuitively means that all frequencies of light are transmitted while not reflected when they go through the constant potential $q=1$.
More generally, if the potential $q$ converges to some limits at infinity sufficiently quickly, e.g. $(1+x^4)(q(x)-e^{i\theta_\pm})\in L^1(\mathbb{R}^\pm)$, then the spectrum of the Lax operator $L$ consists of:

• Essential spectrum $(-\infty, -1]\cup [1,\infty)$;
• Finitely many simple eigenvalues $(\lambda_m)\subset (-1,1)$.

In this case we obtain a solution $\phi$ with the asymptotics \begin{align*} \phi(x) &= T(\lambda) \phi_l(x) + o(1) e^{\textrm{Im}(z) x} \,, \qquad \quad \quad \;\;\, x \rightarrow -\infty \\ \phi(x) &= \phi_l(x) + R(\lambda) \phi_r(x) + o(1) e^{\textrm{Im}(z) x} \,, \quad x \rightarrow \infty \end{align*} Here $R(\lambda)$ is the reflection coefficient and $T(\lambda)$ is the physical transmission coefficient. We call $T^{-1}(\lambda)$ the transmission coefficient instead. In our analogy, this can be interpreted as particles or waves $\phi_l$ travelling from $+\infty$ towards $-\infty$ while interacting with the potential $q$. A part $T(\lambda) \phi_l$ is transmitted while a part $R(\lambda) \phi_r$ is reflected. The scattering data consists of the reflection coefficient $R(\lambda)$, defined for $\lambda$ in the essential spectrum of $L$, together with some renormalization constants $\gamma_m$ which are related to the eigenfunction associated to the eigenvalue $\lambda_m$. As a consequence of the complete integrability of \eqref{GP}, the scattering data $\{R(\lambda), \gamma_m\}$ undergoes a trivial evolution while the transmission coefficient $T^{-1}(\lambda)$ is a conserved quantity, If $q=Q_c$ is the soliton profile, then the spectrum of $L$ consists of $(-\infty, -1]\cup [1,\infty)$ and a single simple eigenvalue $-c$.
We remark that if $q$ satisfies \eqref{ZBC}, then $L$ has essential spectrum $\mathbb{R}$ and no eigenvalues.

If the potential $q\in 1+\mathcal{S}(\mathbb{R})$, then the transmission coefficient $T^{-1}(\lambda)$ is defined for all $\lambda\in \mathbb{C}$. More precisely, if one solves the ODE system (i.e. $L\phi=\lambda\phi$) with the "initial condition" $\phi_l(x)-e^{-iz x}\begin{pmatrix} \alpha \\ \beta \end{pmatrix}\rightarrow 0\hbox{ as }x\rightarrow -\infty$, then $T^{-1}(\lambda)$ is defined as the asymptotic behavior of the solution in the following way: \begin{align*} &\phi_l( x ) = T ^{-1}(\lambda) \cdot \,e^{-iz x} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} +o(1)e^{\textrm{Im}(z)x} \hbox{ as }x\rightarrow +\infty. \end{align*} One can expand the transmission coefficient $T^{-1}(\lambda)$ at infinity as \begin{align*} \ln T^{-1}(\lambda) =\frac{i}{2z}M +\frac{i}{(2z)^2}P+\frac{i}{(2z)^3}E+\cdots+\frac{i}{(2z)^k}H_k+\cdots, \hbox{ as }|\lambda|\rightarrow \infty, \textrm{Im}\lambda>0, \end{align*} where the first three coefficients are the conserved mass, momentum and energy quantities respectively $$M=\int_{\mathbb{R}}(|q|^2-1) dx,\quad P=\textrm{Im}\int_{\mathbb{R}} q \partial_x \bar q dx,\quad E=\|(|q|^2-1, \partial_x)\|_{L^2(\mathbb{R})}^2.$$ Since $T^{-1}(\lambda)$ is conserved by the GP-flow, all the coefficients $H_k$ are all conserved quantities. However, the explicit formulas of these conserved quantities become much complicated for large $k$, and they are in general not positive definite. Obviously the mass $M$ and momentum $P$ are not well-defined for finite-energy potentials, and one has to define the renormalized transmission coefficient for $q\in (X^s, d^s)$ ([KL21, KL22]).

We can rewrite the hydrodynamic formulation \eqref{hGP} of \eqref{GP} for the wave-coordinates $(r,v)$ with $r=\frac12\ln\rho$. We take the long-wave ansatz $$(r,v)(t,x)=(\check{r}, \check{v})(\delta t, \delta x),\quad 0<\delta\ll1,$$ write the (rescaled) equations for $(\check{r}, \check{v})$, and take $\delta\rightarrow 0$ in the (rescaled) equations, to derive formally the Whitham modulation equations as the limit equations: \begin{equation*} \label{whi1} \left\{ \begin{array}{l}\partial_{T} \check{r} = - \partial_{X} \check{v} - 2(\partial_{X} \check{r})\check{v} , \\ \partial_{T} \check{v} = - \partial_{X}(\check{v}^2) - 4 e^{2\check{r}} \partial_{X} \check{r} . \end{array}\right.\end{equation*} This system is (locally-in-time) well-posed in Sobolev spaces, and one can show that the solution $(r,v)$ of the original system \eqref{GP} can be approximated by its WME in the long-wave approximation in the following sense $$ \sup_{t \in [0,T/\delta]} \| (r,v)(x,t) - (\check{r},\check{v}) (\delta x, \delta t) \|_{H^s(\mathbb{R})} \leq C \delta^2. $$ In general, the well-posedness results in Sobolev spaces for the WME system correspond to the so-called Benjamen–Feir stability of the traveling wave solutions of the original system. Here the well-posedness results in Sobolev spaces corresponds to the Benjamin-Feir stability of the standing traveling solution $\psi=e^{2it}$ of \eqref{NLS}, i.e. the constant solution $q=1$ for \eqref{GP}.

Similarly, the \eqref{ccKG} equation possesses a family of traveling wave solutions $$ \Psi_{\gamma,k,\omega}\left( x,t\right) =e^{a+ ikx+i\omega t}, \hbox{ with }a,k,\omega\in \mathbb{R}\hbox{ satisfying }-\gamma e^{2a}=\omega^{2}-k^{2}-1. $$ If $\gamma=-1$ (defocusing case), $k=0$, $\omega=2$, then $\Psi_{1,0,2}=\sqrt{3}e^{2it}$ is a (standing) travelling solution of \eqref{ccKG} (recalling $\psi=e^{2it}$ is a travelling solution of \eqref{NLS}). We rewrite $\Psi=\sqrt{3}e^{r+i\theta+2it}$, and introduce the local spatial wave number $ v = \partial_x \theta$ and the local temporal wave number $ w = \partial_t \theta$. We then write \eqref{ccKG} in the wave-coordinates $(r,v,w)$, make the long-wave ansatz $$(r,v,w)(t,x)=(\check{r}, \check{v}, \check{w})(\delta t, \delta x),\quad 0<\delta\ll1,$$ and take $\delta\rightarrow 0$ to derive the WME for $(\check{r}, \check{v}, \check{w})$. We have verified the WME approximation of \eqref{ccKG} in Gevrey spaces, not only for defocusing case ($\gamma=-1$) but also for focusing case ($\gamma=1$) if $|\omega|>\frac{1}{\sqrt{3}}$ ([HLS22]).