Beals-Coifman’s dbar problem
In what follows, we simply copy the main results from Beals-Coifman-1986.
Let’s consider a linear spectral problem: \begin{align} D_zm(z,x)\equiv (\partial_x+z\hat J)m=q(x)m,\label{spectral} \end{align} where $\hat J=[J,\cdot]$, $z$ is a spectral variable, $q(x)$ is an off-diagonal matrix with complex-valued function entries defined on whole real line[Q1: How to generalize it to periodic case?], and $m$ is a matrix-valued function. In what follows, we also assume $q(x)$ vanishes as $x\rightarrow\infty$.
Now let’s introduce so-called $\bar\partial$ operator. Given any function $f(z)\in C^1(z,\bar z)$, we have $$ df=\partial fdz+\bar\partial fd\bar z, $$ where, if $z=x+iy$, then \begin{align} \partial \equiv \frac{\partial}{\partial z}=\frac{1}{2}(\partial_x-i\partial_y) ,\\ \bar\partial \equiv \frac{\partial}{\partial \bar z}=\frac{1}{2}(\partial_x+i\partial_y). \label{ss} \end{align}
Moreover, the dbar derivative makes sense for distributions too. From the Cauchy-Riemann equations, it is evident that dbar annihilate all holomorphic functions. Apply Green’s formula, one can easily show that $\frac{1}{\pi z}$ is a fundamental solution for $\bar \partial$. We also have that if $f$ is holomorphic off some “nice” simple oritented curve $\Sigma$, and the left/right boundary valued $f_\pm$ exist, then $$ \bar\partial f=\frac{i}{2}(f_+-f_-). $$
Exercise: Prove all the mentioned properties about dbar operator in the sense of distribution.
Back to the linear spectral problem, since $D_z$ is holomorphic with respect to $z$, $[D_z,\bar\partial]=0$. Thus, applying dbar derivative, we have $$ D_z\bar\partial m=q\bar\partial m. $$
Since the system is of first order (derivative), two different solutions are connected by a constant matrix (in classical scattering theory, that is called scattering matrix.) With this intuition, one can show that $$ \bar\partial m=mv, $$ where $D_zv=0$.(Here $v=v(x,z)$!)
The above process can be considered as direct scattering, i.e. from potential $q$ to scattering data $v$.
Next, consider the inverse scattering problem. Suppose $m$ solves the dbar problem with some $v\in \text{Ker}(D_z)$, we now derive a formula to recovery the potential $q$ from the scattering data $v$.
Using the fundamental solution (w.r.t Lebesgue measure) of dbar, we can transfer the dbar problem to an singular integral equation: $$ m=I+CTm, $$ where $Cf=(\frac{1}{\pi z}*f)(z)$ and $Tf=fv$. (Note: this Cauchy operator is with respect to area measure.)
Now let’s compute $D_zm$. In fact, we have \begin{align*} D_zm & =D_z(1-CT)^{-1}I\\ &=[D_z,(1-CT)^{-1}]I\quad (\text{using the fact that $D_zI=0$})\\ &=(1-CT)^{-1}[D_z,CT]m\quad (\text{since $(1-b) [a,(1-b)^{-1}] (1-b)=[a,b]$ })\\ &=(1-CT)^{-1}[D_z,C]Tm\quad (\text{since $[D_z,T]=0$})\\ &=(1-CT)^{-1}[z,C]\hat J Tm\\ &=( [z,C] \hat J Tm)m \quad (\text{using the magic property of the operator $ [z,C] $}). \end{align*} From the last step, we see if we set $q=[z,C] \hat J Tm$, we recovery the potential from the scattering data $v$.
To get a hierarchy of integrable system then is trivial. We will skip it here.
Zahkarov-Manakov’s dressing method
One should compare this method with Dyachenkoa-Zakharov-Zakharov’s paper on primitive potential technique.
In DZZ paper, they consider following scalar dbar problem: $$ \bar \partial \chi=ie^{2ikx}T(k)\chi(-k), $$ where $T(k)$ is a compactly supported distribution, so-called dressing function (following Zakharov-Manakov-1985).
(Intuitively, by put some symmetries on the dressing function, one can reduce from Beals-Coifman’s scheme to Zakharov-Manakov’s dressing method. Under construction)
Application (more details coming soon/or never)
- Primitive potential, see Zakharov-Zakharov
- Soliton gas w.r.t. a curve spectral data, see Tovbis-El
- Soliton gas w.r.t. a compact region spectral data, see Bertola-Grava-Orsatti