%Aug 21, 2003; talk in Helsinki
\input amstex
\NoBlackBoxes
%\footline={\hfil}
\magnification 1500
\input psfig.sty
\font\large=cmr10 scaled \magstep0
\font\larger=cmr10 scaled \magstep1
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\font\largerrrr=cmr10 scaled \magstep4
\font\largerrrrr=cmr10 scaled \magstep5
\define \bR {\bold R}
\define \bCp {\bold C^+}
\define \bCpb {\overline{\bold C^+}}
\define \ds {\displaystyle}
\define \bm {\bmatrix}
\define \endbm {\endbmatrix}
\baselineskip 20 pt
\parskip 10 pt
\phantom{}
\vskip 20 pt

\centerline {{\bf {\largerrr WAVE FOCUSING}}}
\centerline {{\bf {\largerrr IN ONE DIMENSION}}}

\vskip 50 pt

\centerline {{\bf {\largerr Tuncay Aktosun}}}
\centerline {{\bf {\largerr Dept of Mathematics and Statistics}}}
\centerline {{\bf {\largerr Mississippi State University}}}
\centerline {{\bf {\largerr Mississippi State, MS 39762, USA}}}

\vskip 15 pt

\centerline {{\bf {\larger Helsinki, August 2003}}}
\centerline {{\bf {\larger (joint with J. Rose of Iowa State U)}}}

\newpage

\noindent {\bf Plasma-wave eq (PWE)}

%\vskip 15 pt

\noindent 
$\displaystyle 
u_{xx}-u_{tt}=V(x)\,u,\quad x,t\in\bR$ 

\noindent Bayliss, Li, Morawetz,
{\it Math. Comp.} {\bf 52} (1989) 321--338.

\noindent $\bullet$ time-domain approach

\vskip -10 pt

\noindent $\bullet$ assume $V$ has no bound states

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\noindent $\bullet$ assumption on support of $V$

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\noindent $\bullet$ scattering sol behaving like a plane wave as $t\to -\infty$

\newpage

\noindent {\bf Time-domain picture}

\noindent $\bullet$ plane wave incoming from ``free space,"
$x\approx -\infty$

\vskip -10 pt

\noindent $\bullet$ encounters the nonhomogeneity $V(x)$

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\noindent $\bullet$ the wavefront starts ``picking up" a trailing tail

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\noindent $\bullet$ the tail is determined by $V$


\noindent $\bullet$ sol: $u_{\text l}(x,t)=\delta(x-t)+g_{\text l}(x,t)$

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$\delta(x-t)$ \ \ wavefront incident from left

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$g_{\text l}(x,t)$ \ \ trailing tail

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$g_{\text l}(x,t)=0$ for $x>t,$ \ \ causality

\noindent $\bullet$ Example 1: animated example of wave probing

\newpage

\noindent {\bf Our interest}


\noindent Aktosun and Rose, {\it J. Math. Phys.} {\bf 43} (2002) 247--266.

\noindent $\bullet$ include bound states of $V$

\vskip -10 pt

\noindent $\bullet$ no support assumption on $V$

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\noindent $\bullet$ $V$ belongs to the Faddeev class

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real valued, 
measurable, $\ds\int_{-\infty}^\infty
dx\, (1+|x|)\,|V(x)|$ \ \ finite

\vskip -10 pt

\noindent $\bullet$ causal scattering sols that ``focus" to a point

\newpage

\noindent {\bf Focusing sols}


\noindent $\bullet$ incoming from left (or right)

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\noindent $\bullet$ causal, consisting of a planar wavefront and trailing tail

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\noindent $\bullet$ encounters the nonhomogeneity $V(x)$

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\noindent $\bullet$ the tail gets smaller

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\noindent $\bullet$ the tail disappears at a (prescribed) moment, say $t=0$

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\noindent $\bullet$ the wavefront is at $x_0$ when $t=0$

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\noindent $\bullet$ at $t=0$ the wave focuses to the point $x_0$

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\noindent $\bullet$ the tail starts reappearing

\vskip -10 pt

\noindent $\bullet$ the tail at any moment is determined by $V$

\newpage

\noindent {\bf Focusing sol incident from left}

\noindent 
$\displaystyle 
u_{xx}-u_{tt}=V(x)\,u,\quad x,t\in\bR$ 

\noindent
$\displaystyle
u_{\text l}(x,t;x_0)=\delta(x-t-x_0)+g_{\text l}(x,t;x_0)$

\noindent $\bullet$  $\delta(x-t-x_0)$ \ \ planar wavefront moving to right

\vskip -10 pt

\noindent $\bullet$ $g_{\text l}(x,0;x_0)=0$

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\noindent $\bullet$ $g_{\text l}(x,t;x_0)=0$ for $x>t+x_0,$
\ \ causality

\vskip -10 pt

\noindent $\bullet$ $g_{\text l}(x,\cdot ;x_0)\in L^2(\bR)$ if $V$ has no bound states


\noindent $\bullet$ Direct problem:

\vskip -10 pt

given $x_0$ and $V(x)$, construct the tail $g_{\text l}(x,t;x_0)$

\vskip -10 pt

\noindent $\bullet$ Inverse problem:

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how to construct $V(x_0)$ from $g_{\text l}(x,t;x_0)$

\newpage

\noindent {\bf PWE and Schr\"odinger eq (SE)}

\noindent PWE:
\ \ $\displaystyle 
u_{xx}-u_{tt}=V(x)\,u,\qquad x\in\bR,\quad t\in\bR.$ 

\noindent SE:
\ \ $\displaystyle\frac{d^2\psi}{dx^2}+k^2\psi=V(x)\,\psi,\quad x\in\bR.$

\newpage

\noindent {\bf Sols to SE with
$V(x)\sim 0$ as $x\to\pm\infty$}


\noindent $\bullet$ left physical sol

\vskip -10 pt

$\psi_{\text l}(k,x)\sim e^{ikx}+L(k)\,e^{-ikx}, \ \ \ x\to-\infty$

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$\psi_{\text l}(k,x)\sim T(k)\, e^{ikx}, \ \ \ x\to+\infty$

\noindent $\bullet$ right physical sol

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$\psi_{\text r}(k,x)\sim e^{-ikx}+R(k)\,e^{ikx}, \ \ \ x\to+\infty$

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$\psi_{\text r}(k,x)\sim T(k)\, e^{-ikx}, \ \ \ x\to-\infty$

\noindent $\bullet$
$L(k),\ R(k)$ \ \ left and right reflection coeff

\vskip -10 pt

\noindent $\bullet$
$T(k)$ \ \ transmission coeff, with simple poles at $i\kappa_j$

\noindent $\bullet$
\noindent left and right Jost sols

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$f_{\text l}(k,x):=\ds\frac{1}{T(k)}
\psi_{\text l}(k,x), \quad
f_{\text r}(k,x):=\ds\frac{1}{T(k)}
\psi_{\text r}(k,x)$

\noindent $\bullet$ normalized bound-state sols

\vskip -10 pt

$\varphi_j(x):=c_{{\text l}j}\,f_{\text l}(i\kappa_j,x)=c_{{\text r}j}\,
f_{\text r}(i\kappa_j,x),
\quad j=1,\dots,N$

\noindent $\bullet$ left and right norming consts

\vskip -10 pt
$c_{{\text l}j}:=\ds\frac{1}
{\sqrt{\int_{-\infty}^\infty dx\,f_{\text l}(i\kappa_j,x)^2}},
\quad c_{{\text r}j}:=\ds\frac{1}
{\sqrt{\int_{-\infty}^\infty dx\,f_{\text r}(i\kappa_j,x)^2}}$



\newpage 

\noindent {\bf Probing (nonfocusing) sols to PWE}

\noindent PWE: \ \
$\displaystyle
u_{xx}-u_{tt}=V(x)\,u,\quad x,t\in\bR$

$\displaystyle
u_{\text l}(x,t)=
\ds\frac{1}{2\pi}\ds\int_{-\infty}^\infty dk\,
T(k)\,f_{\text l}(k,x)\,e^{-ikt}+\ds\sum_{j=1}^N
c_{{\text r}j}\,\varphi_j(x)\,\ds e^{\kappa_j t}$

$\displaystyle
u_{\text r}(x,t)=
\ds\frac{1}{2\pi}\ds\int_{-\infty}^\infty dk\,
T(k)\,f_{\text r}(k,x)\,e^{-ikt}+\ds\sum_{j=1}^N
c_{{\text l}j}\,\varphi_j(x)\,\ds e^{\kappa_j t}$


\noindent
$\bullet$ if $V\equiv 0$ for $x<0,$ then

\vskip -10 pt

$\displaystyle
u_{\text l}(x,t)=
\delta(x-t)+I(-x-t),\quad x<0$

\vskip -10 pt

$\hat I(t):=\hat L(-t)+\ds\sum_{j=1}^N
c_{{\text r}j}^2\, \ds e^{\kappa_j t}$ impulse response 

\noindent
$\bullet$ inverse problem when $V\equiv 0$ for $x<0$

\vskip -10 pt

find $V(x)$ for $x\in(0,x_1)$ from $\hat I(t)$
for $t\in(0,2x_1)$

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Marchenko method: \ \ $V(x)=2\ds\frac{dK(x,x^-)}
{dx}$

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$K(x,y)+I(x+y)+\ds\int_0^x 
dz\, K(x,z)\,I(z+y)=0$

\newpage

\noindent {\bf Analyticity properties}

\noindent $\bullet$ transmission coeff $T(k)$ 

\vskip -10 pt
meromorphic in $\bold C^+$

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continuous in $\bCpb$
except at simple poles at $k=i\kappa_j$

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$1+O(1/k)$ as $k\to\infty$ in $\bCpb$

\noindent $\bullet$ for each fixed $x,$
Jost sols $f_{\text l}(\cdot,x)$ and 
$f_{\text r}(\cdot,x)$ 

\vskip -10 pt
analytic in $\bCp$ and continuous in $\bCpb$

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$e^{-ikx}f_{\text l}(k,x)=1+O(1/k)$ as $k\to\infty$ in $\bCpb$

\vskip -10 pt

$e^{ikx}f_{\text r}(k,x)=1+O(1/k)$ as $k\to\infty$ in $\bCpb$

\newpage

\noindent {\bf Completeness relation for SE}

\vskip 10 pt

\noindent $\aligned \ds\frac{1}{4\pi}\ds\int_{-\infty}^\infty dk
\left[\psi_{\text l}(k,x)\, \overline{\psi_{\text l}(k,x_0)}\right.
+&\psi_{\text r}(k,x)\left. \overline{\psi_{\text r}(k,x_0)}\right]
\\
&+\ds\sum_{j=1}^N \varphi_j(x)\,\varphi_j(x_0)
=\delta(x-x_0)\endaligned$

\vskip 10 pt

\noindent Unitarity of the scattering matrix leads to

\noindent
$\ds\frac{1}{2\pi}\ds\int_{-\infty}^\infty dk
\,T(k)\,f_{\text l}(k,x)\,f_{\text r}(k,x_0)
+\ds\sum_{j=1}^N \varphi_j(x)\,\varphi_j(x_0)
=\delta(x-x_0)$

\newpage

\noindent {\bf Focusing sols to PWE}

\noindent PWE: \ \ 
$\displaystyle 
u_{xx}-u_{tt}=V(x)\,u,\quad x,t\in\bR$ 

\noindent $\bullet$ 
$\displaystyle
u_{\text l}(x,t;x_0)=\delta(x-t-x_0)+g_{\text l}(x,t;x_0)$

\vskip -10 pt

\noindent $\bullet$ 
$\displaystyle
u_{\text r}(x,t;x_0)=\delta(x+t-x_0)+g_{\text r}(x,t;x_0)$

\vskip 10 pt

\noindent Completeness relation and analyticity properties lead to

\noindent
$\displaystyle
u_{\text l}(x,t;x_0)=
\ds\frac{1}{2\pi}\ds\int_{-\infty}^\infty dk\,
T(k)\,f_{\text l}(k,x)\,f_{\text r}(k,x_0)\,
e^{-ikt}
+\ds\sum_{j=1}^N \varphi_j(x)\,\varphi_j(x_0)\,e^{\kappa_j t}$

\noindent
$u_{\text r}(x,t;x_0)=\ds\frac{1}{2\pi}\ds\int_{-\infty}^\infty dk\,
T(k)\,f_{\text r}(k,x)\,f_{\text l}(k,x_0)\,
e^{-ikt}+\ds\sum_{j=1}^N \varphi_j(x)\,\varphi_j(x_0)\,e^{\kappa_j t}$

\newpage

\noindent {\bf Further properties of focusing waves}

\noindent $\bullet$ if $V\equiv 0$ for $x<0$ or for
$x>0,$ then

\vskip -10 pt

$($focusing sol to $x_0=0$$)$ = (probing physical sol)

\vskip -10 pt

\noindent $\bullet$ time antisymmetries

\vskip -10 pt

\noindent $\bullet$ support of the tail

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\noindent $\bullet$ gap between the tail and the wavefront

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\noindent $\bullet$ kinks of focusing waves, discontinuities of $V$

\vskip -10 pt

\noindent $\bullet$ uniqueness of focusing sols

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\noindent $\bullet$ blurring, approximate focusing sols

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\noindent $\bullet$ relation to Marchenko method

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\noindent $\bullet$ recovery of $V(x_0)$ by focusing to $x_0$

\vskip -10 pt

if $V$ is continuous at $x_0,$ then for any $t\in\bR,$
e.g. $t\sim -\infty$

$V(x_0)=V(x_0+t)+
2\,g_{\text l}(x_0^-+t,t;x_0)^2+4\,\ds\frac{\partial
g_{\text l}(x_0^-+t,t;x_0)
}{\partial x}$

\newpage

\noindent {\bf Generalizations}

\noindent $\bullet$ radial SE: \ \ 
$\displaystyle 
u_{xx}-u_{tt}=V(x)\,u,\quad x>0,\ \ t\in\bR$ 

\vskip -10 pt

\noindent $\bullet$ variable-speed wave eq: \ \ 
$\displaystyle 
u_{xx}-\ds\frac{1}{c(x)^2}
\,u_{tt}=V(x)\,u,\quad x,t\in\bR$ 

\newpage

\noindent {\bf Example 2: focusing within nonhomogeneity}

\centerline{\psfig{figure=pot1.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ vanishing when $x<0$}

\vskip 35 pt

\centerline{\psfig{figure=fig3.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Support of $u_{\text l},$ with $x_0=1$}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex71posx0min2.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex71posx0min1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex71posx0min0.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex71posx0pos1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex71posx0pos2.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with $x_0=1$ at $t=-2,-1,0,1,2.$}

\newpage

\noindent {\bf Example 3: focusing behind nonhomogeneity}

\centerline{\psfig{figure=pot3.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ vanishing when $x>0$}

\vskip 35 pt

\centerline{\psfig{figure=fig1.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Support of $u_{\text l},$ with $x_0=1$}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex76posx0min1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex76posx0min0.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex76posx0pos05.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex76posx0pos1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex76posx0pos2.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with $x_0=1$ at $t=-1,0,0.5,1,2.$}

\newpage

\noindent {\bf Example 4: focusing in front of nonhomogeneity}

\centerline{\psfig{figure=pot1.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ vanishing when $x<0$}

\vskip 35 pt

\centerline{\psfig{figure=fig2.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Support of $u_{\text l},$ with $x_0=-1$}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex72negx00.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex72negx01.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex72negx012.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex72negx02.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex72negx03.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with
$x_0=-1$ at $t=0,1,1.2,2,3.$}

\newpage

\noindent {\bf Example 5: focusing within nonhomogeneity}

\centerline{\psfig{figure=pot3.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ vanishing when $x>0$}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex77negx0min1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex77negx0min0.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex77negx0pos1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex77negx0pos2.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex77negx0pos3.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with $x_0=-1$ at $t=-1,0,1,2,3.$}

\newpage

\noindent {\bf Example 6: focusing with bound states}

\centerline{\psfig{figure=pot2.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ nonvanishing, negative everywhere}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex73posx0min3.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73posx0min2.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73posx0min1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73posx0min0.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73posx0pos1.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with
with $x_0=1$ at $t=-3,-2,-1,0,1.$}

\newpage

\noindent {\bf Example 7: focusing with bound states}

\centerline{\psfig{figure=pot2.ps,height=2.0 truein,width=2.0 truein}}
\centerline{Potential $V,$ nonvanishing, negative everywhere}

\vskip 35 pt

\centerline{\hbox
{\psfig{figure=ex73negx0min2.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73negx0min1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73negx0min0.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73negx0pos1.ps,height=1.3 truein,width=1.3 truein}}
{\psfig{figure=ex73negx0pos2.ps,height=1.3 truein,width=1.3 truein}}}
\centerline{$u_{\text l}$ with
$x_0=-1$ at $t=-2,-1,0,1,2.$}

\newpage

\noindent {\bf Example 8: focusing with Dirac delta distribution}

\newpage

\noindent {\bf Example 9: focusing to edge of potential}

\noindent $\bullet$ if $V\equiv 0$ for $x<0,$ then

\vskip -10 pt

$($focusing sol to $x_0=0$$)$ = (probing physical sol)

\end
