# On compressing sinh-Gordon solutions

David Vegh<EMAIL_ADDRESS>Centre for Research in String Theory, School of
Physics and Astronomy, Queen Mary University of London, 327 Mile End Road,
London E1 4NS, UK

###### Abstract

This paper is concerned with a class of approximate non-linear transformations
that compress solutions of the (generalized) sinh-Gordon equation into
parametrically small regions in two-dimensional spacetime. Given the sinh-
Gordon field near a time-slice, a long Nambu-Goto string can be constructed in
three-dimensional anti-de Sitter space. The string is then approximated to
arbitrary accuracy by a slightly smoothed piecewise linear string of $N$
segments. The corresponding sinh-Gordon field has a comb-like structure and
its size is controlled by the amount of smoothing applied to the segmented
string. In a (singular) large-$N$ limit, the transformation commutes with time
evolution. As an example, a static cosh-Gordon solution is discussed in
detail. The corresponding smooth and segmented string solutions are obtained
and the compressed cosh-Gordon potential is investigated.

††preprint: QMUL-PH-21-13

1\. Introduction. Completely integrable models in two dimensions play an
important role in physics. Since they possess sufficiently many conserved
quantities they are exactly solvable thus providing a controlled laboratory in
which various phenomena can be studied. The focus of this paper is on the
hyperbolic version of the celebrated sine-Gordon equation: the classical
(generalized) sinh-Gordon model. The equation of motion is given by

$\partial_{+}\partial_{-}\alpha+e^{\alpha}-uve^{-\alpha}=0$ (1)

where $\alpha(\sigma^{+},\sigma^{-})$ is the sinh-Gordon field,
$u({\sigma^{-}})$ and $v({\sigma^{+}})$ are auxiliary fields, and
$\sigma^{\pm}={\frac{1}{2}}(\tau\pm\sigma)$ are lightcone coordinates in two
dimensions ($\partial_{-}\equiv\partial_{\sigma^{-}}$ and
$\partial_{+}\equiv\partial_{\sigma^{+}}$). The equation describes constant
mean curvature surfaces in ${\mathbb{{R}}}^{2,1}$, extremal surfaces in AdS3
Pohlmeyer (1976), and it governs the evolution of the mesonic mean field in
the two-dimensional Gross-Neveu model Gross and Neveu (1974); Neveu and
Papanicolaou (1978).

In regions where $uv=0$, eqn. (1) reduces to the Liouville equation. Outside
these regions, after a coordinate transformation given by
$d\tilde{\sigma}^{-}=\sqrt{|u({\sigma^{-}})|}d\sigma^{-}$,
$d\tilde{\sigma}^{+}=\sqrt{|v({\sigma^{+}})|}d{\sigma^{+}}$, the equation
takes the standard sinh-Gordon or cosh-Gordon form

$\tilde{}\partial_{+}\tilde{}\partial_{-}\tilde{\alpha}+e^{\tilde{\alpha}}\pm
e^{-\tilde{\alpha}}=0$ (2)

where $\tilde{\alpha}=\alpha-\log\sqrt{|uv|}$. The variables
$\tilde{\sigma}^{\pm}$ will be called balanced coordinates.

The non-linear sinh-Gordon equation is integrable Pohlmeyer (1976); De Vega
and Sanchez (1993) and possesses singular soliton solutions. Multi-soliton
solutions can be constructed by the inverse scattering method. In this Letter
the relationship between the sinh-Gordon equation and long Nambu-Goto strings
(extremal surfaces) in AdS3 will be exploited to obtain scheme-dependent non-
linear transformations which compress solution of the sinh-Gordon equation
into parametrically small regions. The flowchart of the map is illustrated in
FIG.1.

Figure 1:  Flowchart of the transformation. In a given segmenting and
smoothing scheme the transformed field will depend on two control parameters:
the number of string segments $N$ and a smoothing parameter $\varepsilon$.

2\. Strings in AdS3. In the first step of the transformation one needs to
obtain a smooth string embedding in AdS3. A unit size AdS3 can be immersed
into an ${\mathbb{{R}}}^{2,2}$ ambient space via the equation,

$Y\cdot Y\equiv-Y_{-1}^{2}-Y_{0}^{2}+Y_{1}^{2}+Y_{2}^{2}=-1\qquad
Y\in{\mathbb{{R}}}^{2,2}.$

A part of global AdS3 is covered by the Poincaré patch. The metric on the
patch is given by

$ds^{2}={-dt^{2}+dx^{2}+dy^{2}\over y^{2}}.$

The coordinates $t,x,y$ are related to $Y$ via the following transformation

$(t,\,x,\,y)=\left({Y_{0}\over Y_{2}-Y_{-1}},\ {Y_{1}\over Y_{2}-Y_{-1}},\
{1\over Y_{2}-Y_{-1}}\right),$

whose inverse on the hyperboloid is

$Y=\left({-1+t^{2}-x^{2}-y^{2}\over 2y},\,{t\over y},\,{x\over
y},\,{1+t^{2}-x^{2}-y^{2}\over 2y}\right).$

The boundary of AdS is located at $y=0$. In the ambient space the boundary is
the set of points that satisfy $Y^{2}=0$ with the identification $Y\cong aY$
(with $a\in{\mathbb{{R}}}^{+}$).

The string can be mapped into AdS3 by taking the target space to be
${\mathbb{{R}}}^{2,2}$ and then forcing the string to lie on the hyperboloid
using a Lagrange multiplier. In conformal gauge the action is given by

$S=-{T\over 2}\int d\tau
d\sigma(\partial_{\sigma}Y^{\mu}\partial_{\sigma}Y_{\mu}-\partial_{\tau}Y^{\mu}\partial_{\tau}Y_{\mu}+\lambda(Y^{2}+1)),$

where $T$ is the string tension and $Y(\tau,\sigma)\in{\mathbb{{R}}}^{2,2}$ is
the embedding function. The equation of motion in lightcone coordinates is

$\partial_{+}\partial_{-}Y-(\partial_{+}Y\cdot\partial_{-}Y)Y=0\,.$

Due to the gauge choice, the equations are supplemented by the Virasoro
constraints

$\partial_{-}Y\cdot\partial_{-}Y=\partial_{+}Y\cdot\partial_{+}Y=0\,.$

From the string embedding the sinh-Gordon field can be computed via

$\alpha=\log|\partial_{-}Y\cdot\partial_{+}Y|\,.$ (3)

If the unit normal vector $N$ is defined such that

$N_{a}=e^{-\alpha}\epsilon_{abcd}Y^{b}\partial_{-}Y^{c}\partial_{+}Y^{d}$

then the auxiliary fields are given by

$\displaystyle u$ $\displaystyle=$
$\displaystyle-N\cdot\partial_{-}\partial_{-}Y=\partial_{-}N\cdot\partial_{-}Y=-Y\cdot\partial_{-}\partial_{-}N$
$\displaystyle v$ $\displaystyle=$ $\displaystyle
N\cdot\partial_{+}\partial_{+}Y=-\partial_{+}N\cdot\partial_{+}Y=Y\cdot\partial_{+}\partial_{+}N$
(4)

The string equations of motion guarantee that $u=u({\sigma^{-}})$ and
$v=v({\sigma^{+}})$ and that the quantities satisfy the generalized sinh-
Gordon equation Pohlmeyer (1976); De Vega and Sanchez (1993). Note that while
changing $N\to-N$ flips the signs of both $u$ and $v$, their product which
appears in (1) remains invariant.

3\. Auxiliary linear system. Given a solution of the generalized sinh-Gordon
equation, one would like to construct the string embedding. In order to do
this one has to solve an auxilliary scattering problem. In terms of
$\alpha,u,v$ the $SL(2)$ Lax matrices are Alday and Maldacena (2009)

$\displaystyle B^{-}_{L}=\begin{pmatrix}{1\over
4}\partial_{-}\alpha&{1\over\sqrt{2}}{e^{{\alpha\over
2}}}\cr{u\over\sqrt{2}}{e^{-{\alpha\over 2}}}&-{1\over
4}\partial_{-}\alpha\end{pmatrix}$ $\displaystyle
B^{+}_{L}=\begin{pmatrix}-{1\over
4}\partial_{+}\alpha&-{v\over\sqrt{2}}{e^{-{\alpha\over
2}}}\cr-{1\over\sqrt{2}}e^{{\alpha\over 2}}&{1\over
4}\partial_{+}\alpha\end{pmatrix}$ $\displaystyle
B^{-}_{R}=\begin{pmatrix}-{1\over
4}\partial_{-}\alpha&{-{u\over\sqrt{2}}e^{-{\alpha\over
2}}}\cr{1\over\sqrt{2}}{e^{{\alpha\over 2}}}&{1\over
4}\partial_{-}\alpha\end{pmatrix}$ $\displaystyle
B^{+}_{R}=\begin{pmatrix}{1\over
4}\partial_{+}\alpha&-{1\over\sqrt{2}}{e^{{\alpha\over
2}}}\cr{v\over\sqrt{2}}{e^{-{\alpha\over 2}}}&-{1\over
4}\partial_{+}\alpha\end{pmatrix}\,.$

It is easy to check that the flatness conditions

$\displaystyle\partial_{-}B^{+}_{L}-\partial_{+}B^{-}_{L}+[B^{-}_{L},B^{+}_{L}]$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\partial_{-}B^{+}_{R}-\partial_{+}B^{-}_{R}+[B^{-}_{R},B^{+}_{R}]$
$\displaystyle=$ $\displaystyle 0$

are in fact equivalent to (1). Consider the left and right auxiliary linear
systems

$\displaystyle\partial_{-}\psi^{L}_{\alpha}+(B^{-}_{L})_{\alpha}^{~{}\beta}\psi^{L}_{\beta}=0\,,$
$\displaystyle\partial_{+}\psi^{L}_{\alpha}+(B^{+}_{L})_{\alpha}^{~{}\beta}\psi^{L}_{\beta}=0\,,$
$\displaystyle\partial_{-}\psi^{R}_{\dot{\alpha}}+(B^{-}_{R})_{\dot{\alpha}}^{~{}\dot{\beta}}\psi^{R}_{\dot{\beta}}=0\,,$
$\displaystyle\partial_{+}\psi^{R}_{\dot{\alpha}}+(B^{+}_{R})_{\dot{\alpha}}^{~{}\dot{\beta}}\psi^{R}_{\dot{\beta}}=0\,.\quad$
(5)

Figure 2:  Approximating the smooth embedding (yellow surface) with a
segmented string. After discretizing the string on a time-slice (three black
dots on a line) one can determine future null-separated points on the
worldsheet (two red dots). The resulting zigzag (dashed lines) constitutes the
initial condition for the segmented string.

where $\psi^{L}_{\alpha}$ and $\psi^{R}_{\dot{\alpha}}$ are two-component
spinors111The index $\alpha$ is not to be confused with the sinh-Gordon
potential.. Each of these systems have two linearly independent solutions,
denoted by $\psi^{L}_{\alpha,a}$ and $\psi^{R}_{\dot{\alpha},\dot{a}}$, (with
$a,\dot{a},\alpha,\dot{\alpha}=1,2)$. These can be normalized so that

$\epsilon^{\beta\alpha}\psi^{L}_{\alpha,a}\psi^{L}_{\beta,b}=\epsilon_{ab},~{}~{}~{}~{}~{}\epsilon^{\dot{\beta}\dot{\alpha}}\psi^{R}_{\dot{\alpha},\dot{a}}\psi^{R}_{\dot{\beta},\dot{b}}=\epsilon_{\dot{a}\dot{b}}$
(6)

where $\epsilon$ is the $2\times 2$ Levi-Civita tensor. Finally, the string
embedding is given by222For the scattering problem in an $SU(1,1)$ basis, see
Jevicki et al. (2008).

$Y_{a\dot{a}}\equiv\begin{pmatrix}Y_{-1}+Y_{2}&Y_{1}-Y_{0}\cr
Y_{1}+Y_{0}&Y_{-1}-Y_{2}\end{pmatrix}_{a,\dot{a}}=\psi^{L}_{\alpha,a}M^{\alpha\dot{\beta}}\psi^{R}_{\dot{\beta},\dot{a}}$
(7)

where $M=\textrm{diag}(1,1)$.

4\. Segmented strings. Once the smooth333In general the embedding will only be
smooth almost everywhere. Namely, the string may contain cusps which
correspond to singular solitons in the sinh-Gordon theory. string embedding is
obtained, one can proceed by approximating it with a segmented string.
Segmented strings are AdS generalizations of piecewise linear strings in flat
space Vegh (2018); Callebaut et al. (2015). They consist of elementary AdS2
segments which themselves are linear subspaces in ${\mathbb{{R}}}^{2,2}$.
Kinks between the segments move with the speed of light which ensures that the
string remains segmented at all times. The scalar curvature of the induced
metric is constant everywhere, except for points on the worldsheet where left-
and right-moving kinks collide. At these locations the curvature diverges
(similarly to the case of a polyhedron).

The segmentation procedure is depicted in FIG. 2. The diagram shows the string
worldsheet (yellow surface) embedded into AdS3. A time-slice on the worldsheet
is indicated by a black line. One can discretize this curve according to any
preferred scheme. This will result in a set of points $\\{P_{i}\\}$.

Figure 3:  Adjacent string segments have different inverse radii. The jump can
be smoothed by an appropriate function (dashed line) over a region of size
$\varepsilon$ (shaded area) so that the parameters of the two AdS2 segments
outside the region do not change.

These points are indicated by black dots in the figure. For a full set of
initial data, one needs to further specify another set of points denoted by
$\\{Q_{i}\\}$ (red dots in FIG. 2) which are null-separated from their
neighbors. This condition means that they must lie at the triple intersection
of future lightcones and the smooth worldsheet as in the figure. The initial
data for the segmented string will then be defined by the null zigzag
$P_{1},Q_{1},P_{2},Q_{2},\ldots$. If the points are taken to be in
${\mathbb{{R}}}^{2,2}$, then any three consecutive points define a linear AdS2
subspace: a segment of the string444The segmented string is an exact Nambu-
Goto string solution. Time evolution can be obtained by reflecting the points
on their two neighbors (e.g. $P_{2}$ on $Q_{1}$ and $Q_{2}$) as described in
Vegh (2018); Callebaut et al. (2015); Gubser et al. (2016); Vegh (2016, 2021).
. On the Poincaré patch the segments are given by the equations

$(x-x_{i})^{2}+y^{2}-(t-t_{i})^{2}=R_{i}^{2}$ (8)

where $i$ labels the segments. Each segment has three parameters
$\\{x_{i},t_{i},R_{i}\\}$. The total number of segments will be denoted by
$N$.

4\. Smoothing. One can compute the generalized sinh-Gordon fields
corresponding to a segmented string. Inside the AdS2 patches $u=v=0$, on the
kink worldlines and at the kink collision points (vertices) they are ill-
defined. In order for these fields to take finite values, the segmented string
has to be smoothed. This can be achieved by smoothing the initial conditions,
for instance by using Mikhailov’s solution Mikhailov (2003). The embedding is
given in terms of the position function of the string endpoint on the
boundary,

$\displaystyle t(t_{r},y)$ $\displaystyle=$ $\displaystyle
t_{r}+{y\over\sqrt{1-x_{0}^{\prime}(t_{r})^{2}}}$ $\displaystyle x(t_{r},y)$
$\displaystyle=$ $\displaystyle
x_{0}(t_{r})+{y\,x_{0}^{\prime}(t_{r})\over\sqrt{1-x_{0}^{\prime}(t_{r})^{2}}}$

Figure 4:  A patch of the worldsheet of the smoothed segmented string. Kinks
between AdS2 string segments (blue regions) occupy finite size regions (gray
areas). In balanced coordinates, their linear size is proportional to
$\sqrt{\varepsilon}$. If the smoothing does not affect the AdS2 patches, then
$u=v=0$ in these regions. Hence in balanced coordinates they are compressed
into points, kinks are squeezed into lines, and the vertex regions (orange
diamonds) fill the entire worldsheet.

where $x_{0}(t_{r})$ specifies the endpoint of the string in terms of the
retarded time $t_{r}$. The resulting string will have non-linear waves moving
in one direction.

Consider the motion given by

$x_{0}(t_{r})=\theta(\sqrt{1+t^{2}}-1)\,,$

where $\theta$ is the Heaviside function. Mikhailov’s embedding gives a
segmented string that consists of two pieces: a vertical segment at $x=0$
attached to another one with radius $R=1$. The relativistic acceleration of
the string endpoint (the inverse radius of the segment) jumps as seen in FIG.
3. One can smooth the kink by choosing a new $\tilde{x}_{0}(t_{r})$ function
which is equal to $x_{0}(t_{r})$ outside the region $(-{\varepsilon\over
2},{\varepsilon\over 2})$ and whose acceleration is smooth (red dashed line in
the figure). The average slope of the acceleration function will be of the
order of $\Delta a/\varepsilon$ where $\Delta a$ is the jump in the
acceleration. In order to have an estimate for the $u$ field across this
region, the smooth motion of the endpoint can be approximated by taking
$\tilde{x}_{0}(t_{r})={\Delta at_{r}^{3}/\varepsilon}$ which will produce a
linear acceleration function for small positive $t_{r}$. After changing to
lightcone coordinates in Mikhailov’s embedding, one gets $u\approx{6\Delta
a/\varepsilon}$. This means that in balanced coordinates the width of the
smoothed kink is (see FIG. 4)

$L\propto\sqrt{u}\,\varepsilon\propto\sqrt{\Delta a\varepsilon}\,.$

If there are $N$ segments, then $\Delta a$ scales as $N^{-1}$ and
$\varepsilon$ can also be taken to scale in the same way. The balanced linear
size on the worldsheet is then proportional to $N^{0}\sqrt{\varepsilon}$.

In the final step of the transformation, the sinh-Gordon fields $\alpha,u,v$
are computed. As $N\to\infty$, the segmented string approaches the original
smooth string and thus their time evolution will be the same.

Figure 5:  Scaling string embeddings for various values for the parameter $k$.
The figure corresponds to $t=1$ on the Poincaré patch. In these coordinates,
the size of the string increases linearly with time, but its shape does not
change. The dashed line indicates the lightcone emanating from the event at
$x=y=t=0$.

5\. An example. Consider the function

$\alpha=-\log\left[k\,\textrm{sn}^{2}\left({\sigma\over\sqrt{2k}}|-k^{2}\right)\right]$
(9)

where sn$(u|m)$ is the Jacobi elliptic function and $k\geq 1$ is a parameter.
It solves (1) with $u=-v=1$, i.e. it is a static cosh-Gordon solution.

In order to compress the function, the first step is constructing the
corresponding string embedding. The spinor solutions of (5) are found to be

$\displaystyle\psi^{L}_{1}$ $\displaystyle=$ $\displaystyle
e^{\omega_{-}}\left(\sqrt{\frac{2}{\text{sn}}+2k\,\text{sn}},\
-\frac{\sqrt{2}\left(\text{cn}\,\text{dn}+\text{sn}\sqrt{k^{2}-1}\right)}{\sqrt{\text{sn}+k\text{sn}^{3}}}\right)$
$\displaystyle\psi^{L}_{2}$ $\displaystyle=$ $\displaystyle
e^{-\omega_{-}}\left({\sqrt{\frac{1+k\text{sn}^{2}}{8\text{sn}\left(k^{2}-1\right)}}},\
\frac{\text{sn}\sqrt{k^{2}-1}-\text{cn}\,\text{dn}}{\sqrt{8\left(k^{2}-1\right)\left(\text{sn}+k\text{sn}^{3}\right)}}\right)$
$\displaystyle\psi^{R}_{1}$ $\displaystyle=$ $\displaystyle
e^{\omega_{+}}\left(-\frac{\sqrt{2}\left(\text{cn}\,\text{dn}+\text{sn}\sqrt{k^{2}-1}\right)}{\sqrt{\text{sn}-k\text{sn}^{3}}},\
\sqrt{\frac{2}{\text{sn}}-2k\,\text{sn}}\right)$ $\displaystyle\psi^{R}_{2}$
$\displaystyle=$ $\displaystyle
e^{-\omega_{+}}\left(\frac{-\text{sn}\sqrt{k^{2}-1}+\text{cn}\,\text{dn}}{\sqrt{8\left(k^{2}-1\right)\left(\text{sn}-k\text{sn}^{3}\right)}},\
-{\sqrt{\frac{1-k\text{sn}^{2}}{8\left(k^{2}-1\right)\text{sn}}}}\right)$

where

$\displaystyle\Pi_{\pm}$ $\displaystyle\equiv$ $\displaystyle\Pi\left(\pm
k;\text{am}\left(\frac{\sigma}{\sqrt{2k}}|-k^{2}\right)|-k^{2}\right)$
$\displaystyle\omega_{\pm}$ $\displaystyle\equiv$
$\displaystyle\sqrt{k-k^{-1}}\left(-\sqrt{k}\Pi_{\pm}+{\tau+\sigma\over
2\sqrt{2}}\right)$

and sn, cn and dn denote Jacobi elliptic functions with arguments
$\left(\frac{\sigma}{\sqrt{2k}}|-k^{2}\right)$, and am is the Jacobi amplitude
function. If one eliminates $\tau$ in favor of Poincaré time $t$, then the
embeddings will have the form $x(\sigma,t)=t\,\tilde{x}(\sigma)$ and
$y(\sigma,t)=t\,\tilde{y}(\sigma)$. Thus they scale linearly with time while
their shape is constant.

Figure 6:  Segmented string (with $N=13$ segments) approximating a static
smooth embedding (gray curve in the background) on the Poincaré patch. Kinks
move with the speed of light in the direction indicated by the arrows.
Smoothed segmented string.

The shape $\tilde{x}(\tilde{y})$ of such an embedding satisfies the equation
of motion

$\tilde{y}\left(\tilde{x}^{2}+\tilde{y}^{2}-1\right)\tilde{x}^{\prime\prime}+\left(2-2\tilde{y}^{2}\right)\tilde{x}^{\prime
3}+4\tilde{x}\tilde{y}\tilde{x}^{\prime
2}+\left(2-2\tilde{x}^{2}\right)\tilde{x}^{\prime}=0$

which can be derived from the Nambu-Goto action. FIG. 6 (left) shows the
embedding on the Poincaré patch for various values of $k$. As $k\to 1^{+}$, at
$t=1$ the string is mapped into a vanishingly small region near the origin and
its size increases slowly as time evolves. By applying an $SO(2,2)$ isometry
transformation the size of the string can be kept finite as the limit is
taken. Thus at $k=1$ the string is static. In the following, let us
concentrate on this special case. Its shape $x(y)$ satisfies the equation

$2x^{\prime}+2(x^{\prime})^{3}-yx^{\prime\prime}=0$

which is indeed invariant under rescaling the coordinates by a constant
factor. The explicit solution centered at $x=0$ has two branches555The Ricci
scalar of the induced metric is $R=-2\left(1+(y/y_{0})^{4}\right)$. Since
$R<R_{0}$ (where $R_{0}=-2$ is the curvature of AdS3) the solution in balanced
coordinates satisfies the cosh-Gordon equation. Close to the boundary ($y\to
0$) the solution asymptotes to AdS2.

$x(y)=\mp x_{0}\pm{y^{3}\over
3y_{0}^{2}}\,_{2}F_{1}\left({\frac{1}{2},\frac{3}{4};\frac{7}{4}};{y^{4}\over
y_{0}^{4}}\right)$

where
$x_{0}\equiv\frac{\sqrt{\pi}\Gamma\left(\frac{7}{4}\right)}{3\Gamma\left(\frac{5}{4}\right)}y_{0}$
and $y_{0}$ is a parameter. The string touches the boundary at $x=\pm x_{0}$.
In FIG. 6 (middle) the embedding is plotted (background gray curve).

Figure 7:  Two adjacent AdS2 patches on the worldsheet. Their normal vectors
are denoted by $N_{1}$ and $N_{2}$.

One can switch to balanced coordinates via the coordinate transformation given
by

$t={y_{0}\tau\over\sqrt{2}}\qquad
y=y_{0}\,\textrm{sn}\left({\sigma\over\sqrt{2}},-1\right)\,.$

In these coordinates $u=-v=1$ and it is easy to check that $\alpha$ is indeed
given by (9) with $k=1$.

The next step is to segment the string. Since in an appropriate frame the
string is static, one can apply the shooting technique described in Appendix B
of Vegh (2021). The only parameter is the number $N$ of string segments. The
resulting segmented string is periodic in Poincaré time. An example with
$N=13$ is depicted in FIG. 6 (middle). The black curves are circular arcs
given by (8). Red and black dots indicate left- and right-moving kinks between
the segments.

The third step is smoothing the segmented string. This can be done in the way
described earlier, but in practice it is easier to do the following. One can
subdivide the string into many pieces by adding “null kinks” Vegh (2016) which
do not further break the individual segments (i.e. they have the same
parameters in (8)). If each segment is subdivided into $n$ pieces then one has
$nN$ segments in total. After the subdivision, a smoothing filter can be
applied on the points. For instance one can repeatedly apply the
transformation

$P_{i}^{\prime}\propto\lambda(P_{i-1}+\lambda P_{i+1})+P_{i}\quad\textrm{such
that}\quad(P_{i}^{\prime})^{2}=-1\,,$

for points in the refined segmented string, and the same for $Q_{i}$. This
step smoothes the points similarly to a discretized heat equation. After an
iteration $Q^{\prime}_{i}-P^{\prime}_{i}$ and
$Q^{\prime}_{i}-P^{\prime}_{i+1}$ are generically not null, but one can define
new $Q^{\prime\prime}_{i}$ positions such that they lie in the AdS2 spanned by
$\\{P^{\prime}_{i},Q^{\prime}_{i},P^{\prime}_{i+1}\\}$ and that
$(Q^{\prime\prime}_{i}-P^{\prime}_{i})^{2}=(Q^{\prime\prime}_{i}-P^{\prime}_{i+1})^{2}=0$,
and the points $P^{\prime\prime}_{i}=P^{\prime}_{i}$ are not changed. In the
next iteration one smoothes the initial data given by the null zigzag
$\\{P^{\prime\prime}_{i},Q^{\prime\prime}_{i}\\}$. Let $q$ denote the number
of smoothing iterations on the string. For sufficiently large $n$, the
resulting string is a good approximation for a smooth string. An example is
depicted in FIG. 6 (right).

Figure 8:  Top: Density plot of the Ricci scalar of the induced metric on the
worldsheet of a smoothed segmented string. The function is approximately
periodic in time. Middle: The $u(\sigma^{-})$ function. Bottom: The compressed
cosh-Gordon potential $\alpha$ in balanced $(\tilde{\tau},\tilde{\sigma})$
coordinates. Peaks move with the speed of light and when they collide the
$e^{\alpha}$ area density becomes large. These points (orange) correspond to
the AdS2 patches of the smoothed segmented string. The blue diamonds
correspond to smoothed vertices.

Finally, the generalized sinh-Gordon fields $\alpha,u,v$ can be computed using
worldsheet points and normal vectors which all take values in
${\mathbb{{R}}}^{2,2}$. For instance in FIG. 7 for the patch with normal
vector $N_{1}$ one has

$e^{\alpha}=a^{-2}(V_{10}-V_{00})(V_{01}-V_{00})$ (10)

where $a$ is the lattice spacing. A discrete version of (4) is given by

$v\approx a^{-2}(N_{1}-N_{2})(V_{01}-V_{00})$ (11)

and an approximate $u$ function can similarly be computed. FIG. 8 (middle)
shows an example for $v$ computed this way for a slightly smoothed string with
$N=30$ segments. After switching to balanced coordinates, the cosh-Gordon
field is seen to have peaks traveling with the speed of light, see FIG. 8
(bottom). Time-slices are show in FIG. 9. For comparison, the original
potential given in (9) with $k=1$ is plotted in black. The difference between
the two figures is the number of smoothing iterations applied on the
subdivided segmented string. Without any smoothing the balanced width of the
string would be zero and the spikes would diverge.

Finally, one can check that the balanced width $L$ of the compressed function
indeed scales as expected. Based on the properties of the heat equation, one
expects that an appropriate smoothing parameter can be defined as
$\varepsilon=\sqrt{q}/n$ and that $L\propto\sqrt{\varepsilon}$. This is indeed
born out by numerical calculations: for an example with $N=10$ segments a
numerical fit gave $L\propto n^{\gamma}$ with $\gamma=-0.495\pm 0.01$ and
$L\propto q^{\delta}$ with $\delta=0.246\pm 0.01$, confirming the
expectations.

Figure 9:  Top: A static cosh-Gordon solution from eqn. (9) with $k=1$ (black
curve) and its compressed version (blue curve) on a time-slice. The compressed
function is obtained from a smoothed segmented string with $N=30$ segments.
The arrow indicates the reduction of the balanced width. The snapshot is taken
at the moment when left- and right-moving peaks collide. Bottom: A smaller
amount of smoothing yields a more compressed (and more singular) potential.

6\. Discussion. This paper presented a transformation of the generalized sinh-
Gordon field that approximately commutes with time-evolution. The
transformation has been defined by constructing from the sinh-Gordon field a
long Nambu-Goto string in AdS3. The (almost everywhere) smooth string is
approximated by a segmented string. The segmented string is smoothed and the
new sinh-Gordon field is computed from the embedding. The new function
occupies a smaller region in balanced coordinates in which the equation has a
standard sinh- or cosh-Gordon form. The transformed function has peaks (see
FIG. 9) which travel with the speed of light.

For the transformation, one can use various segmentation and smoothing
schemes. In a given scheme one is still left with two parameters: the number
of string segments $N$, and the smoothing parameter $\varepsilon$. It is
plausible that in an appropriate large-$N$ limit the scheme-dependence goes
away. It would be interesting to see if one can define the transformation in a
closed form similarly to an auto-Bäcklund transformation.

The equations of motion for a string in AdS3 have an internal $SO(2,2)$
symmetry acting on the vectors $\\{Y,e^{-{\alpha\over
2}}\partial_{-}Y,e^{-{\alpha\over 2}}\partial_{+}Y,N\\}$ Alday and Maldacena
(2009). Note that this symmetry has been broken by both smoothing procedures
discussed in the paper.

The results made use of the equivalence of the sinh-Gordon equation and the
string equation of motion in AdS3. Similar equivalences exist in higher
dimensional AdS spacetimes which allows for the extension of the compressing
transformation to other systems, such as the $B_{2}$ Toda theory which is
related to strings moving in AdS4 Jevicki and Jin (2010).

The transformed function contains $N$ peaks which superficially look similar
to boosted sinh-Gordon solitons. Solitons can be thought of as particles whose
motion is governed by the hyperbolic Ruijsenaars-Schneider model with a
special coupling value Ruijsenaars and Schneider (1986). Since the balanced
size of the transformed domain vanishes as $\varepsilon\to 0$, it is plausible
to think that this gives an interesting limit of the model. Perhaps this limit
leads to a dual description of the string.

_Acknowledgments._ This work is supported by the STFC Ernest Rutherford grant
ST/P004334/1.

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