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\bf{PHY 711 -- Notes on Hydrodynamics -- ("Solitary Waves"\cite{FW10})}
\end{center}

\subsubsection*{Basic assumptions}
We assume that we have in incompressible fluid ($\rho =$ constant)  a velocity potential of the form $\Phi(x,z,t)$.  The surface of the fluid is described by $h+\zeta(x,t)=z$.  The fluid is contained in a tank with a structureless bottom (defined by the plane $z = 0$) and is filled to a vertical height $h$ at equilibrium. These functions satisfy the following conditions.
\begin{equation}
{\mbox{Poisson equation:    }}\frac{\partial^2 \Phi(x,z,t)}{\partial x^2} + \frac{\partial^2 \Phi(x,z,t)}{\partial z^2} = 0   \label{poisson}
\end{equation}
\begin{equation}
{\mbox{Zero vertical velocity at bottom of the tank:    }}
\frac{\partial \Phi(x,0,t)}{\partial z} = 0  \label{bot}
\end{equation}
\begin{equation}
{\mbox{Bernoulli's equation:    }}
\left. -\frac{\partial \Phi(x,z,t)}{\partial t} + \frac{1}{2}\left(\frac{\partial \Phi(x,z,t)}{\partial x}\right)^2 +g\zeta(x,t)\right\rfloor_{z=h+\zeta} = 0  \label{bern}
\end{equation}
\begin{equation}
{\mbox{Surface equation:    }}
\left. -\frac{\partial \Phi(x,z,t)}{\partial z} + \frac{\partial \Phi(x,z,t)}{\partial x} \frac{\partial \zeta(x,t)}{\partial x} - \frac{\partial \zeta(x,t)}{\partial t}\right\rfloor_{z=h+\zeta} = 0  \label{surf}
\end{equation}



In this treatment, we assume seek the form of surface waves traveling along the $x-$ direction and assume that the effective wavelength is much larger than the height of the surface $h$.   This allows us to approximate the $z-$ dependence of $\Phi(x,z,t)$ by means of a Taylor series expansion:
\begin{equation}
\Phi(x,z,t) \approx \Phi(x,0,t)
+ z \frac{\partial \Phi}{\partial z}(x,0,t)
+ \frac{z^2}{2} \frac{\partial^2 \Phi}{\partial z^2}(x,0,t)
+  \frac{z^3}{3!} \frac{\partial^3 \Phi}{\partial z^3}(x,0,t) \cdots  \label{tay1}
\end{equation}

and

\begin{equation}
\frac{\partial \Phi(x,z,t)}{\partial z} \approx
+  \frac{\partial \Phi}{\partial z}(x,0,t)
+ z \frac{\partial^2 \Phi}{\partial z^2}(x,0,t)
+  \frac{z^2}{2} \frac{\partial^3 \Phi}{\partial z^3}(x,0,t)
+  \frac{z^3}{3!} \frac{\partial^4 \Phi}{\partial z^4}(x,0,t)
\cdots . \label{tay2}
\end{equation}

These expansions can be simplified because of the bottom boundary condition (\ref{bot}) and the Poisson equation (\ref{poisson}).  In particular, (\ref{bot}) and (\ref{poisson}) ensure that all odd derivatives $\frac{\partial^n \Phi}{\partial z^n}(x,0,t)$ vanish from the Taylor expansions (\ref{tay1}) and (\ref{tay2}). Taking the time derivative of \ref{tay1} and evaluating it at the surface, $z=h+\zeta(x,t)$, and  using the Poisson equation (\ref{poisson}) to convert all derivatives with respect to $z$ to derivatives with respect to $x$, we find:

\begin{equation}
\frac{\partial \Phi(x,z,t)}{\partial t} \approx \frac{\partial \Phi(x,0,t)}{\partial t}
- \frac{\partial}{\partial t} \left( \frac{\left(h+\zeta(x,t)\right)^2}{2} \frac{\partial^2 \Phi}{\partial x^2}(x,0,t) \right)   \cdots ,\label{tay1a}
\end{equation}

and simplifying (\ref{tay2}), we find:

\begin{equation}
\frac{\partial \Phi(x,z,t)}{\partial z} \approx
- \left(h+\zeta(x,t)\right) \frac{\partial^2 \Phi}{\partial x^2}(x,0,t)  +  \frac{\left(h+\zeta(x,t)\right)^3}{3!} \frac{\partial^4 \Phi}{\partial x^4}(x,0,t)   \cdots .\label{tay2a}
\end{equation}


We can now use these results to approximate the boundary conditions implied by Bernoulli's equation (\ref{bern} and the time derivative the of the surface equation (\ref{surf}).   Keeping all terms up to leading order in non-linearity and up to fourth order derivatives in the linear terms we find that the surface definition equation becomes:
\begin{equation}
 \frac{\partial}{\partial x} \left( (h+\zeta(x,t))\frac{\partial \Phi(x,0,t)}{\partial x} \right) - \frac{h^3}{3!}\frac{\partial^4 \Phi(x,0,t)}{\partial x^4}-\frac{\partial \zeta(x,t)}{\partial t} = 0,   \label{surfa}
\end{equation}
and the Bernoulli's equation becomes:
\begin{equation}
 -\frac{\partial \Phi(x,0,t)}{\partial t} +\frac{h^2}{2}\frac{\partial^3 \Phi(x,0,t)}{\partial t \partial x^2}  + \frac{1}{2} \left( \frac{\partial \Phi(x,0,t)}{\partial x} \right)^2+g\zeta(x,t) = 0.   \label{berna}
\end{equation}

We would like to solve Eqs. (\ref{surfa}-\ref{berna}) for a traveling wave of the form:
\begin{equation}
\Phi(x,0,t) = \phi(x-ct) {\mbox{     and     }} \zeta(x,t) = \psi(x-ct), \label{wave}
\end{equation}
where the speed of the wave $c$ will be determined.
Letting $u \equiv x-ct$, Eqs. (\ref{surfa} and \ref{berna}) become:
\begin{equation}
\frac{d}{d u}\left( (h + \psi(u)) \frac{d \phi(u)}{d u} \right)
-\frac{h^3}{6}\frac{d^4 \phi(u)}{d u^4} +c\frac{d \psi(u)}{d u} =
0,\label{surfb}
\end{equation}
and
\begin{equation}
c\frac{d \phi(u)}{d u} - \frac{ch^2}{2}\frac{d^3 \phi(u)}{d u^3}
+\frac{1}{2}\left(\frac{d \phi(u)}{d u} \right)^2 + g \psi(u) = 0.
\label{bernb}
\end{equation}

The modified surface equation (\ref{surfb}) can be integrated once with respect to u, choosing the constant of integration to be zero and giving the new form for the surface condition:
\begin{equation}
(h+\psi) \phi^{\prime}-\frac{h^3}{6}\phi^{\prime \prime \prime} +c
\psi = 0, \label{surfc}
\end{equation}

where we have abreviated derivatives with respect to $u$ with the
``$^\prime$" symbol. This equation, and the modified Bernoulli
equation (\ref{bernb}) are now two coupled non-linear equations.
In order to solve them, we use, the modified Bernoulli equation to
approximate $\phi^{\prime}(u)$ and its higher drivatives in terms
of the surface function $\psi(u)$.  Equation (\ref{bernb}) becomes
approximately:

\begin{equation}\label{bernba}
\phi^{\prime} = -\frac{g}{c}\psi +\frac{h^2}{2}\phi^{\prime \prime
\prime} - \frac{1}{2c}(\phi^{\prime})^2 \approx -\frac{g}{c}\psi -
\frac{h^2g}{2c}\psi^{\prime \prime} -\frac{g^2}{2c^3}\psi^2.
\end{equation}
Using similar approximations, we can eliminate  $\phi^{\prime}(u)$
and its higher drivatives from the surface equation (\ref{surfc}):
\begin{equation}\label{surfca}
(h+\psi) \left( -\frac{g}{c}\psi - \frac{h^2g}{2c}\psi^{\prime
\prime} -\frac{g^2}{2c^3}\psi^2 \right) + \frac{h^3 g}{6
c}\psi^{\prime \prime} + c \psi = 0,
\end{equation}

where some terms involving non-linearity of higher than 2 or
involving higher order derivatives have been  discarded.
Collecting the leading terms, we obtain:
\begin{equation}\label{kew1}
  \left(1 -\frac{gh}{c^2}\right) \psi - \frac{g h^3}{3 c^2}
  \psi^{\prime \prime} - \frac{g}{c^2} \left(1 + \frac{gh}{2c^2} \right) \psi^2 = 0.
\end{equation}

For the second two terms, {\em{Fetter and Walecka}} argue that it
is consistent to approximate $gh \approx c^2$, which reduces
(\ref{kew1}) to


\begin{equation}\label{kew2}
\left( 1 - \frac{hg}{c^2} \right) \psi(u)  - \frac{h^2}{3}
\psi^{\prime \prime}(u) -   \frac{3}{2h} \left[ \psi(u) \right]^2
= 0.
\end{equation}

Your text shows that a solution to Eq. (\ref{kew2}), with the
initial condition $\psi(0) = \psi_0$ and $\psi^{\prime}(0)=0$, is
the solitary wave form:
\begin{equation}
\zeta(x,t)=\psi(x-ct)=\psi_0 \; {\mbox{\rm{sech}}}^2  \left(
\sqrt{\frac{3\psi_0}{h}}\;\frac{x-ct}{2h}\right), \label{solitary}
\end{equation}
with
\begin{equation}
c = \sqrt{\frac{gh}{1-\psi_0/h}} \approx \sqrt{gh} \left(
1+\frac{\psi_0}{2h} \right).
\end{equation}

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\bibitem{FW10} Alexander L. Fetter and John Dirk Walecka, {\bf{Theoretical Mechanics of Particles and Continua}}, (McGraw Hill, 1980), Chapt. 10.
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