The initial value problem for a third order dispersive equation on the two-dimensional torus
Hiroyuki CHIHARA
We present the necessary and sufficient condition for the $L^2$-well-posedness of the initial problem for a third order linear dispersive equation on the two dimensional torus. Birkhoff's method of asymptotic solutions is used to prove the necessity. Some properties of a system for quadratic algebraic equations associated to the principal symbol play crucial role in proving the sufficiency.
math0404006.tex
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\title[dispersive equation]{The initial value problem for a third order dispersive equation on the two-dimensional torus}
\author[H.~Chihara]{Hiroyuki CHIHARA}
\address{Mathematical Institute,  
         Tohoku University, 
         Sendai 980-8578, Japan}
\email{chihara@math.tohoku.ac.jp}
\subjclass[2000]{35G10}
%\thanks{Supported by JSPS Grant-in-Aid for Scientific Research
%\#12740086.}
%\date{} 
%\dedicatory{}
%\keywords{}
\begin{abstract}
We present the necessary and sufficient condition for 
the $L^2$-well-posedness of the initial problem 
for a third order linear dispersive equation 
on the two dimensional torus. 
Birkhoff's method of asymptotic solutions is 
used to prove the necessity. 
Some properties of a system for 
quadratic algebraic equations associated to 
the principal symbol 
play crucial role in proving the sufficiency.  
\end{abstract}
\maketitle
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%%%%%% Document
%%%%%%
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%%%%%% Section 1
%%%%%%
\section{Introduction}
\label{section:1}
This paper is concerned with the initial value problem of the form 
%
%
\begin{alignat}{2}
Lu&=f(t,x)
&
\quad\text{in}\quad
&
\mathbb{R}\times\mathbb{T}^2,
\label{equation:pde}
\\
u(0,x)&=u_0(x)
&
\quad\text{in}\quad
&
\mathbb{T}^2,
\label{equation:data}
\end{alignat}
%
%
where 
$u(t,x)$ is a real-valued unknown function of 
$(t,x)=(t,x_1,x_2)\in\mathbb{R}\times\mathbb{T}^2$, 
$\mathbb{T}^2=\mathbb{R}^2/2\pi\mathbb{Z}^2$, 
$u_0(x)$ and $f(t,x)$ are given real-valued functions, 
%
%
$$
L
=
\p_t+p(\p)
+
\sum_{\lvert\alpha\rvert=2}
\frac{2!}{\alpha!}
a_{\sigma(\alpha)}(x)\p^\alpha
+
\vec{b}(x)\cdot\nabla
+
c(x),
$$
%
%
$\p_t=\p/\p{t}$, 
$\p_j=\p/\p{x_j}$, 
$\p=\nabla=(\p_1,\p_2)$, 
$p(\xi)=\xi_1\xi_2(\xi_1+\xi_2)$, 
$\alpha=(\alpha_1,\alpha_2)$ is a multi-index, 
$\lvert\alpha\rvert=\alpha_1+\alpha_2$, 
$\alpha!=\alpha_1!\alpha_2!$, 
$\sigma(\alpha)=(\alpha_1-\alpha_2)/2$, 
$\vec{b}(x)=(b_1(x),b_2(x))$, 
and 
$a_{\sigma(\alpha)}$, $b_j(x)$, $c(x)$ 
are real-valued smooth functions on $\mathbb{T}^2$. 
Such operators arise 
in the study of gravity wave of deep water. 
See \cite{bks}, \cite{dysthe}, \cite{hogan} 
and the references therein. 
%
\par
%
The purpose of this paper is to present 
the necessary and sufficient condition of 
the existence of a unique solution to 
\eqref{equation:pde}-\eqref{equation:data}. 
To state our results, 
we introduce notation and the definition of $L^2$-well-posedness.  
$C^\infty(\mathbb{T}^2)$ 
is the set of all smooth functions on
$\mathbb{T}^2$. 
$L^2(\mathbb{T}^2)$ 
is the set of all square-integrable functions on
$\mathbb{T}^2$. 
For $g{\in}L^2(\mathbb{T}^2)$, set 
%
%
$$
\lVert{g}\rVert
=
\left(
\int_{\mathbb{T}^2}\lvert{g(x)}\rvert^2dx
\right)^{1/2}. 
$$
%
%
$C(\mathbb{R};L^2(\mathbb{T}^2))$ 
is the set of all $L^2(\mathbb{T}^2)$-valued 
continuous functions on $\mathbb{R}$, 
and similarly,  
$L^1_{\text{loc}}(\mathbb{R};L^2(\mathbb{T}^2))$ 
is the set of all $L^2(\mathbb{T}^2)$-valued 
locally integrable functions on $\mathbb{R}$. 
Here we state the definition of $L^2$-well-posedness. 
%
%
\begin{definition}
\label{definition:l2wp}
The initial value problem 
{\rm \eqref{equation:pde}-\eqref{equation:data}} 
is said to be $L^2$-well-posed 
if for any $u_0{\in}L^2(\mathbb{T}^2)$ 
and $f{\in}L^1_{\text{loc}}(\mathbb{R};L^2(\mathbb{T}^2))$, 
{\rm \eqref{equation:pde}-\eqref{equation:data}} 
possesses a unique solution 
$u{\in}C(\mathbb{R};L^2(\mathbb{T}^2))$. 
\end{definition}
%
%
In view of Banach's closed graph theorem, 
if \eqref{equation:pde}-\eqref{equation:data} is $L^2$-well-posed, 
then for any $T>0$, there exists $C_T>0$ such that 
all the solutions satisfy the energy inequality 
%
%
\begin{equation}
\lVert{u(t)}\rVert
\leqslant
C_T
\left(
\lVert{u_0}\rVert
+
\left\lvert
\int_0^t\lVert{f(s)}\rVert{ds}
\right\rvert
\right),
\quad
(\lvert{t}\rvert\leqslant{T}). 
\label{equation:energy}
\end{equation}
%
%
\par
%
Our results are the following.
%
%
\begin{theorem}
\label{theorem:main}
The following conditions are mutually equivalent. 
\\
{\rm I}. \quad
{\rm \eqref{equation:pde}-\eqref{equation:data}} is $L^2$-well-posed. 
\\
{\rm II}. \quad
For any $x\in\mathbb{T}^2$ and for any $\xi\in\Lambda$, 
%
%
\begin{equation}
 \int_0^{2\pi}
 \sum_{\lvert\alpha\rvert=2}
 \frac{2!}{\alpha!}
 a_{\sigma(\alpha)}(x+tp^\prime(\xi))\xi^\alpha
 dt
 =0, 
\label{equation:vanish}
\end{equation}
%
%
where 
$
p^\prime(\xi)=\nabla_\xi{p(\xi)}
=(\xi_2(2\xi_1+\xi_2),\xi_1(2\xi_2+\xi_1))$ 
and 
$
\Lambda
=
\{
\xi\in\mathbb{R}^2 
\vert 
p^\prime(\xi)\in\mathbb{Z}^2
\}
$. 
\\
{\rm III}. \quad
$a_0(x)=a_1(x)+a_{-1}(x)$,
and there exists $\phi(x){\in}C^\infty(\mathbb{T}^2)$ 
such that 
$\nabla\phi(x)=(a_{-1}(x),a_1(x))$.
\end{theorem}
%
%
Here we explain the background of our problem. 
There are many papers dealing with the well-posedness of 
the initial value problem for dispersive equations. 
Generally speaking, 
it is difficult to characterize the well-posedness. 
In fact, results on the characterization are very limited. 
In \cite{mizohata} Mizohata studied 
the initial value problem for 
Schr\"odinger-type operator of the form 
%
%
$$
S
=
\p_t-iq(\p)+\vec{b}(x)\cdot\nabla+c(x)
$$
%
%
on $\mathbb{R}^n$, 
where $i=\sqrt{-1}$ and $q(\xi)=\xi_1^2+\dotsb+\xi_n^2$. 
He gave the necessary condition for 
the $L^2$-well-posedness
%
%
\begin{equation}
\sup_{(T,x,\xi)\in\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n}
\left\lvert
\int_0^T
\im\vec{b}(x+tq^\prime(\xi))\cdot\xi
dt
\right\rvert
<+\infty. 
\label{equation:mizohata}
\end{equation}
%
%  
\eqref{equation:mizohata} 
gives the upper bound of the admissible bad first order term 
$(\im\vec{b}(x))\cdot\nabla$. 
In other words, 
\eqref{equation:mizohata} 
is necessary so that $(\im\vec{b}(x))\cdot\nabla$ 
can be controled by the local smoothing effect of 
$e^{itq(\p)}$. 
In \cite{ichinose1} 
Ichinose generalized \eqref{equation:mizohata} 
for the Schr\"odinger-type equation on a 
complete Riemannian manifold. 
When $n=1$, 
\eqref{equation:mizohata} 
is also the sufficient condition for $L^2$-well-posedness. 
Moreover, in \cite{tarama1} and \cite{tarama2} 
Tarama characterized the $L^2$-well-posedness 
for third order equations on $\mathbb{R}$. 
%
\par
%
For equations on compact manifolds, 
the local smoothing effect breaks down 
everywhere in the manifold. 
Then, the restriction on the bad lower order terms 
becomes stronger, 
and the characterization of $L^2$-well-posedness 
seems to be relatively easier. 
In fact, 
the $L^2$-well-posedness for $S$ on $\mathbb{T}^n$ is characterized. 
See \cite{chihara}, \cite{ichinose2} and \cite{tsukamoto}. 
%
\par
%
$L$ is the simplest example of 
higher order dispersive operators on 
higher dimensional spaces. 
$p(\xi)$ satisfies 
$p^\prime(\xi)\ne0$ and 
$\det{p^{\prime\prime}(\xi)}\ne0$ 
for $\xi\ne0$. 
The symbol of the Laplacian $q(\xi)$ also satisfies 
the same conditions. 
It is interesting that our method of the proof 
does not work if we replace $p(\xi)$ 
by $r(\xi)=\xi_1^3+\xi_2^3$. 
This seems to be due to the degeneracy of 
$\det{r^{\prime\prime}}(\xi)=36\xi_1\xi_2$ for $\xi\ne0$. 
%
\par
%
The organization of this paper is as follows. 
In Sections~\ref{section:2} and \ref{section:3} 
we prove I $\Longrightarrow$ II and 
II $\Longrightarrow$ III respectively. 
We omit the proof of III $\Longrightarrow$ I. 
In deed, under the condition 
$a_0=a_{1}+a_{-1}$, $L$ becomes 
%
%
$$
L
=
\p_t
+
p(\p)
+
a_{-1}(x)\frac{\p p}{\p\xi_1}(\p)
+
a_{1}(x)\frac{\p p}{\p\xi_2}(\p)
+
\vec{b}(x)\cdot\nabla
+
c(x).
$$
%
%
In view of $\nabla\phi=(a_{-1},a_1)$, 
we have 
%
%
$$
e^\phi{L}e^{-\phi}
=
\p_t+A, 
\quad
A
=
p(\p)
+
\tilde{b}_1(x)\p_1
+
\tilde{b}_2(x)\p_2
+
\tilde{c}(x),
$$
%
%
where 
$\tilde{b}_1(x)$, 
$\tilde{b}_2(x)$ 
and  
$\tilde{c}(x)$ 
are real-valued smooth functions on $\mathbb{T}^2$. 
It is easy to see that the initial value problem for 
$e^\phi{L}e^{-\phi}$ is $L^2$-well-posed since 
%
%
$$
A+A^\star
=
-\frac{\p\tilde{b}_1}{\p{x_1}}(x)
-\frac{\p\tilde{b}_2}{\p{x_2}}(x). 
$$
%%%%%%
%%%%%% Section 2
%%%%%%
\section{The Proof of I $\Longrightarrow$ II}
\label{section:2}
We begin with the reduction of \eqref{equation:vanish}. 
Set 
%
%
$$
a(x,\xi)
=
\sum_{\lvert\alpha\rvert=2}
\frac{2!}{\alpha!}
a_{\sigma(\alpha)}(x)\xi^\alpha
$$
%
%
for short. If 
%
%
\begin{equation}
\sup_{(T,x,\xi)\in\mathbb{R}\times\mathbb{T}^2\times\mathbb{R}^2}
\left\lvert
\int_0^T
a(x+tp^\prime(\xi),\xi)
dt
\right\rvert
<+\infty, 
\label{equation:real}
\end{equation}
%
%
then \eqref{equation:vanish} holds 
since $a(x+tp^\prime(\xi),\xi)$ is a $2\pi$-periodic function in
$t\in\mathbb{R}$ for any $(x,\xi)\in\mathbb{T}^2\times\Lambda$. 
\eqref{equation:real} is equivalent to 
an apparently weaker condition 
%
%
\begin{equation}
\sup_{(T,x,\xi)\in\mathbb{R}\times\mathbb{T}^2\times\mathbb{Q}^2}
\left\lvert
\int_0^T
a(x+tp^\prime(\xi),\xi)
dt
\right\rvert
<+\infty. 
\label{equation:rational}
\end{equation}
%
%
For $\xi\in\mathbb{Q}^2$, there exist 
$\alpha\in\mathbb{Z}^2$ and $l\in\mathbb{N}$ such that 
$\xi=\alpha/l$. 
Changing the variable by $t=l^2s$, we have 
%
%
$$
\int_0^T
a(x+tp^\prime(\xi),\xi)
dt
=
\int_0^{T/l^2}
a(x+sp^\prime(\alpha),\alpha)
dt.
$$
%
%
Then, \eqref{equation:rational} is equivalent to 
%
%
\begin{equation}
\sup_{(T,x,\alpha)\in\mathbb{R}\times\mathbb{T}^2\times\mathbb{Z}^2}
\left\lvert
\int_0^T
a(x+tp^\prime(\alpha),\alpha)
dt
\right\rvert
<+\infty. 
\label{equation:integer}
\end{equation}
%
%
To prove I $\Longrightarrow$ II. 
we shall show the contraposition of 
I $\Longrightarrow$ \eqref{equation:integer}. 
%
\par
%
Suppose that \eqref{equation:integer} fails to hold, 
that is, 
for any $n\in\mathbb{N}$, 
there exist 
$T_n\in\mathbb{R}$, 
$x_n\in\mathbb{T}^2$ 
and 
$\alpha\in\mathbb{Z}^2$ 
such that 
%
%
$$
\left\lvert
\int_0^{T_n}
a(x_n+tp^\prime(\alpha),\alpha)
dt
\right\rvert
\geqslant
2n. 
$$
%
%
We shall show that the energy inequality 
\eqref{equation:energy} fails to hold. 
Firstly, we consider the case that $T_n>0$ and 
%
%
$$
\int_0^{T_n}
a(x_n+tp^\prime(\alpha),\alpha)
dt
\geqslant
2n. 
$$
%
%
Since $[0,T_n]\times\mathbb{T}^2$ is compact and 
%
%
$$
(s,x)\in[0,T_n]\times\mathbb{T}^2
\longmapsto
\int_0^s
a(x+tp^\prime(\alpha),\alpha)
dt
$$
%
%
is continuous, 
there exist $(T,y)\in(0,T_n]\times\mathbb{T}^2$ such that 
%
%
$$
\max_{(s,x)\in[0,T_n]\times\mathbb{T}^2}
\int_0^s
a(x+tp^\prime(\alpha),\alpha)
dt
=
\int_0^T
a(y+tp^\prime(\alpha),\alpha)
dt. 
$$
%
%
Pick up $\psi{\in}C^\infty(\mathbb{T}^2)$ such that 
$\lVert\psi\rVert=1$ and 
%
%
$$
\int_0^T
a(x+tp^\prime(\alpha),\alpha)
dt
\geqslant
n
\quad\text{in}\quad
\supp[\psi].
$$
%
%
\par
%
Let $u$ be a complex-valued solution to 
\eqref{equation:pde} 
with a complex-valued given function $f(t,x)$. 
Then, $L\re{u}=\re{f}$ and $L\im{u}=\im{f}$ 
since all the coefficients in $L$ are real-valued. 
We construct a sequence of complex-valued 
asymptotic solutions to $Lu=0$. 
For $l\in\mathbb{N}$, set 
%
%
$$
u_l(t,x)
=
e^{itp(l\alpha)+il\alpha\cdot{x}+\phi_l(t,x)}
\psi(x+(t-T/l^2)p^\prime(l\alpha)),
$$
%
%
$$
\phi_l(t,x)
=
\int_0^{l^2t}
a(x+sp^\prime(\alpha),\alpha)
ds.  
$$
%
%
Then, $u_l{\in}C^\infty(\mathbb{R}\times\mathbb{T}^2)$, 
%
%
\begin{align}
  \lVert{u_l}(0)\rVert
& =
  \lVert\psi\rVert
  =1,
\label{equation:initial}
\\
  \lVert{u_l}(T/l^2)\rVert
& =
  \lVert\exp(\phi_l(T/l^2,\cdot))\psi(\cdot)\rVert
  \geqslant
  e^n.
\label{equation:evolution}
\end{align}
%
%
\par
%
Next we compute $Lu_l$. 
Set $b(x,\xi)=\vec{b}(x)\cdot\xi$ and 
$v_l(t,x)=e^{-itp(l\alpha)-il\alpha\cdot{x}}u_l(t,x)$ 
for short. 
We deduce 
%
%
\begin{align}
  e^{-itp(l\alpha)-il\alpha\cdot{x}}Lu_l
& =
  (\p_t+ip(l\alpha))v_l
  +
  p(\p+il\alpha)v_l
\nonumber
\\
& +
  a(x,\p+il\alpha)v_l
  +
  b(x,\p+il\alpha)v_l
  +
  c(x)v_l,
\label{equation:lu} 
\\
  (\p_t+ip(l\alpha))v_l
& =
  ip(l\alpha)v_l+a(x+tp^\prime(l\alpha),l\alpha))v_l
\nonumber
\\
& +
  e^{\phi_l}
  p^\prime(l\alpha)\cdot\nabla\psi(x+(t-T/l^2)p^\prime(l\alpha))
\label{equation:part1}
\\
  p(\p+il\alpha)v_l
& =
  (p(\p)+il\alpha\cdot{p^\prime(\p)}-p^\prime(l\alpha)\cdot\nabla-ip(l\alpha))v_l
\nonumber
\\
& =
  (p(\p)+il\alpha\cdot{p^\prime(\p)})v_l-ip(l\alpha)v_l
\nonumber
\\
& -
  \left(
  \int_0^{l^2t}
  p^\prime(l\alpha)\cdot\nabla(a(x+sp^\prime(\alpha),\alpha))
  ds
  \right)
  v_l
\nonumber
\\
& -
  e^{\phi_l}
  p^\prime(l\alpha)\cdot\nabla\psi(x+(t-T/l^2)p^\prime(l\alpha))
\nonumber
\\
& =
  (p(\p)+il\alpha\cdot{p^\prime(\p)})v_l-ip(l\alpha)v_l
\nonumber
\\
& -
  \left(
  \int_0^{l^2t}
  \frac{d}{ds}(a(x+sp^\prime(\alpha),l\alpha))
  ds
  \right)
  v_l
\nonumber
\\
& -
  e^{\phi_l}
  p^\prime(l\alpha)\cdot\nabla\psi(x+(t-T/l^2)p^\prime(l\alpha))
\nonumber
\\
& =
  (p(\p)+il\alpha\cdot{p^\prime(\p)})v_l-ip(l\alpha)v_l
\nonumber
\\
& -
  a(x+tp^\prime(l\alpha),l\alpha))v_l
  +
  a(x,l\alpha)v_l
\nonumber
\\
& -
  e^{\phi_l}
  p^\prime(l\alpha)\cdot\nabla\psi(x+(t-T/l^2)p^\prime(l\alpha)),
\label{equation:part2}
\\
  a(x,\p+il\alpha)v_l
& =
  a(x,\p)v_l-a(x,l\alpha)v_l
\nonumber
\\
& +
  2il(\alpha_1a_1(x)+\alpha_2a_0(x))\p_1v_l
\nonumber
\\
& +
  2il(\alpha_1a_0(x)+\alpha_2a_{-1}(x))\p_2v_l,
\label{equation:part3}
\\
  b(x,\p+il\alpha)v_l
& =
  b(x,\p)v_l-ib(x,l\alpha)v_l.
\label{equation:part4} 
\end{align}
%
%
Substituting 
\eqref{equation:part1}, 
\eqref{equation:part2}, 
\eqref{equation:part3} 
and 
\eqref{equation:part4} 
into
\eqref{equation:lu}, 
we obtain 
%
%
$$
\lvert{Lu_l(t,x)}\rvert
\leqslant
Cl\lvert\alpha\rvert
\sum_{\lvert\beta\rvert\leqslant3}
\lvert\p^\beta{v_l(t,x)}\rvert.
$$
%
%
Then, we deduce 
%
%
\begin{equation}
\lvert{Lu_l(t,x)}\rvert
\leqslant
C_0l\lvert\alpha\rvert
(1+T\lvert\alpha\rvert^2)^3
\exp\left(
    \int_0^T
    a(y+sp^\prime(\alpha),\alpha)
    ds
    \right) 
\label{equation:part5}
\end{equation}
%
%
for $t\in[0,T/l^2]$. 
Integrating the $L^2(\mathbb{T}^2)$-norm of 
\eqref{equation:part5} over $[0,T/l^2]$, 
we have 
%
%
$$
\int_0^{T/l^2}
\lVert{Lu_l(t)}\rVert
dt
\leqslant
\frac{A_\alpha}{l},
$$
%
%
$$
A_\alpha
=
2\pi{C_0T}\lvert\alpha\rvert
(1+T\lvert\alpha\rvert^2)^3
\exp\left(
    \int_0^T
    a(y+sp^\prime(\alpha),\alpha)
    ds
    \right). 
$$
%
%
If we take $l$ satisfying $A_\alpha\leqslant{l}$, then 
%
%
\begin{equation}
\int_0^{T/l^2}
\lVert{Lu_l(t)}\rVert
dt
\leqslant
1.
\label{equation:rhs}
\end{equation}
%
%
Combining 
\eqref{equation:initial}, 
\eqref{equation:evolution} 
and 
\eqref{equation:rhs}, 
we obtain 
%
%
$$
\lVert{u_l(T/l^2)}\rVert
\geqslant
e^n
>
2
\geqslant
\lVert{u_l(0)}\rVert
+
\int_0^{T/l^2}
\lVert{Lu_l(t)}\rVert
dt,
$$
%
%
which breaks the energy inequality \eqref{equation:energy}.
%
\par
%
When $T_n>0$ and 
%
%
$$
\int_0^{T_n}
a(x_n+tp^\prime(\alpha),\alpha)
dt
\leqslant
-2n,  
$$
%
%
we employ a sequence of asymptotic solutions of the form 
%
%
$$
u_l(t,x)
=
e^{-itp(l\alpha)-il\alpha\cdot{x}-\phi_l(t,x)}
\psi(x+(t-T/l^2)p^\prime(l\alpha)).
$$
%
%
When $T_n<0$, the proof above works also in $[T,0]$ 
for some $T\in[T_n,0]$. 
The proof of I $\Longrightarrow$ II finished. 
%%%%%%
%%%%%% Section 3
%%%%%%
\section{The Proof of II $\Longrightarrow$ III}
\label{section:3}
To prove II $\Longrightarrow$ III, 
we need to know the properties of $\Lambda$. 
%
%
\begin{lemma}
\label{theorem:alg}
For any $\alpha\in\mathbb{Z}^2$, 
there exists $\xi(\alpha)\in\Lambda$ such that 
$p^\prime(\pm\xi(\alpha))=\alpha$. 
Moreover, 
$\xi(0)=0$, 
and for $\alpha\ne0$, $\xi(\alpha)\ne0$ and 
%
%
\begin{equation}
\xi_1(\alpha)\xi_2(\alpha)
+
\xi_1(-\alpha)\xi_2(-\alpha)
\ne0. 
\label{equation:trick}
\end{equation}
%
%
\end{lemma}
%
%
\begin{proof}
For the sake of intelligibility, 
we express two-vectors by the entries as 
$(\xi,\eta)\in\Lambda$ 
and 
$(\alpha,\beta)\in\mathbb{Z}^2$. 
We solve a system for quadratic algebraic equations 
%
%
$$
\eta(2\xi+\eta)=\alpha, 
\quad
\xi(2\eta+\xi)=\beta.
$$
%
%
{\bf Case} $(\alpha,\beta)=(0,0)$.\quad
Suppose $\eta(2\xi+\eta)=0$ and $\xi(2\eta+\xi)=0$. 
Then $\eta=0$ or $2\xi+\eta=0$, 
and $\xi=0$ or $2\eta+\xi=0$. 
In any case, 
$(\xi,\eta)=0$ is a unique solution. 
\vspace{6pt}\\
{\bf Case} $\alpha=0$, $\beta\ne0$.\quad
Suppose $\beta\ne0$, 
$\eta(2\xi+\eta)=0$ and $\xi(2\eta+\xi)=\beta$. 
Then, $\eta=0$ or $\eta=-2\xi$, 
and $\xi(2\eta+\xi)=\beta$. 
If $\eta=0$, then $\xi^2=\beta$, 
which implies 
$\beta>0$ and $(\xi,\eta)=(\pm\sqrt{\beta},0)$. 
If $\eta=-2\xi$, then $-3\xi^2=\beta$, 
which implies 
$\beta<0$ and 
$(\xi,\eta)=\pm(\sqrt{-\beta/3},-2\sqrt{-\beta/3})$. 
Then, we have 
%
%
$$
(\xi(0,\beta),\eta(0,\beta))
=
\begin{cases}
\pm(\sqrt{\beta},0) & (\text{if}\quad\beta>0)
\\
\pm\left(\sqrt{-\dfrac{\beta}{3}},-2\sqrt{-\dfrac{\beta}{3}}\right) & (\text{if}\quad\beta<0) 
\end{cases}
$$ 
%
%
$$
\xi(0,\beta)\eta(0,\beta)
+
\xi(0,-\beta)\eta(0,-\beta)
=
-\frac{2}{3}\lvert\beta\rvert\ne0.
$$
%
%
\vspace{6pt}\\
{\bf Case} $\alpha\ne0$, $\beta=0$.\quad
In the same way as the case $\alpha=0$ and $\beta\ne0$, 
we have 
%
%
$$
(\xi(\alpha,0),\eta(\alpha,0))
=
\begin{cases}
\pm(0,\sqrt{\alpha}) & (\text{if}\quad\alpha>0)
\\
\pm\left(-2\sqrt{-\dfrac{\alpha}{3}},\sqrt{-\dfrac{\alpha}{3}}\right) & (\text{if}\quad\alpha<0) 
\end{cases}
$$ 
%
%
$$
\xi(\alpha,0)\eta(\alpha,0)
+
\xi(\alpha,0)\eta(0,-\alpha)
=
-\frac{2}{3}\lvert\alpha\rvert\ne0.
$$
%
%
\vspace{6pt}\\
{\bf Case} $\alpha\beta\ne0$.\quad
Suppose  $\alpha\beta\ne0$, 
$\eta(2\xi+\eta)=\alpha$ and $\xi(2\eta+\xi)=\beta$. 
$\xi\eta\ne0$ 
since 
$\alpha\beta=\xi\eta(2\eta+\xi)(2\xi+\eta)\ne0$. 
Substituting 
$\eta=-\xi/2+\beta/2\xi$ 
into 
$\eta(2\xi+\eta)=\beta$, 
we have 
$3\xi^4+2(2\alpha-\beta)\xi^2-\beta^2=0$. 
Then, 
%
%
$$
\xi^2
=
\frac{\beta-2\alpha\pm\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}.
$$
%
%
Since 
$\lvert\beta-2\alpha\rvert<\sqrt{(\beta-2\alpha)^2+3\beta^2}$ 
and $\xi^2>0$, 
%
%
$$
\xi^2
=
\frac{\beta-2\alpha+\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}. 
$$
%
%
Then, we have 
%
%
$$
\xi(\alpha,\beta)
=
\pm
\sqrt{
\frac{\beta-2\alpha+\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}
}. 
$$
%
%
Using $2\xi\eta=\beta-\xi^2=\alpha-\eta^2$, we get 
%
%
\begin{align}
  2\xi\eta
& =
  \frac{2(\alpha+\beta)-\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}, 
\label{equation:xieta}
\\
  \eta^2
& =
  \frac{\alpha-2\beta+\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}. 
\label{equation:eta}
\end{align}
%
%
Here we remark that 
$\eta^2>0$ is satisfied in \eqref{equation:eta} 
since 
%
%
$$
(\sqrt{(\beta-2\alpha)^2+3\beta^2})^2-(\alpha-2\beta)^2
=
3\alpha^2>0.
$$
%
%
Using \eqref{equation:xieta} and \eqref{equation:eta}, 
we deduce 
%
%
\begin{align*}
  2\lvert\xi\eta\rvert
& =
  \chi(\alpha,\beta)
  \frac{2(\alpha+\beta)-\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3},
\\
  \chi(\alpha,\beta)
& =
  \sgn\left(2(\alpha+\beta)-\sqrt{(\beta-2\alpha)^2+3\beta^2}\right).
\end{align*}
%
%
$\chi(\alpha,\beta)$ makes sense 
for $\alpha\beta\ne0$ since 
%
%
$$
4(\alpha+\beta)^2
-
\left(\sqrt{(\beta-2\alpha)^2+3\beta^2}\right)^2
=
12\alpha\beta\ne0.
$$
%
%
Thus, we have 
%
%
$$
\begin{bmatrix}
\xi(\alpha,\beta)
\\
\eta(\alpha,\beta)
\end{bmatrix}
=
\pm
\begin{bmatrix}
\sqrt{
\dfrac{\beta-2\alpha+\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}
} 
\\
\chi(\alpha,\beta)
\sqrt{
\dfrac{\alpha-2\beta+\sqrt{(\beta-2\alpha)^2+3\beta^2}}{3}
}
\end{bmatrix}.
$$
%
%
It follows from \eqref{equation:xieta} that 
%
%
$$
\xi(\alpha,\beta)\eta(\alpha,\beta)
+
\xi(-\alpha,-\beta)\eta(-\alpha,-\beta)
=
-\frac{1}{3}
\sqrt{(\beta-2\alpha)^2+3\beta^2}
\ne0.
$$
%
%
This completes the proof.
\end{proof}
%
%
Finally, we prove II $\Longrightarrow$ III. 
Express $a_{\sigma(\alpha)}(x)$ by the Fourier series of the form
%
%
$$
a_{\sigma(\alpha)}(x)
=
\sum_{\beta\in\mathbb{Z}^2}
a_{\sigma(\alpha),\beta}
\exp(i\beta\cdot{x}). 
$$
%
%
Substitute the Fourier series into 
\eqref{equation:vanish}. 
Then, for $(x,\xi)\in\mathbb{T}^2\times\Lambda$, 
%
%
\begin{align*}
  0
& =
  \sum_{\beta\in\mathbb{Z}^2}
  \sum_{\lvert\alpha\rvert=2}
  \frac{2!}{\alpha!}
  a_{\sigma(\alpha),\beta}\xi^\alpha
  e^{i\beta\cdot{x}}
  \int_0^{2\pi}
  e^{it\beta\cdot{p^\prime(\xi)}}
  dt
\\
& =
  2\pi
  \sum_{\substack{\beta\in\mathbb{Z}^2
                  \\
                  \beta\cdot{p^\prime(\xi)}=0}}
  (a_{1,\beta}\xi_1^2
   +
   2a_{0,\beta}\xi_1\xi_2
   +
   a_{-1,\beta}\xi_2^2)
  e^{i\beta\cdot{x}}
\\
& =
  2\pi
  \sum_{\substack{\beta\in\mathbb{Z}^2
                  \\
                  \beta\cdot{p^\prime(\xi)}=0}}
  \{
  (a_{-1,\beta},a_{1,\beta})\cdot{p^\prime(\xi)}
  +
  2(a_{0,\beta}-a_{1,\beta}-a_{-1,\beta})\xi_1\xi_2
  \}
  e^{i\beta\cdot{x}}.
\end{align*}
%
% 
It follows that if
$\beta\cdot{p^\prime(\xi)}=0$ 
and 
$\xi\in\Lambda$, 
then 
%
%
$$ 
(a_{-1,\beta},a_{1,\beta})\cdot{p^\prime(\xi)}
+
2(a_{0,\beta}-a_{1,\beta}-a_{-1,\beta})\xi_1\xi_2
=0. 
$$
%
%
In view of Lemma~\ref{theorem:alg}, we have 
%
%
\begin{align}
  (a_{-1,\beta},a_{1,\beta})\cdot\alpha
  +
  2(a_{0,\beta}-a_{1,\beta}-a_{-1,\beta})
  \xi_1(\alpha)\xi_2(\alpha)
& = 0, 
\label{equation:part11}
\\
  -
  (a_{-1,\beta},a_{1,\beta})\cdot\alpha
  +
  2(a_{0,\beta}-a_{1,\beta}-a_{-1,\beta})
  \xi_1(-\alpha)\xi_2(-\alpha)
& = 0, 
\label{equation:part12}
\end{align}
%
%
for $\alpha\in\mathbb{Z}^2$ 
satisfying $\alpha\cdot\beta=0$. 
The sum of \eqref{equation:part11} 
and \eqref{equation:part12} is 
%
%
$$
2(a_{0,\beta}-a_{1,\beta}-a_{-1,\beta})
(\xi_1(\alpha)\xi_2(\alpha)
 +
 \xi_1(-\alpha)\xi_2(-\alpha))
=0.
$$
%
%
In view of \eqref{equation:trick}, 
we get $a_{0,\beta}=a_{1,\beta}+a_{-1,\beta}$ 
for all $\beta\in\mathbb{Z}^2$. 
Thus, $a_0(x)=a_1(x)+a_{-1}(x)$, and 
\eqref{equation:part11} becomes 
%
%
\begin{equation}
(a_{-1,\beta},a_{1,\beta})\cdot\alpha=0
\quad\text{if}\quad
\alpha\cdot\beta=0.
\label{equation:part13}
\end{equation}
%
%
$(a_{-1,0},a_{1,0})=0$ 
since $\alpha\cdot0=0$ 
for all $\alpha\in\mathbb{Z}^2$. 
For $\beta\ne0$, \eqref{equation:part13} implies that 
there exists $\phi_\beta\in\mathbb{C}$ such that 
$(a_{-1,\beta},a_{1,\beta})=i\phi_\beta\beta$. 
If we set 
%
%
$$
\phi(x)
=
\sum_{\beta\ne0}
\phi_\beta
e^{i\beta\cdot{x}},
$$
%
%
then 
%
%
$$
\nabla\phi(x)
=
\sum_{\beta\ne0}
i\phi_\beta\beta
e^{i\beta\cdot{x}}
=
\sum_{\beta\ne0}
(a_{-1,\beta},a_{1,\beta})
e^{i\beta\cdot{x}}
=
(a_{-1}(x),a_1(x)),
$$
%
%
which is desired. 
The proof of II $\Longrightarrow$ III finished. 
%%%%%%
%%%%%% Reference
%%%%%%
\bibliographystyle{amsplain}
\begin{thebibliography}{10}
\bibitem{bks}M.~Ben-Artzi, H.~Koch and J.-C.~Saut, 
{\it Dispersion estimates for third order equations in two dimensions}, 
Comm.\ Partial Differential Equations 
{\bf 28} (2003), 1943--1974. 
\bibitem{chihara}H.~Chihara, 
{\it The initial value problem for Schr\"odinger equations on the torus}. 
Int.\ Math.\ Res.\ Not.\ {\bf 2002}, 789--820.
\bibitem{dysthe}K.~B.~Dysthe, 
{\it Note on a modification to the nonlinear Schr\"odinger equation for application to deep water}, 
Proc.\ Roy.\ Soc.\ London Ser.\ A {\bf 369} (1979), 
105--114.
\bibitem{hogan}S.~J.~Hogan, 
{\it The fourth-order evolution equation for deep-water gravity-capillary waves}, 
Proc.\ Roy.\ Soc.\ London Ser.\ A {\bf 402} (1985), 
359--372.
\bibitem{ichinose1}W.~Ichinose, 
{\it On $L^2$ well posedness of the Cauchy problem 
for Schr\"odinger type equations 
on the Riemannian manifold and the Maslov theory}, 
Duke Math.\ J. {\bf 56} (1988), 549--588.
\bibitem{ichinose2}\bysame, 
{\it A note on the Cauchy problem for Schr\"odinger type equations on the Riemannian manifold}, 
Math.\ Japon.\ {\bf 35} (1990), 205--213.
\bibitem{mizohata}S.~Mizohata, 
``On the Cauchy problem'', 
Notes and Reports in Mathematics in Science and Engineering {\bf 3}, 
Academic Press, Inc., Orlando, FL; 
Science Press, Beijing, 1985.
\bibitem{tarama1}S.~Tarama, 
{\it On the wellposed Cauchy problem for some dispersive equations}, 
J. Math.\ Soc.\ Japan {\bf 47} (1995), 143--158.
\bibitem{tarama2}\bysame, 
{\it Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations},  
J. Math.\ Kyoto Univ.\ {\bf 37} (1997), 757--765.
\bibitem{tsukamoto}C.~Tsukamoto, 
{\it Integrability of infinitesimal Zoll deformations}, 
Geometry of geodesics and related topics (Tokyo, 1982), 
97--104, Adv.\ Stud.\ Pure Math.\ {\bf 3}, 
North-Holland, Amsterdam, 1984.
\end{thebibliography}
%%%%%%
%%%%%% End
%%%%%%
\end{document}
%%%%%%
%%%%%%
%%%%%%
math0404006.extracted_bib
\begin{thebibliography}{10}
\bibitem{bks}M.~Ben-Artzi, H.~Koch and J.-C.~Saut, 
{\it Dispersion estimates for third order equations in two dimensions}, 
Comm.\ Partial Differential Equations 
{\bf 28} (2003), 1943--1974. 
\bibitem{chihara}H.~Chihara, 
{\it The initial value problem for Schr\"odinger equations on the torus}. 
Int.\ Math.\ Res.\ Not.\ {\bf 2002}, 789--820.
\bibitem{dysthe}K.~B.~Dysthe, 
{\it Note on a modification to the nonlinear Schr\"odinger equation for application to deep water}, 
Proc.\ Roy.\ Soc.\ London Ser.\ A {\bf 369} (1979), 
105--114.
\bibitem{hogan}S.~J.~Hogan, 
{\it The fourth-order evolution equation for deep-water gravity-capillary waves}, 
Proc.\ Roy.\ Soc.\ London Ser.\ A {\bf 402} (1985), 
359--372.
\bibitem{ichinose1}W.~Ichinose, 
{\it On $L^2$ well posedness of the Cauchy problem 
for Schr\"odinger type equations 
on the Riemannian manifold and the Maslov theory}, 
Duke Math.\ J. {\bf 56} (1988), 549--588.
\bibitem{ichinose2}\bysame, 
{\it A note on the Cauchy problem for Schr\"odinger type equations on the Riemannian manifold}, 
Math.\ Japon.\ {\bf 35} (1990), 205--213.
\bibitem{mizohata}S.~Mizohata, 
``On the Cauchy problem'', 
Notes and Reports in Mathematics in Science and Engineering {\bf 3}, 
Academic Press, Inc., Orlando, FL; 
Science Press, Beijing, 1985.
\bibitem{tarama1}S.~Tarama, 
{\it On the wellposed Cauchy problem for some dispersive equations}, 
J. Math.\ Soc.\ Japan {\bf 47} (1995), 143--158.
\bibitem{tarama2}\bysame, 
{\it Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations},  
J. Math.\ Kyoto Univ.\ {\bf 37} (1997), 757--765.
\bibitem{tsukamoto}C.~Tsukamoto, 
{\it Integrability of infinitesimal Zoll deformations}, 
Geometry of geodesics and related topics (Tokyo, 1982), 
97--104, Adv.\ Stud.\ Pure Math.\ {\bf 3}, 
North-Holland, Amsterdam, 1984.
\end{thebibliography}