# On the first bifurcation of solitary waves

Vladimir Kozlov  Department of Mathematics, Linköping University, SE-581 83
Linköping, Sweden

###### Abstract.

We consider solitary water waves on the vorticity flow in a two-dimensional
channel of finite depth. The main object of study is a branch of solitary
waves starting from a laminar flow and then approaching an extreme wave. We
prove that there always exists a bifurcation point on such branches. Moreover,
the crossing number of the first bifurcation point is $1$, i.e. the
bifurcation occurs at a simple eigenvalue.

##  1\. Introduction

We consider steady water waves in a two-dimensional channel bounded below by a
flat, rigid bottom and above by a free surface that does not touch the bottom.
The surface tension is neglected and the water motion can be rotational. In
appropriate Cartesian coordinates $(X,Y)$, the bottom coincides with the
$X$–axis and gravity acts in the negative $Y$–direction. We choose the frame
of reference so that the velocity field is time-independent as well as the
free-surface profile which is supposed to be the graph of $Y=\xi(X)$,
$x\in\mathbb{R}$, where $\xi$ is a positive and continuous unknown function.
Thus

$D=D_{\xi}=\\{X\in\mathbb{R},0<Y<\xi(X)\\},\;\;S=S_{\xi}=\\{X\in\mathbb{R},\;Y=\xi(X)\\}$

is the water domain and the free surface respectively. We will use the stream
function $\Psi$, which is connected with the velocity vector $({\bf u},{\bf
v})$ as $\Psi_{Y}=c-{\bf u}$ and ${\bf v}=\Psi_{X}$, where $c$ is a constant
wave speed.

We assume that $\xi$ is a positive, even function. Since the surface tension
is neglected, $\Psi$ and $\xi$ after a certain scaling satisfy the following
free-boundary problem (see, for example, [14]):

$\displaystyle\Delta\Psi+\omega(\Psi)=0\;\;\mbox{in $D_{\xi}$},$
$\displaystyle\frac{1}{2}|\nabla\Psi|^{2}+\xi=R\;\;\mbox{on $S_{\xi}$},$
$\displaystyle\Psi=1\;\;\mbox{on $S_{\xi}$},$
$\displaystyle\Psi=0\;\;\mbox{for $Y=0$},$ (1.1)

where $\omega\in C^{1,\alpha}$, $\alpha\in(0,1)$, is a vorticity function and
$R$ is the Bernoulli constant. We assume that $\Psi$ is even in $X$ and

$\Psi_{Y}>0\;\;\mbox{on $\overline{D_{\xi}}$},$ (1.2)

which means that the flow is unidirectional.

The main our subject of study is the solitary waves. The solitary wave
solution satisfies

$\;\;\xi(X)\to d,\;\;\Psi(X,Y)\to U(Y)\;\;\mbox{as $X\to\pm\infty$}$ (1.3)

uniformly in $Y$. Here $(U,d)$ is a uniform stream solution with the Bernoulli
constant $R=d+U^{\prime}(d)^{2}/2$. Moreover the Froude number

$F=\Big{(}\int_{0}^{d}\frac{dY}{U^{\prime}(Y)^{2}}\Big{)}^{-2}$ (1.4)

for this uniform stream solution must be $>1$. Every non-trivial solitary wave
satisfies $\xi(X)>d$ and it is symmetric, after a certain translation, and
monotonic decreasing for positive $X$. In what follows we assume that such
translation is already done and we consider even solitary waves.

If the Froude number $F$ is equal to $1$ or equivalently $R=R_{c}$, then the
only solution of (1) is the uniform stream solution $(U_{*}(Y),d_{*})$, where
$d_{*}=d(s_{c})$ (see Sect. 2 for details).

The Frechet derivative for the problem is evaluated for example in [20], [11]
in the periodic case. The same evaluation can be used to obtain the Frechet
derivative in the case, when the solution is a solitary wave. The
corresponding eigenvalue problem for the Frechet derivative has the form

$\displaystyle\Delta w+\omega^{\prime}(\Psi)w+\mu w=0\;\;\mbox{in $D_{\xi}$},$
$\displaystyle\partial_{n}w-\rho w=0\;\;\mbox{on $S_{\xi}$},$ $\displaystyle
w=0\;\;\mbox{for $Y=0$},$ (1.5)

where $n$ is the unite outward normal to $Y=\xi(X)$ and

$\rho=\rho(X)=\frac{(1+\Psi_{X}\Psi_{XY}+\Psi_{Y}\Psi_{YY})}{\Psi_{Y}(\Psi_{X}^{2}+\Psi_{Y}^{2})^{1/2}}\Big{|}_{Y=\xi(X)}.$
(1.6)

The function $w$ in (1) is supposed also to be even.

Let us introduce several function spaces. For $\alpha\in(0,1)$ and
$k=0,1,\ldots$, the space $C^{k,\alpha}(D)$ consists of bounded functions in
$D$ such that the norms $C^{k,\alpha}(D_{a,a+1})$ are uniformly bounded with
respect to $a\in\mathbb{R}$. Here

$D_{a,a+1}=\\{(X,Y)\in\overline{D},\;:\,a\leq x\leq a+1\\}.$

The space $C^{k,\alpha}_{0}(D)$ consists of function in $C^{k,\alpha}(D)$
vanishing for $Y=0$. The space $C^{k,\alpha}_{0,e}(D)$ consists of even
functions from $C^{k,\alpha}_{0}(D)$. Similarly we define the space
$C^{k,\alpha}_{e}(\mathbb{R})$) consisting of even functions in
$C^{k,\alpha}(\mathbb{R})$.

We will consider a branch of solitary water waves depending on a parameter
$t>0$, i.e.

$\Psi=\Psi(X,Y;t),\;\;\xi=\xi(X,t),\;\;R=R(t),\;\;t\in(0,\infty),$ (1.7)

where $R$ is the Bernoulli constant.

We assume that the above sequence of solitary waves satisfies the conditions
of the following theorem, which follows from [30], if we take into account
Theorem 6.1 in [31].

######  Theorem 1.1.

Fix a Hölder exponent $\alpha\in(0,1/2]$. There exists a continuous curve
(1.7) of solitary wave solutions to (1) with the regularity

$(\Psi(t),\xi(t),R(t)))\in C^{2,\alpha}(D(t))\times
C^{2,\alpha}({\mathbb{R}})\times(R_{c},\infty),$ (1.8)

where $D(t)$ denotes the fluid domain corresponding to $\xi(t)$. The solution
curve satisfies the following property (critical laminar flow, i.e. $R=R_{c}$)

$\lim_{t\to 0}(\Psi(t),\xi(t),R(t))=(U_{*},d_{*},R_{c}).$ (1.9)

Furthermore, there exists a sequence $\\{t_{j}\\}_{j=1}^{\infty}$ such that
$t_{j}\to\infty$ as $j\to\infty$ and $(X_{j},Y_{j})\in D(t_{j})$ such that the
following is valid

$(\Psi(t_{j}))_{Y}(X_{j},Y_{j})\to 0\;\;\mbox{as
$j\to\infty$}\;\;\;\mbox{(Stagnation)}.$ (1.10)

######  Remark 1.2.

(i) The convergence in (1.9) is understood in the spaces

$C^{2,\alpha}(D(t))\times C^{2,\alpha}({\mathbb{R}})\times\mathbb{R}$

(ii) The dependence on $t\in(0,\infty)$ in (1.8) is analytic in the variables
$(q,p)$ using in partial hodograph transformation, see [31], Sect.6.

(iii) There are two more options in the above theorem, see Theorem 1.1 in
[30]. The first one says that the Froude number (equivalently the Bernoulli
constant) can be arbitrary large along the curve (1.7). This option is
excluded by Proposition 2.4. The second one includes existence of non-trivial
solitary waves with the Froude number $1$ or equivalently with the Bernoulli
constant $R_{c}$. By Theorem 1 in [16] the only solution of (1) is the uniform
stream solution. This fact excludes the second option.

(iv) In paper [30] the Froude number $F(t)$ is used instead of $R(t)$. Since
the dependence $R$ on $F$ is monotone and analytic (see Proposition 2.2) we
use here the function $R(t)$.

In Sect. 3 we prove the following clarification of the stagnation condition
(1.10):

######  Proposition 1.3.

The stagnation condition (1.10) is equivalent to the following options

(i) If $R_{0}=\infty$ then there exists a sequence $\\{t_{j}\\}_{j\geq 1}$,
$t_{j}\to\infty$ as $j\to\infty$ such that

$\lim_{j\to\infty}|\xi^{\prime}(t_{j})(0)|=\infty$ (1.11)

or

$\lim_{j\to\infty}R(t_{j})=R\;\;\mbox{and}\;\;\lim_{j\to\infty}\xi(t_{j})(0)=R.$
(1.12)

(ii) If $R_{0}<\infty$ then besides the options in (i) there is one more
option:

$\lim_{j\to\infty}R(t_{j})=R\;\;\mbox{and}\;\;\lim_{j\to\infty}\sup_{X\in\mathbb{R}}\Psi_{Y}(t_{j})(X_{j},Y_{j})=0,$
(1.13)

where $Y_{j}\to 0$ as $j\to\infty$.

Denote by ${\mathcal{A}}(t)={\mathcal{A}}_{(\Psi,\xi)}$ the unbounded operator
in $L^{2}(D_{\xi})$ corresponding to the problem (1), i.e.
${\mathcal{A}}_{(\Psi,\xi)}=-\Delta-\omega^{\prime}(\Psi)$ with the domain

${\mathcal{D}}(t)={\mathcal{D}}({\mathcal{A}}_{(\Psi,\xi)})=\\{w\in
H^{2}(D_{\xi})\,:\,w(x,0)=0,\;\;\partial_{\nu}w-\rho
w=0\,\mbox{for}\;Y=\xi(X)\\}.$

For every $t>0$ we denote by

$(U(t),d(t),R(t))$ (1.14)

the limit uniform stream solution and Bernoulli constant in the asymptotics
(1.3).

Consider the eigenvalue problem

$\displaystyle-v^{{}^{\prime\prime}}-\omega^{\prime}(U(t))v=\nu v\;\;\mbox{on
$(0,d(t))$},$ $\displaystyle
v(0)=0\;\;\mbox{and}\;\;v^{\prime}(d(t))-\rho_{0}(t)v(d(t))=0,$

where

$\rho_{0}=\frac{1+U_{Y}U_{YY}}{U_{Y}^{2}}\Big{|}_{Y=d}.$ (1.15)

As it is shown in Lemma 4.1, the eigenvalues of this problem are positive for
all $t>0$. We denote by $\nu_{0}=\nu_{0}(t)$ the lowest eigenvalue. This
eigenvalue is simple and corresponding eigenfunctions does not change sign on
$(0,d(t)]$.

The following proposition is possibly known but we present its proof in Sect.
4.2 for readers convenience

######  Proposition 1.4.

The operator (Frechet derivative) ${\mathcal{A}}$ has a continuous spectrum
$[\nu_{0},\infty)$. The spectrum on the half line $(-\infty,a]$ consists of
isolated eigenvalues of finite multiplicity for any $a<\nu_{0}$.

The main result of this paper is the following

######  Theorem 1.5.

(i) The lowest eigenvalue of the operator ${\mathcal{A}}(t)$ for $t>0$ is
always negative and simple with the eigenfunction which does not change sign
in $D(t)$. We denote it by $\mu_{0}(t)$.

(ii) For small positive $t$ the spectrum of the operator ${\mathcal{A}}(t)$ on
the half line $\mu\leq 0$ consists of the eigenvalue $\mu_{0}(t)$ only.

(iii) If for a certain $t_{1}>0$ and $t\in(0,t_{1})$ the operator
${\mathcal{A}}(t)$ has the only negative eigenvalue $\mu_{0}$ and
${\mathcal{A}}(t_{1})$ has the eigenvalue $\mu=0$ then this eigenvalue is
simple.

Denote by $\mu_{1}(t)$ the second eigenvalue of the operator
${\mathcal{A}}(t)$ on the half-line $(-\infty,\nu_{0})$ if it exists,
otherwise we put $\mu_{1}(t)=\nu_{0}(t)$. Assume that

$\displaystyle\mbox{the existence of point $t_{*}>0$ such that: (i)
$\mu_{1}(t)>0$ for $t\in(0,t_{*})$},$ $\displaystyle\mbox{ $\mu_{1}(t)<0$ for
small positive $t-t_{*}$}.$ (1.16)

According to Theorem 1.5(iii) the eigenvalue $\mu_{1}(t_{*})=0$ is simple.
This implies that the crossing number of the Frechet derivative is $1$ at
$t_{*}$.

######  Proposition 1.6.

Let the condition (1) hold. Then $t_{*}$ is a bifurcation point and
bifurcating solution builds a connected continuum.

The proof of this assertion follows from Theorem I.16.4,[9], (see also [8]).
First one must use the partial hodograph transformation to obtain a problem
with an analytic operator function and then apply result of [9] on the local
bifurcation under the condition (1). Now one can extend this local bifurcation
to a global bifurcation branch by using bifurcation results proved in [30] and
[31].

In the next proposition we give a condition when (1) is fulfilled

######  Proposition 1.7.

Let there exists a sequence $\\{t_{j}\\}_{j=1}^{\infty}$ such that the curve
(1.7) approach an extreme solitary wave with the angle111Actually the
condition on the angle can be removed, see Remark 6.1. Since the paper [22] is
not published yet we will keep this angle requirement. $120^{\circ}$ at the
crest along this sequence then there exists a point $t_{*}$ which satisfies
the condition (1).

Indeed, by using [20] together with Theorem 3.1 in [11] one can show that the
number of negative eigenvalues of the operator ${\mathcal{A}}(t_{j})$ tends to
infinity when $j\to\infty$. This implies (1) and hence Proposition 1.6.

The first study of bifurcations on the branches of solitary water waves was
done by Plotnikov in [26] in the irrotational case. In the case of vortical
flows one cannot apply the same technique as in irrotational case since it is
based on the complex analysis approach. The important contributions allowing
to approach the bifurcation problems are papers [30] and [31], where the
global branches of solutary waves were constructed, and the papers [20]
together with [11] and [10], where the bifurcation analysis for branches of
Stokes waves were presented.

##  2\. Uniform stream solution, dispersion equation, the Bernoulli and Froude
constants

###  2.1. Uniform stream solution and dispersion equation

The uniform stream solution $\Psi=U(Y)$ with the constant depth $\xi=d$
satisfies the problem

$\displaystyle U^{{}^{\prime\prime}}+\omega(U)=0\;\;\mbox{on $(0,d)$},$
$\displaystyle U(0)=0,\;\;U(d)=1,$
$\displaystyle\frac{1}{2}U^{\prime}(d)^{2}+d=R.$ (2.1)

To find solutions to this problem we introduce a parameter $s=U^{\prime}(0)$.
We assume that $s>s_{0}:=\sqrt{2\max_{\tau\in[0,1]}\Omega(\tau)}$, where

$\Omega(\tau)=\int_{0}^{\tau}\omega(p)dp.$

Then the problem (2.1) has a solution $(U,d)$ with a strongly monotone
function $U$ for

$R={\mathcal{R}}(s):=\frac{1}{2}s^{2}+d(s)-\Omega(1).$ (2.2)

The solution is given by

$Y=\int_{0}^{U}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}},\;\;d=d(s)=\int_{0}^{1}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}}.$
(2.3)

The function $d$ is strongly decreasing. Since
$U^{\prime}(Y)^{2}=s^{2}-2\Omega(U)(Y)$ and $0\leq U(Y)\leq 1$, we have

$s^{2}-s_{0}^{2}\leq U^{\prime}(Y)^{2}\leq
s^{2}-2\min_{\tau\in[0,1]}\Omega(\tau).$ (2.4)

If we consider (2.2) as the equation with respect to $s$ then it is solvable
if $R\geq R_{c}$, where

$R_{c}=\min_{s\geq s_{0}}{\mathcal{R}}(s),$ (2.5)

and it has two solutions if $R\in(R_{c},R_{0})$, where

$R_{0}={\mathcal{R}}(s_{0}).$ (2.6)

We denote by $s_{c}$ the point where the minimum in (2.5) is attained.

######  Proposition 2.1.

([16], [15], [24])

(a) If $R<R_{c}$ then the problem (1) has no solutions;

(b) If $R=R_{c}$ then the only solution to (1) is $(U_{*}(Y;d_{*}),d_{*})$;

(c) If $R_{0}$ is finite and $R\geq R_{0}$ then there are no non-trivial
solitary wave solutions to (1);

(d) If $R\in(R_{c},R_{0})$ then a solitary wave solution to (1) satisfies

$\xi(0)>d_{+}.$

One more formula for the Froude number is the following

$\frac{1}{F^{2}(s)}=-\frac{d^{\prime}(s)}{s},$ (2.7)

where the Froude number $F(s)$ corresponds to the uniform stream solution
$(U(Y;s),d(s))$ and $R={\mathcal{R}}(s)$. One can verified directly by using
(2.3), that

$\Big{(}\frac{d^{\prime}(s)}{s}\Big{)}^{\prime}>0.$

Therefore the function $F(s)$ is increasing and $F(s)\to\infty$ as
$s\to\infty$. Furthermore

${\mathcal{R}}^{\prime}(s)=s(1-F^{-2}(s))$ (2.8)

The above formula implies that the function ${\mathcal{R}}$ is a strongly
increasing function of $s$, $s>s_{c}$, and ${\mathcal{R}}(s)\to\infty$ as
$s\to\infty$. Therefore every parameter $s>s_{c}$, $R>R_{c}$ or $F>1$ can be
expressed through another and can be used for parametrisation of the solitary
waves. In this case

$s\in(s_{c},\infty),\;\;R\in(R_{c},\infty)\;\;\mbox{and}\;\;F\in(1,\infty).$

The equation ${\mathcal{R}}(s)=R$ with $R\leq R_{c}$ has exactly one solution
on $[s_{c},\infty)$. We denote it by $s_{-}$. We put

$d_{-}=d(s_{-}),\;\;F_{-}=F(s_{-}).$

######  Proposition 2.2.

The Bernoulli constant $R\in[R_{c},\infty)$ can be considered as a continuous
function of the Froude number $F\in[1,\infty)$. This function is strongly
increasing, analytical on $(1,\infty)$ and $R(1)=R_{c}$. Moreover

$R(F)=\frac{1}{2}F^{4/3}+O(1)\;\;\mbox{as $F\to\infty$}.$

###### Proof.

The function $d(s)$ is strongly decreasing and analytic for $s>s_{0}$.
Furthermore $d(s)=s^{-1}(1+O(s^{-2})$ for large $s$. Therefore,
${\mathcal{R}}(s)$ is strongly increasing and analytic for $s>s_{c}$. For
large $s$, ${\mathcal{R}}(s)=s^{2}/2+O(1)$. From (2.79 it follows that $F(s)$
ia stronglu increasing and analytic for $s>s_{c}$. For large $s$,
$F(s)=s^{3/2}(1+O(s^{-2})$. The above considerations lead to the proof of our
proposition.

∎

Existence of small amplitude Stokes waves is determined by the dispersion
equation (see, for example, [14]). It is defined as follows. The strong
monotonicity of $U$ guarantees that the problem

$\gamma^{{}^{\prime\prime}}+\omega^{\prime}(U)\gamma-\tau^{2}\gamma=0,\;\;\gamma(0,\tau)=0,\;\;\gamma(d,\tau)=1$
(2.9)

has a unique solution $\gamma=\gamma(y,\tau)$ for each $\tau\in\mathbb{R}$,
which is even with respect to $\tau$ and depends analytically on $\tau$.
Introduce the function

$\sigma(\tau)=\kappa\gamma^{\prime}(d,\tau)-\kappa^{-1}+\omega(1),\;\;\kappa=U^{\prime}(d).$
(2.10)

It depends also analytically on $\tau$ and it is strongly increasing with
respect to $\tau>0$. Moreover, it is an even function. The dispersion equation
is the following

$\sigma(\tau)=0.$ (2.11)

It has a positive solution if

$\sigma(0)<0.$ (2.12)

By [14] this is equivalent to $s+d^{\prime}(s)<0$ or what is the same

$1<\int_{0}^{d}\frac{dY}{U^{\prime 2}(Y)}.$ (2.13)

By (1.4) the inequality (2.13) means that $F<1$, which is well-known condition
for existence of Stokes waves of small amplitude. Another equivalent
formulation is given by requirement (see, for example [17])

$s\in(s_{0},s_{c}).$ (2.14)

The existence of such $s$ is guaranteed by $R\in(R_{c},R_{0})$. If (2.12)
holds then equation (2.11) has a unique positive root, which will be denoted
by $\tau_{*}$. It is connected with the corresponding period $\Lambda_{0}$ by
the relation

$\tau_{*}=\frac{2\pi}{\Lambda_{0}}.$

The value $\sigma(0)$ admits the following representation (see [14]):

$\sigma(0)=-\frac{3}{2\kappa}\frac{{\mathcal{R}}^{\prime}(s)}{d^{\prime}(s)}=\frac{3(F^{2}(s)-1)}{2\kappa}.$

If $F>1$ then

$\int_{0}^{d}(|v^{\prime}|^{2}-\omega^{\prime}(U)|v|^{2})dY-\rho_{0}|v(d)|^{2}>0\;\;\mbox{for
all $v\in H^{1}(0,d)$\,:\, $v(0)=0$}.$ (2.15)

To give another representation of the function $\sigma$ we introduce

$\rho_{0}=\frac{1+U^{\prime}(d)U^{{}^{\prime\prime}}(d)}{U^{\prime}(d)^{2}}$
(2.16)

and note that

$\frac{1+U^{\prime}(d)U^{{}^{\prime\prime}}(d)}{U^{\prime}(d)^{2}}=\kappa^{-2}-\frac{\omega(1)}{\kappa}.$

Hence another form for (2.10) is

$\sigma(\tau)=\kappa\gamma^{\prime}(d,\tau)-\kappa\rho_{0}.$ (2.17)

The following problem will be used in asymptotic analysis of a branch of
Stokes waves of small amplitude (see Sect. 5.3 and 5.4):

$\displaystyle
v^{{}^{\prime\prime}}+\omega^{\prime}(U)v-\tau^{2}v=f\;\;\mbox{on $(0,d)$},$
(2.18) $\displaystyle v^{\prime}(d)-\rho_{0}v(d)=g\;\;\mbox{and}\;\;v(0)=0.$

######  Proposition 2.3.

Let $\tau\geq 0$ and $\tau\neq\tau_{*}$. Let also $f\in C^{1,\alpha}([0,d])$
and $g$ be a constant. Then the problem (2.18) has a unique solution $v\in
C^{3,\alpha}$. If $\tau=\tau_{*}$ then the problem (2.18) has the one
dimensional kernel which consists of function

$c\gamma(Y;\tau_{*}).$

###  2.2. Estimates for the Bernoulli constant and the Froude number

The aim of this section is to prove the following.

######  Proposition 2.4.

(i) If $R_{0}<\infty$ then for existence of non-trivial solitary wave
solutions to (1) the inequality $R<R_{0}$ must be valid.

(ii) Let $R_{0}=\infty$. There exist constants $\widehat{R}$ and $\widehat{F}$
depending on $\omega$ such that if $R>\widehat{R}$ (or $F>\widehat{F}$), then
the problem (1) has no solitary wave solutions satisfying (1.2).

###### Proof.

We start from estimating the Bernoulli constant.

The following classification of vorticity functions was introduced in [17].
According to that classification the following three options describing
vorticity behaviour are possible:

(i) Either $\max_{\tau\in[0,1]}\Omega(\tau)$ is attained at an inner point of
$(0,1)$ or the maximum is attained at one of the end-points (or at both), and
in the latter case, one (or both) of the following conditions holds:

$\omega(1)=0\mbox{\;\; when $\Omega(1)>\Omega(\tau)$ for $\tau\in(0,1)$};$
$\omega(0)=0\;\;\mbox{when $\Omega(0)>\Omega(\tau)$ for $\tau\in(0,1)$}.$

(ii) The function $\Omega$ attains its maximum only at the interval’s left
end-point in which case

$\Omega(0)>\Omega(\tau)\;\;\mbox{ for $\tau\in(0,1]$ and $\omega(0)<0$}.$

(iii) The function $\Omega$ attains its maximum when $\tau=1$, that is,
$\Omega(\tau)<\Omega(1)$ for $\tau\in(0,1)$, and $\omega(1)>0$.

In the cases (ii) and (iii) $d(s_{0})<\infty$ and the number
$R_{0}=\mathcal{R}(s_{0})$ is also finite. It was proved in [16] that if
$R\geq R_{0}$ then the problem (1) has no nontrivial solutions. This implies
the estimate $R<R_{0}$ for existence of non-trivial solutions to (1) and in
particular solitary waves. This proves the assertion (i) of Proposition 2.4.

In the case (i), $d(s_{0})=\infty$ and the equation $\mathcal{R}(s)=R$ has two
roots $s_{+}<s_{c}<s_{-}$ for all $R>R_{c}$. The Froude number depends on the
uniform stream solution. If we fix the vorticity and the Bernoulli constant
$R$ then there are two solutions to the problem (2.1). We denote then by
$(U_{-},d_{-})$ and $(U_{+},d_{+})$. They correspond to $s=s_{+}$ and
$s=s_{-}$ respectively. Corresponding Froude numbers we denote by $F_{\pm}$,
then

$\frac{1}{F_{\pm}^{2}}=\int_{0}^{d_{\pm}}\frac{dY}{U_{\pm}^{\prime 2}(Y)}.$
(2.19)

Hence

$F_{+}<1<F_{-}.$

The depths

$d_{-}=d(s_{-}),\;\;d_{+}=d(s_{+})$

are called supercritical and subcritical respectively.

Let us turn to a nontrivial solitary solution $(\Psi,\xi)$. By [16, 15]

$\xi(0)>d_{+}.$ (2.20)

Consider the flow forth invariant

$S=\int_{0}^{\xi(X)}\Big{(}\Psi_{Y}^{2}-\Psi_{X}^{2}+R-Y+\int_{\Psi}^{1}\omega(\tau)d\tau\Big{)}dY.$
(2.21)

This expression does not depend on $X$. Evaluating it for $X=0$ and for
$X\to\infty$ we get

$\int_{0}^{d+a}\Big{(}\Psi_{Y}^{2}+R-Y+\int_{\Psi}^{1}\omega(\tau)d\tau\Big{)}dY=\int_{0}^{d}\Big{(}U_{y}^{2}+R-Y+\int_{U}^{1}\omega(\tau)d\tau\Big{)}dY,$
(2.22)

where the function $\Psi$ is evaluated at the point $X=0$, $d+a=\xi(0)$ and
$d=d_{-}$. We write (2.22) as

$\int_{0}^{d+a}\Psi_{Y}^{2}dY+Ra-\frac{a(2d+a)}{2}=\int_{0}^{d}\Big{(}U_{y}^{2}+\int_{U}^{1}\omega(\tau)d\tau\Big{)}dY-\int_{0}^{d+a}\int_{\Psi}^{1}\omega(\tau)d\tau
dY.$ (2.23)

Using inequalities $d+a\leq R$ and

$\Big{(}\int_{0}^{d+a}\Psi_{Y}dY\Big{)}^{2}\leq(d+a)\int_{0}^{d+a}\psi_{Y}^{2}dY,$

we get

$\frac{1}{d+a}\leq\int_{0}^{d+a}\Psi_{Y}^{2}dY.$

Therefore (2.23) implies

$\frac{1}{d+a}+\frac{aR}{2}-\frac{ad}{2}\leq\int_{0}^{d}\Big{(}U_{Y}^{2}+\int_{U}^{1}\omega(\tau)d\tau\Big{)}dY-\int_{0}^{d+a}\int_{\Psi}^{1}\omega(\tau)d\tau
dY.$ (2.24)

Next

$\displaystyle\int_{0}^{d}\int_{U}^{1}\omega(\tau)d\tau
dY-\int_{0}^{d+a}\int_{\psi}^{1}\omega(\tau)d\tau
dY=\int_{0}^{d}\int_{U}^{1}\omega(\tau)d\tau
dY-\int_{0}^{d}\int_{\psi}^{1}\omega(\tau)d\tau dY$
$\displaystyle+\int_{0}^{d}\int_{\psi}^{1}\omega(\tau)d\tau
dY-\int_{0}^{d+a}\int_{\psi}^{1}\omega(\tau)d\tau dY\leq
a\omega_{0}+\int_{0}^{d}\int_{\psi}^{U}\omega(\tau)d\tau dY$
$\displaystyle\leq a\omega_{0}+d\omega_{0}.$ (2.25)

Therefore

$\frac{1}{d+a}+\frac{aR}{2}-\frac{ad}{2}\leq\int_{0}^{d}U_{Y}^{2}dY+a\omega_{0}++d\omega_{0}$
(2.26)

or

$\frac{aR}{2}\leq\frac{ad}{2}+\int_{0}^{d}\Big{(}U_{Y}^{2}-\frac{1}{d^{2}}\Big{)}dY+\frac{1}{d}-\frac{1}{d+a}+(a+d)\omega_{0},$
(2.27)

which implies

$\frac{R}{2}\leq\frac{d}{2}+\frac{1}{a}\int_{0}^{d}\Big{(}U_{Y}^{2}-\frac{1}{d^{2}}\Big{)}dY+\frac{1}{d(d+a)}+\frac{(a+d)\omega_{0}}{a}.$
(2.28)

Using (2.20), we get

$\frac{R}{2}\leq\frac{d}{2}+\frac{1}{d_{+}-d}\int_{0}^{d}\Big{(}U_{Y}^{2}-\frac{1}{d^{2}}\Big{)}dY+\frac{1}{dd_{+}}+\Big{(}1+\frac{d}{d_{+}-d}\Big{)}\omega_{0}.$
(2.29)

For large $R$ the equation

${\mathcal{R}}(s)=R$

has two solutions $s_{+}<s_{c}<s_{-}$ and they are

$s_{-}=\sqrt{2R}+O\Big{(}\frac{1}{R}\Big{)}$ (2.30)

and

$d(s_{+})=R+O(1),$ (2.31)

where $s_{+}\to s_{0}$ as $R\to\infty$. From (2.30) and (2.3) it follows

$d=d(s_{-})=\frac{1}{s_{-}}+O\Big{(}\frac{1}{s_{-}^{3}}\Big{)}=\frac{1}{\sqrt{2R}}+O\Big{(}\frac{1}{R^{3/2}}\Big{)}.$
(2.32)

Hence

$\frac{1}{d^{2}}=s_{-}^{2}+O(1).$

The relations (2.31) and (2.32) imply

$dd_{+}=\frac{\sqrt{R}}{\sqrt{2}}+O\Big{(}\frac{1}{\sqrt{R}}\Big{)}.$

Furthermore,

$U_{Y}^{2}=s_{-}^{2}-2\Omega(U)$

and so

$U_{Y}^{2}(Y)=\frac{1}{d^{2}}+O(1).$

Applying these estimates to (2.28), we obtain

$R\leq O(\omega_{0})+O\Big{(}R^{-1/2}\Big{)}$

which implies the boundedness of $R$.

Boundedness of $F$ follows from Proposition 2.2. ∎

##  3\. Proof of Proposition 1.3

###### Proof.

Let $R_{0}=\infty$.Assume that the options (1.11) and (1.12) are not valid.
Then there exist constants $M>0$ and $\delta>0$ such that

$|\xi^{\prime}(t)(X)|\leq M\;\;\mbox{for all $t>0$}$ (3.1)

and

$|R(t)-\xi(t)(0)|\geq\delta\;\;\mbox{for all $t\geq 1$}.$ (3.2)

These inequalities implies

$\Psi_{Y}^{2}(X,\xi(X))(1+\xi^{\prime}(X)^{2})=2(R-\xi(X)).$

Hence

$\Psi_{Y}^{2}(X,\xi(X))\geq\frac{2\delta}{1+M^{2}}.$ (3.3)

Furthermore, by Proposition 3.1[19] and by (2.4),

$\Psi_{Y}(X,0)\geq\delta_{1}:=\sqrt{s^{2}-s_{0}^{2}},$ (3.4)

where $s$ is the root of the equation ${\mathcal{R}}(s)=R$. Since the constant
$R(t)$ is uniformly bounded we obtain that $\delta_{1}$ does not depend on
$t$. Let us prove the following

######  Lemma 3.1.

There exists a constant $c_{0}>0$ depending on $M$, $\delta$, $R$ such that

$\Psi_{Y}(X,Y)\geq c_{0}\;\;\mbox{for all $(X,Y)\in D_{\xi}$}.$ (3.5)

###### Proof.

By Proposition 3.1[19] there exists constants $\alpha\in(0,1)$ and $C>0$
depending on $M$, $\delta$ and $R$ such that

$||\Psi||_{C^{2,\alpha}(D_{\xi})}\leq C.$ (3.6)

Now the estimate (3.5) follows from (3.3), (3.4) and (3.6) by using the
Harnack principle.

∎

Since the estimate (3.5) contradicts to (1.10), we conclude that one of
options (1.11) or (1.12) must be valid.

Now let $R_{0}<\infty$. Then instead of (3.4) we have only positivity of
$\Psi_{Y}(X,0)$. Using again the estimates (3.3) and (3.6) together with the
Harnack principle, we get that for every $\epsilon>0$ there exists $\delta>0$
such that $\Psi_{Y}>\delta$ if $Y>\epsilon$. This implies the second assertion
of Proposition 1.3.

∎

##  4\. Spectral problems

###  4.1. Spectral problems on the interval and in the strip

######  Lemma 4.1.

Let $(U,d)$ be the stream solution with the Froude number $>1$ and let
$\rho_{0}$ be given by (1.15). Then the lowest eigenvalue of the eigenvalue
problem

$\displaystyle-v^{{}^{\prime\prime}}-\omega^{\prime}(U)v=\nu v\;\;\mbox{on
$(0,d)$},$ $\displaystyle
v(0)=0\;\;\mbox{and}\;\;v^{\prime}(d)-\rho_{0}v(d)=0$ (4.1)

is positive. This eigenvalue is simple and corresponding eigenfunctions does
not change sign in $(0,d]$.

###### Proof.

Let $\nu=-\tau^{2}$. Then it is an eigenvalue if and only if $\sigma(\tau)=0$.
Since $F>1$ then the function $\sigma$ is positive and hence there are no non-
positive eigenvalues of the spectral problem (4.1). The assertion on
simplicity and sign of the corresponding eigenfunction is quite standard.

∎

We denote the lowest eigenvalue of the problem (4.1) by
$\nu_{0}=\nu_{0}(U,d)$. We have used also the notation $\nu_{0}(t)$ when this
eigenvalue is associated with the branch of solitary waves. Then

$\nu_{0}(U,d)>0$ (4.2)

and this eigenvalue continuously depends on $(U,d)$.

Let

$\Pi_{d}=\\{(X,Y)\,:\,X\in\mathbb{R},\;0<Y<d\\}.$

Consider the spectral problem

$\displaystyle-\Delta v-\omega^{\prime}(U)v=\mu v\;\;\mbox{in $\Pi_{d}$},$
$\displaystyle\partial_{Y}v(X,d)-\rho_{0}v(X,d)=0\;\;\mbox{and}\;\;v(X,0)=0.$
(4.3)

We assume that the Froude number corresponding to the uniform stream solution
$(U,d)$ is greater than $1$. Consider the unbounded in $L^{2}(\Pi_{d})$
operator ${\mathcal{A}}=-\Delta-\omega^{\prime}(U)$ introduced in Introduction
with the domain

${\mathcal{D}}({\mathcal{A}})=\\{v\in
H^{2}(\Pi_{d})\,:\,v(X,0)=0\;\;\mbox{and}\;\;\partial_{Y}v(X,d)-\rho_{0}v(X,d)=0\\}.$

Using the Fourier transform and Lemma 4.1, we obtain that

$({\mathcal{A}}v,v)_{L^{2}(\Pi_{d})}=\int_{\Pi_{d}}(|\nabla
v|^{2}-\omega^{\prime}(U)|v|^{2})dXdY-\int_{\mathbb{R}}\rho_{0}|v|^{2}dX\geq\nu_{0}\int_{\Pi_{d}}|v|^{2}dXdY$

for $v\in{\mathcal{D}}(\Pi_{d})$. Therefore the spectrum of the operator
${\mathcal{A}}$ is continuous and coincides with the half-line
$[\nu_{0},\infty)$.

###  4.2. Spectrum of the Frechet derivative (proof of Proposition 1.4 )

Let $(\Psi,\xi)$ be a general solitary solution of (1). Then

$\xi(X)=d+O(e^{-\sqrt{\nu_{0}}|X|})\;\;\mbox{as $X\to\pm\infty$}$ (4.4)

and

$\Psi(X,Y)=U(Y)+O(e^{-\sqrt{\nu_{0}}|X|})\;\;\mbox{as $X\to\pm\infty$},$ (4.5)

where $(U,d)$ is a uniform stream solution with the Froude number $>1$.

###### Proof.

(of Proposition 1.4) If the free surface is horizontal and defined by the
uniform stream solution $(U,d)$ with $F>1$ then the result follows from Sect.
4.1.

Since the domain $D_{\xi}\setminus\Pi_{d}$ has the Rellich property (see [1])
the result follows from the preservation of continuous spectrum under a
compact perturbation of the domain.

To prove the second part we assume that there are infinitely many
eigenfunctions $\\{\Phi_{j}\\}_{j=1}^{\infty}$ corresponding to eigenvalues
$\leq a$ and such that

$||\Phi_{j}||_{L^{2}(D_{\xi})}=1,\;j=1,\ldots\;\;\mbox{and}\;\;(\Phi_{j},\Phi_{k})_{L^{2}(D_{\xi})}=0\;\;\mbox{for
$j\neq k$}.$

Since the operator ${\mathcal{A}}$ is bounded from below, the eigenvalues are
located in a bounded interval separated from the half-line $[\nu_{0},\infty)$.
Using local estimates we get

$||\Phi_{j}||_{C^{2,\alpha}(D_{\xi})}\leq C_{1}.$

Using asymptotical results for solutions in domains which are asymptotically a
strip, we get that

$|\Phi_{j}(X,Y)|\leq C_{2}e^{-b|X|}\;\;\mbox{in $D_{\xi}$}$

with a positive $b$. These two estimates imply compactness of the set
$\\{\Phi_{j}\\}_{j=1}^{\infty}$ in $L^{2}(D_{\xi})$, which contradicts to its
definition.

∎

Differentiating (1) with respect to $X$, we get

$\displaystyle\Delta\Psi_{X}+\omega^{\prime}(\Psi)\Psi_{X}=0\;\;\mbox{in
${\mathcal{D}}_{\xi}$},$
$\displaystyle\partial_{\nu}\Psi_{X}-\rho\Psi_{X}=0\;\;\mbox{on
${\mathcal{S}}_{\xi}$},$ $\displaystyle\Psi_{X}=0\;\;\mbox{for $Y=0$}.$ (4.6)

From (4.5) it follows that $\Psi_{X}\in H^{1}({\mathcal{D}}_{\xi})$,

$|\Psi_{X}(X,Y)|\leq Ce^{-\sqrt{\nu_{0}}|X|},$

and $\Psi_{X}$ is an eigenfunction corresponding to the zero eigenvalue but
this function is odd with respect to $X$. Nevertheless it can be used in the
forthcoming analysis of even eigenfunctions of the Frechet derivative.

We introduce the form on $D_{a}=\\{u\in H^{1}(D_{\xi})\,:\,u(X,0)=0\\}$:

$a(u,v)=\int_{D_{\xi}}(\nabla
u\cdot\nabla\overline{v}-\omega^{\prime}(\Psi)u\overline{v})dXdY-\int_{-\infty}^{\infty}\rho
u\overline{v}dS,\;\;dS=\sqrt{1+\xi^{\prime 2}}dX.$ (4.7)

The next proposition contains the proof of the first assertion of Theorem 1.5.

######  Proposition 4.2.

(i) The operator ${\mathcal{A}}$ has at least one negative eigenvalue.

(ii) If $\mu_{0}$ the lowest eigenvalue and $\Phi_{0}$ is a corresponding
eigenfunction, then $\Phi_{0}(X,Y)>0$ for $Y>0$.

###### Proof.

We choose $a\in(0,\nu_{0})$. Then all eigenfunctions of the operator
${\mathcal{A}}$ corresponding to eigenvalues $\leq a$ satisfy the estimate

$\int_{D_{\xi}}e^{-2b|X|}(|\Phi(X,Y)|^{2}+|\nabla\Phi|^{2})dXdY\leq
C\;\;\mbox{for certain positive $b$}$ (4.8)

and the minimal eigenvalue satisfies the relation

$\mu_{0}=\inf\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}}^{2}},$ (4.9)

where infimum is taken over all non-zero $\Phi\in D_{a}$ satisfying (4.8).
This set is not empty since the function $|\Psi_{X}|$ belongs to it. This
implies in particular that $\mu_{0}\leq 0$. If $\mu_{0}=0$ then $|\Psi_{X}|$
is an eigenfunction but this function is not sufficiently smooth. Therefore
$\mu_{0}<0$. This proves (i).

If the eigenfunction $\Phi_{0}$ change sign then both functions

$u_{\pm}=\max(0,\pm\Phi)$

also satisfy (4.9) and hence they are also eigenfunctions. But they are
vanishing on a certain open set inside $D_{\xi}$, which leads that they are
identically zero. This contradiction proves (ii). ∎

By Proposition 4.2 we can numerate all eigenvalues of the operator
${\mathcal{A}}$ according to the increasing order

$\mu_{0}<\mu_{1}\leq\cdots$

The eigenvalue $\mu_{0}$ is simple and $\mu_{0}<0$ according to Proposition
4.2. Our aim is to investigate the eigenvalue $\mu_{1}$.

###  4.3. On simplicity of the second eigenvalue of the Frechet derivative

Since we are dealing with even functions instead the problem (1) defined for
all $X$ we can consider the same problem only for $X>0$ by adding the
homogeneous Neumann boundary condition $\partial_{Y}u=0$ for $X=0$. Let

$a_{+}(u,v)=\int_{D_{+}}(\nabla
u\cdot\nabla\overline{v}-\omega^{\prime}(\Psi)u\overline{v})dXdY-\int_{0}^{\infty}\rho
u\overline{v}dS,$ (4.10)

where

$D_{+}=\\{(X,Y)\in D\,:\,X>0\\}.$

We denote by ${\mathcal{A}}_{+}$ the unndounded operator corresponding to the
form $a_{+}$. Clearly, the spectra of the operators ${\mathcal{A}}$ and
${\mathcal{A}}_{+}$ coincide and the eigenfunctions of ${\mathcal{A}}_{+}$ are
restrictions to $D_{+}$ of the eigenfunctions of the operator ${\mathcal{A}}$.
The assumption (1.2) implies

$a_{+}(u,u)>0\;\;\mbox{for nonzero $u\in H^{1}(D_{+})$ satisfying
$u(X,\xi(X))=u(X,0)=0$}.$ (4.11)

In what follows we shall use the function

$u_{*}(X,Y):=\Psi_{X}(X,Y),$ (4.12)

which satisfies the boundary value problem (4.2) and which is odd, belongs to
$C^{2,\alpha}(\overline{D})$ and $u_{*}>0$ in $D_{+}$, see [30], Proposition
2.2.

The following Green’s formula for the form $a$ will be useful in the proof of
the nest proposition:

$\displaystyle-\int_{D_{+}}(\Delta
u+\omega^{\prime}(\Psi)u)\overline{v}dXdY+\int_{0}^{\infty}(\partial_{\nu}u-\rho(X)u)\overline{v}dS$
$\displaystyle=a_{+}(u,v)+\int_{0}^{\xi(0)}u_{X}\overline{v}|_{X=0}dY.$ (4.13)

Here the function $u$ belongs to the space $H^{2}_{0}(D)$ consisting of
functions from $H^{2}(D)$ vanishing for $Y=0$, and $v\in H^{1}_{0}(D)$.

######  Proposition 4.3.

Let $\xi$ be not identically constant. If the eigenvalue $\mu_{1}$ is equal to
zero then $\mu_{1}$ is simple. Moreover the corresponding eigenfunction has
exactly two nodal sets and the nodal line separating these nodal sets has one
of its end points on the curve $S_{+}=\\{Y=\xi(X),\;\;X>0\\}$.

###### Proof.

Denote by ${\mathcal{X}}$ the space of real eigenfunctions corresponding to
the eigenvalue $\mu_{1}=0$. Since all eigenfunctions from ${\mathcal{X}}$
orthogonal to $\Phi_{0}$, each eigenfunction must change sign inside $D$. Let
us show that $\dim{\mathcal{X}}=1$. The proof consists of several steps.

(a) First we show that every nonzero eigenfunction $w\in{\mathcal{X}}$ has
exactly two nodal sets. Assume that there is an eigenfunction
$w\in{\mathcal{X}}$ which has more then two nodal sets, say $Y_{j}$,
$j=1,\ldots,N$, $N>2$. Introduce the functions $u_{j}(X,Y)=w(X,Y)$ for
$(X,Y)\in Y_{j}$ and zero otherwise. Then $u_{j}\in H^{1}_{0}(D)$ and

$a_{+}(u_{j},u_{k})=0\;\;\mbox{for all $k,j=1,\ldots,N$}.$ (4.14)

We choose the constant $\alpha$ such that

$(u_{1}-\alpha u_{2},\Phi_{0})_{L^{2}(D_{+})}=0$

and observe that $a_{+}(u_{1}-\alpha u_{2},u_{1}-\alpha u_{2})=0$. Since

$\mu_{1}=\min a_{+}(w,w)$

where $\min$ is taken over $w\in H^{1}_{0}(D_{+})$ satisfying
$(w,\Phi_{0})_{L^{2}(D_{+})}=0$ and $||w||_{L^{2}(D_{+})}=1$, the minimum is
attained on the function $(u_{1}-\alpha u_{2})/||u_{1}-\alpha
u_{2}||_{L^{2}(D_{+})}$. So we conclude that $u_{1}-\alpha u_{2}$ is an
eigenfunction corresponding to the eigenvalue $\mu_{1}=0$, which is impossible
since this function is zero on $Y_{j}$ for $j>2$.

(b) Let $u\in{\mathcal{X}}$. Assume, than the nodal line of $u$ does not touch
the surface $S_{+}$. Then one of nodal sets (say $X_{1}$) contains a
neighborhood of $S_{+}$. The second nodal set (say $X_{2}$) is separated from
$S$. Consider the function $w(X,Y)=u(X,Y)$ for $(X,Y)\in X_{2}$ and zero
otherwise, Then $a(w,w)=0$ but by (4.11) this form is positive definite.
Therefore $w=0$. If both end points of $\gamma$ lie outside $S$ then introduce
$w_{2}$ which coincides with $w$ on the nodal set separated from $S$ and
vanishes otherwise. Then $a(w_{2},w_{2})=0$ and hence $w_{2}=0$ also. This
proves that one of end points of the line separated nodal sets must lie on
$S$.

(c) Now we are in position to prove that $\dim{\mathcal{X}}=1$. If
$\dim{\mathcal{X}}>1$ then there is an eigenfunction, say $w_{*}$ which is
zero at the point $z_{1}=(0,\xi(0))$, which must be one of the end points of
the nodal line separating two nodal sets. By b) another end-point $z_{2}$ of
the nodal line lies on $S_{+}$. Denote by $Y_{1}$ the nodal set attached to
the part of $S$ between $z_{1}$ and $z_{2}$. Let also $Y_{2}$ be the remaining
nodal domain. We can assume that $w_{*}<0$ in $Y_{1}$ and $w_{*}>0$ in
$Y_{2}$.

Let $u_{*}$ be the function introduced by (4.12). Using that both functions
$w_{*}$ and $u_{*}$ satisfy the problem (1) but the first one satisfies
$\partial_{X}w_{*}=0$ for $X=0$ and the second is subject to $u_{*}=0$ for
$X=0$, we have, by using (4.3),

$\int_{0}^{\xi(0)}\partial_{X}v_{*}(0,Y)w_{*}(0,Y)dY=0.$ (4.15)

Consider the function

$U=w_{*}+\beta u_{*},\;\;\mbox{where $\beta>0$}.$

Since $u_{*}$ is a positive function inside $D_{+}$, for small $\beta$ the
function $U$ has also two nodal sets $\widetilde{Y}_{1}$ and
$\widetilde{Y}_{2}$, separated by a nodal line $\widetilde{\gamma}$ with end
points $z_{1}=(0,\eta(0))$ and $\widetilde{z}_{2}\in S_{+}$ which is close to
$z_{2}$. We assume that the nodal set $\widetilde{Y}_{1}$ is attached to the
part of $S$ between the points $z_{1}$ and $\widetilde{z}_{2}$. Introduce the
functions

$\displaystyle U_{1}(X,Y)=U(X,Y)\;\;\mbox{if $(X,Y)\in\widetilde{Y}_{1}$ and
zero otherwise},$ $\displaystyle U_{2}(X,Y)=U(X,Y)\;\;\mbox{if
$(X,Y)\in\widetilde{Y}_{2}$ and zero otherwise}.$

We choose the constant $\theta$ such that

$(U_{1}+\theta U_{2},\Phi_{0})_{L^{2}(D)}=0.$ (4.16)

It is clear that $\theta\neq 0$ because of

$a_{+}(U_{1},\Phi_{0})=\mu_{0}(U_{1},\Phi_{0})_{L^{2}(D)}\neq 0.$

Here we have used that $\Phi_{0}>0$ inside $D_{+}$ by Proposition 4.2.
Furthermore due to (4.15) and the fact that $w_{*}$ is an eigenfunction
corresponding to $\mu_{1}=0$, we have

$a_{+}(U_{1}+\theta U_{2},U_{1}+\theta U_{2})=0.$

This together with (4.16) implies that $U_{1}+\theta U_{2}$ is an
eigenfunction corresponding to the eigenvalues $\mu_{1}=0$ of the spectral
problem (1). But this contradicts to the smoothness properties of the function
$U_{1}+\theta U_{2}$. (Compare with the argument used in a) and b)). ∎

##  5\. Positivity of $\mu_{1}(t)$ for small $t$

We take $R$ sufficiently close to $R_{c}$, what is equivalent to $F$ is close
to $1$. According to Sect. 2.1 the equation ${\mathcal{R}}(s)=R$ has two roots
$s_{+}<s_{c}<s_{-}$. The root $s_{-}$ corresponds to the Froude number
$F_{-}=F(s_{-})>1$ and the root $s_{+}$ has the Froude number
$F_{+}=F(s_{+})<1$. In the latest case the dispersion equation (2.11) has a
unique solution $\tau=\tau_{*}$. Here the Froude numbers $F_{\pm}$ are defined
by (2.19). Furthermore, there are two uniform stream solutions corresponding
to this value of $R$: $(U_{-},d_{-})$ and $(U_{+},d_{+})$, where
$d_{-}=d(s_{-})$ and $d_{+}=d(s_{+})$.

In order to investigate the sign of $\mu_{1}(t)$ for small $t$ we proceed as
follows.

So we consider the branch of Stokes waves starting from the uniform stream
solution $(U(s_{+}),d(s_{+}))$ and having the same Bernoulli constant $R$, by
considering the period $\Lambda$ as a parameter and

$\Lambda_{0}:=\Lambda(0)=\frac{2\pi}{\tau_{*}},$

where $\tau_{*}$ is the root of the equation (2.11). According to [19] and
[21] there exists a branch of Stokes waves

$(\psi(\tilde{t}),\xi(\tilde{t}),\Lambda(\tilde{t}))\in
C^{k,\alpha}_{0,e,\Lambda}({\mathcal{D}})\times
C^{k,\alpha}_{e,\Lambda}({\mathbb{R}}),\;\;\tilde{t}\in[0,\infty).$ (5.1)

Here $C^{k,\alpha}_{0,e}({\mathcal{D}})$ is the subspace of even functions
vanishing for $Y=0$ in $C^{k,\alpha}({\mathcal{D}})$ and
$C^{k,\alpha}_{0,e,\Lambda}({\mathcal{D}})$ is a subspace of
$C^{k,\alpha}_{0,e}({\mathcal{D}})$ of functions of period $\Lambda$.
Similarly we can define the subspace $C^{k,\alpha}_{e,\Lambda}({\mathbb{R}})$.

We will consider solutions to (1) satisfying

$|\xi^{\prime}(X)|\leq M\;\;\mbox{for all $x\in\mathbb{R}$}.$ (5.2)

We formulate a consequence of Theorem 1 and 2 from [18].

######  Theorem 5.1.

For every positive $M$ there exists $R_{M}\in(R_{c},R_{0})$ (depending also on
$\omega$) such that the following assertions are true.

(I) For every $R\in(R_{0},R_{M})$ problem (1) has one and only one non-trivial
solitary-wave solution $(\widehat{\Psi},\widehat{\xi})$.

(II) The following inequalities hold,

$d_{-}<\xi(0;\tilde{t})\leq\widehat{\xi}(0).$

Moreover the mapping

$[0,\infty)\ni\tilde{t}\to\xi(0;\tilde{t})\in[d_{-},\widehat{\xi}(0))$

is isomorphism and

$\xi(0;\tilde{t})\to\widehat{\xi}(0)\;\;\mbox{and}\;\;\Lambda(t)\to\infty\;\;\mbox{as
$\tilde{t}\to\infty$}.$

(III) There are no other solutions to (1) satisfying (5.2) exept solutions
described by (5.1) and above theorem.

Denote by $\widetilde{\mathcal{A}}(\tilde{t})$ the Frechet derivative of the
Stokes branch (5.1) at the point $\tilde{t}$. As it was shown in Proposition
2.1[10], the first eigenvalue is always negative and the second one we denote
by $\tilde{\mu}_{1}(\tilde{t})$. We have $\tilde{\mu}_{1}(0)=0$. Our first
goal is to show that $\tilde{\mu}_{1}(\tilde{t})>0$ for small positive
$\tilde{t}$.

We will use the partial hodograph transform for this purpose.

###  5.1. Partial hodograph transform

In what follows we will study branches of Stokes waves
$(\Psi(X,Y;t),\xi(X;t))$ of period $\Lambda(t)$, $t\geq 0$, started from the
uniform stream solution at $t=0$. The existence of such branches is
established in [7] with fixed period but variable $R$ and in [19] for variable
$\Lambda$ and fixed $R$. In our case of variable $\Lambda$ it is convenient to
make the following change of variables

$x=\lambda X,\;\;y=Y,\;\;\lambda=\frac{\Lambda_{0}}{\Lambda(\tilde{t})}$ (5.3)

in order to deal with the problem with a fixed period. As the result we get

$\displaystyle\Big{(}\lambda^{2}\partial_{x}^{2}+\partial_{y}^{2}\Big{)}\psi+\omega(\psi)=0\;\;\mbox{in
$D_{\eta}$},$
$\displaystyle\frac{1}{2}\Big{(}\lambda^{2}\psi_{x}^{2}+\psi_{y}^{2}\Big{)}+\eta=R\;\;\mbox{on
$B_{\eta}$},$ $\displaystyle\psi=1\;\;\mbox{on $B_{\eta}$},$
$\displaystyle\psi=0\;\;\mbox{for $y=0$},$ (5.4)

where

$\psi(x,y;t)=\Psi(\lambda^{-1}x,y;t)\;\;\mbox{and}\;\;\eta(x;t)=\xi(\lambda^{-1}x;t).$

Here all functions have the same period $\Lambda_{0}$, $D_{\eta}$ and
$B_{\eta}$ are the domain and the free surface after the change of variables
(5.3).

We assume that

$\psi_{y}>0\;\;\mbox{in $\overline{D_{\eta}}$}$

and use the variables

$q=x,\;\;p=\psi.$

Then

$q_{x}=1,\;\;q_{y}=0,\;\;p_{x}=\psi_{x},\;\;p_{y}=\psi_{y},$

and

$\psi_{x}=-\frac{h_{q}}{h_{p}},\;\;\psi_{y}=\frac{1}{h_{p}},\;\;dxdy=h_{p}dqdp.$
(5.5)

System (5.1) in the new variables takes the form

$\displaystyle\Big{(}\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+\Omega(p)\Big{)}_{p}-\lambda^{2}\Big{(}\frac{h_{q}}{h_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$
$\displaystyle\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+h=R\;\;\mbox{for
$p=1$},$ $\displaystyle h=0\;\;\mbox{for $p=0$}.$ (5.6)

Here

$Q=\\{(q,p)\,:\,q\in\mathbb{R}\,,\;\;p\in(0,1)\\}.$

The uniform stream solution corresponding to the solution $U$ of (2.1) is

$H(p)=\int_{0}^{p}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}},\;\;s=U^{\prime}(0)=H_{p}^{-1}(0).$
(5.7)

One can check that

$H_{pp}-H_{p}^{3}\omega(p)=0$ (5.8)

or equivalently

$\Big{(}\frac{1}{2H_{p}^{2}}\Big{)}_{p}+\omega(p)=0.$ (5.9)

Moreover it satisfies the boundary conditions

$\frac{1}{2H_{p}^{2}(1)}+H(1)=R,\;\;H(0)=0.$ (5.10)

The Froude number in new variables can be written as

$\frac{1}{F^{2}}=\int_{0}^{1}H_{p}^{3}dp.$

Then according to Theorem 2.1, [19] there exists a branch of solutions to
(5.1)

$h=h(q,p;t):[0,\infty)\rightarrow
C^{2,\gamma}_{pe}(\overline{Q}),\;\;\lambda=\lambda(t):[0,\infty)\rightarrow(0,\infty),$
(5.11)

which has a real analytic reparametrization locally around each $t\geq 0$.

###  5.2. Bifurcation equation

In order to find bifurcation points and bifuracating solutions we put $h+w$
instead of $h$ in (5.1) and introduce the operators

$\displaystyle{\mathcal{F}}(w;\tilde{t})=\Big{(}\frac{1+\lambda^{2}(h_{q}+w_{q})^{2}}{2(h_{p}+w_{p})^{2}}\Big{)}_{p}-\Big{(}\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}\Big{)}_{p}$
$\displaystyle-\lambda^{2}\Big{(}\frac{h_{q}+w_{q}}{h_{p}+w_{p}}\Big{)}_{q}+\lambda^{2}\Big{(}\frac{h_{q}}{h_{p}}\Big{)}_{q}$

and

${\mathcal{G}}(w;\tilde{t})=\frac{1+\lambda^{2}(h_{q}+w_{q})^{2}}{2(h_{p}+w_{p})^{2}}-\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+w$

acting on $\Lambda_{0}$-periodic, even functions $w$ defined in $Q$. After
some cancelations we get

${\mathcal{F}}={\mathcal{J}}_{p}+{\mathcal{I}}_{q},\;\;{\mathcal{G}}={\mathcal{J}}+w,$

where

${\mathcal{J}}=\frac{\lambda^{2}h_{p}^{2}(2h_{q}+w_{q})w_{q}-(2h_{p}+w_{p})(1+\lambda^{2}h_{q}^{2})w_{p}}{2h_{p}^{2}(h_{p}+w_{p})^{2}}$

and

${\mathcal{I}}=-\lambda^{2}\frac{h_{p}w_{q}-h_{q}w_{p}}{h_{p}(h_{p}+w_{p})}.$

Both these functions are well defined for small $w_{p}$. Then the problem for
finding solutions close to $h$ is the following

$\displaystyle{\mathcal{F}}(w;\tilde{t})=0\;\;\mbox{in $Q$}$
$\displaystyle{\mathcal{G}}(w;\tilde{t})=0\;\;\mbox{for $p=1$}$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (5.12)

Furthermore, the Frechet derivative (the linear approximation of the functions
${\mathcal{F}}$ and ${\mathcal{G}}$) is the following

$Aw=A(\tilde{t})w=\Big{(}\frac{\lambda^{2}h_{q}w_{q}}{h_{p}^{2}}-\frac{(1+\lambda^{2}h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}_{p}-\lambda^{2}\Big{(}\frac{w_{q}}{h_{p}}-\frac{h_{q}w_{p}}{h_{p}^{2}}\Big{)}_{q}$
(5.13)

and

${\mathcal{N}}w={\mathcal{N}}(\tilde{t})w=(Nw-w)|_{p=1},$ (5.14)

where

$Nw=N(\tilde{t})w=\Big{(}-\frac{\lambda^{2}h_{q}w_{q}}{h_{p}^{2}}+\frac{(1+\lambda^{2}h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}\Big{|}_{p=1}.$
(5.15)

The eigenvalue problem for the Frechet derivative, which is important for the
analysis of bifurcations of the problem (5.2), is the following

$\displaystyle A(\tilde{t})w=\mu w\;\;\mbox{in $Q$},$
$\displaystyle{\mathcal{N}}(\tilde{t})w=0\;\;\mbox{for $p=1$},$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (5.16)

For $\tilde{t}=0$ and $\mu=0$ this problem becomes

$\displaystyle
A_{0}w:=-\Big{(}\frac{w_{p}}{H_{p}^{3}}\Big{)}_{p}-\Big{(}\frac{w_{q}}{H_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$ $\displaystyle B_{0}w:=-\frac{w_{p}}{H_{p}^{3}}+w=0\;\;\mbox{for
$p=1$},$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$ (5.17)

Since the function $H$ depends only on $p$ this problem admits the separation
of variables and its solutions are among the functions

$v(q,p)=\alpha(p)\cos(\tau
q),\;\;\mbox{where}\;\;\tau=k\tau_{*},\;\;k=0,1,\ldots.$ (5.18)

According to [11] the function (5.18) solves (5.2) if and only if

$\alpha(p)=\gamma(H(p);\tau)H_{p},$

where the function $\gamma(Y;\tau)$ solves the euation (2.9) and
$\sigma(\tau)=0$. Therefore if $\tau\neq\tau_{*}$ then the problem (5.2) has
no non-trivial solutions. If $\tau=\tau_{*}$ then the kernel of the above
operator is one dimensional in the class of $\Lambda_{0}$ periodic, even
function and it is given by

$v=\alpha(p)\cos(\tau_{*}q),\alpha(p)=\gamma(H(p);\tau_{*})H_{p}.$

We will need also the problem

$\displaystyle-\Big{(}\frac{u_{p}}{H_{p}^{3}}\Big{)}_{p}+\frac{\tau^{2}u}{H_{p}}=F\;\;\mbox{on
(0,d)}$ $\displaystyle u(0)=0,\;\;-\frac{u_{p}}{H_{p}^{3}}+u=c\;\;\mbox{for
$p=1$},$ (5.19)

where $F\in C^{0,\alpha}([0,1])$ and $c$ is a constant. Clearly this problem
is elliptic and uniquely solvable for all $\tau\geq 0$, $\tau\neq\tau_{*}$,
the problem (5.2) has a unique solution in $C^{2,\alpha}([0,1])$. This
solution is given by

$u(p)=v(H(p))H_{p}(p),$

where $v(Y)$ solves the problem (2.18) with $f=F(H(y))$ and $g=c$.

The functions $\tilde{\mu}_{1}(\tilde{t})$ and $\lambda(\tilde{t})$ have the
representations near $\tilde{t}=0$:

$\tilde{\mu}_{1}(\tilde{t})=\mu_{2}\tilde{t}^{2}+O(\tilde{t}^{3})$

and

$\lambda(\tilde{t})=1+\lambda_{2}\tilde{t}^{2}+O(\tilde{t}^{3}).$ (5.20)

It was proved in [13] that

$-4\lambda_{2}\tau_{*}^{2}\int_{0}^{d}\gamma(Y;\tau_{*})^{2}dY=\mu_{2}\int_{0}^{d}\gamma(Y;\tau_{*})^{2}\frac{dY}{\Psi_{Y}},$
(5.21)

This implies that the sign of $\mu_{2}$ is opposite to the sign of
$\lambda_{2}$. Our main result of this section is the following

######  Proposition 5.2.

There exists $R_{1}>R_{c}$ such that

$\mu_{2}>0\;\;\mbox{for $R\in(R_{c},R_{1})$}.$

Due to (5.21) it is sufficient to prove that $\lambda_{2}<0$ for
$R\in(R_{c},R_{1})$. We note that both quantities $\lambda_{2}$ and $\mu_{2}$
depemds on $\tau_{*}$, which depends on $R$ and $\omega$. If $R$ is close to
$R_{c}$ then $\tau_{*}$ is close to zero.

###  5.3. Stokes waves for small $\tilde{t}$

Here we consider asymptotics of solutions of (5.11) for small $t$. For this
purpose we take

$h=H(p)$

and represent the solutions in the form

$H(p)+w(q,p,\tilde{t}),\;\;w=\tilde{t}v,$

where

$v(q,p;\tilde{t})=v_{0}(q,p)+\tilde{t}v_{1}(q,p)+\tilde{t}^{2}v_{2}(q,p)+\cdots$
(5.22)

In this case

${\mathcal{J}}=A_{1}\Big{(}1+\frac{w_{p}}{H_{p}}\Big{)}^{-2}+A_{2}\Big{(}1+\frac{w_{p}}{H_{p}}\Big{)}^{-2},$

where

$A_{1}=-\frac{w_{p}}{H_{p}^{3}}$

and

$A_{2}=\frac{\lambda^{2}w_{q}^{2}}{2H_{p}^{2}}-\frac{w_{p}^{2}}{2H_{p}^{4}}.$

Therefore

${\mathcal{J}}={\mathcal{J}}_{1}+{\mathcal{J}}_{2}+{\mathcal{J}}_{3}+O(t^{4}),$

where

${\mathcal{J}}_{1}=A_{1},$
${\mathcal{J}}_{2}=A_{2}-2\frac{w_{p}}{H_{p}}A_{1}=\frac{\lambda^{2}w_{q}^{2}}{2H_{p}^{2}}+\frac{3}{2}\frac{w_{p}^{2}}{H_{p}^{4}}$

and

${\mathcal{J}}_{3}=3\frac{w_{p}^{2}}{H_{p}^{2}}A_{1}-2\frac{w_{p}}{H_{p}}A_{2}=-2\frac{w_{p}^{3}}{H_{p}^{5}}-\frac{w_{p}w_{q}^{2}}{H_{p}^{3}}.$

Furthermore

${\mathcal{I}}=-\lambda^{2}\frac{w_{q}}{H_{p}}\Big{(}1+\frac{w_{p}}{H_{p}}\Big{)}^{-1}={\mathcal{I}}_{1}+{\mathcal{I}}_{2}+{\mathcal{I}}_{3}+O(t^{4}).$

Here

${\mathcal{I}}_{1}=-\lambda^{2}\frac{w_{q}}{H_{p}},\;\;{\mathcal{I}}_{2}=\lambda^{2}\frac{w_{q}w_{p}}{H^{2}_{p}},\;\;{\mathcal{I}}_{3}(w)=-\lambda^{2}\frac{w_{q}w_{p}^{2}}{H^{3}_{p}}.$

In what follows we shall use the notation

$\tau_{*}=\frac{2\pi}{\Lambda_{*}},\;\;\Lambda_{*}=\Lambda(0).$

Inserting (5.22) and (5.20) into (5.2) and equating terms of the same power
with respect to $t$, we get

$\displaystyle
A_{0}v_{0}:=-\Big{(}\frac{v_{0p}}{H_{p}^{3}}\Big{)}_{p}-\Big{(}\frac{v_{0q}}{H_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$ $\displaystyle
B_{0}v_{0}:=-\frac{v_{0p}}{H_{p}^{3}}+v_{0}=0\;\;\mbox{for $p=1$},$
$\displaystyle v_{0}=0\;\;\mbox{for $p=0$}.$

As we have shown in previous section the kernel of the above operator is one
dimensional and is generated by the function

$v_{0}=\alpha_{0}(p)\cos(\tau_{*}q),\;\;\alpha_{0}=\gamma(H(p);\tau_{*})H_{p}.$
(5.23)

The next term in the asymptotics satisfies the boundary value problem

$\displaystyle
A_{0}v_{1}+\Big{(}\frac{v_{0q}^{2}}{2H_{p}^{2}}+\frac{3}{2}\frac{v_{0p}^{2}}{H_{p}^{4}}\Big{)}_{p}+\Big{(}\frac{v_{0q}v_{0p}}{H^{2}_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$ $\displaystyle
B_{0}v_{1}+\frac{v_{0q}^{2}}{2H_{p}^{2}}+\frac{3}{2}\frac{v_{0p}^{2}}{H_{p}^{4}}=0\;\;\mbox{for
$p=1$},$ $\displaystyle v_{1}=0\;\;\mbox{for $p=0$}.$ (5.24)

The solution of this problem, orthogonal to $v_{0}$ in $L^{2}$, is given by

$v_{1}=\alpha_{1}(p)+\beta_{1}(p)\cos(2\tau_{*}q),$ (5.25)

where $\alpha_{1}$ and $\beta_{1}$ satisfy the problem (5.2) with $\tau=0$ and
$\tau=2\tau_{*}$ respectively with certain right-hand sides. Further, the term
$v_{2}$ is fond from the following problem

$\displaystyle
A_{0}v_{2}+\Big{(}\frac{v_{0q}v_{1q}}{H_{p}^{2}}+\frac{3v_{0p}v_{1p}}{H_{p}^{4}}+{\mathcal{J}}_{3}(v_{0})\Big{)}_{p}+\Big{(}\frac{v_{1q}v_{0p}+v_{0q}v_{1p}}{H^{2}_{p}}+{\mathcal{I}}_{3}(v_{0})\Big{)}_{q}=2\lambda_{2}\Big{(}\frac{v_{0q}}{H_{p}}\Big{)}_{q}\;\;\mbox{on
$Q$},$ $\displaystyle
B_{0}v_{2}+\frac{v_{0q}v_{1q}}{H_{p}^{2}}+\frac{3v_{0p}v_{1p}}{H_{p}^{4}}+{\mathcal{J}}_{3}(v_{0})=0\;\;\mbox{for
$p=1$}$ $\displaystyle v_{2}(q,0)=0.$

The solvability condition for the last problem has the form

$\displaystyle
2\lambda_{2}\int_{\Omega}\frac{v_{0q}^{2}}{H_{p}}dqdp-\int_{\Omega}\Big{(}\Big{(}\frac{v_{0q}v_{1q}}{H_{p}^{2}}+\frac{3v_{0p}v_{1p}}{H_{p}^{4}}\Big{)}v_{0p}+\frac{v_{0q}v_{1p}+v_{1q}v_{0p}}{H^{2}_{p}}v_{0q}\Big{)}dqdp$
$\displaystyle+\int_{\Omega}\Big{(}\Big{(}\frac{2v_{0p}^{3}}{H_{p}^{5}}+\frac{v_{0p}v_{0q}^{2}}{H_{p}^{3}}\Big{)}v_{0p}+\frac{v_{0p}^{2}v_{0q}}{H_{p}^{3}}v_{0q}\Big{)}dqdp=0.$
(5.26)

This relation can be used to find $\lambda_{2}$ for small $\tau_{*}$. The
function $v_{2}$ has the form

$v_{2}=\alpha_{2}(p)\cos(\tau_{*}q)+\beta_{2}(p)\cos(3\tau_{*}q),$ (5.27)

where $\alpha_{2}$ and $\beta_{2}$ satisfy the problem (5.2) with
$\tau=\tau_{*}$ and $\tau=3\tau_{*}$ respectively with certain right-hand
sides.

Thus we have shown that $\lambda$ and $v$ have the form (5.20) and (5.22)
respectively. More exactly $v_{0}$ is given by (5.23), $v_{1}$ is represented
as (5.25) and $v_{2}$ by (5.27).

###  5.4. Coefficient $\lambda_{2}$ for small $\tau_{*}$ and proof of
Proposition 5.2

Here we assume that $0<\tilde{t}\ll\tau_{*}\ll 1$. The relation (5.3) for
finding $v_{1}$ has the form

$\displaystyle
A_{0}v_{1}+\frac{3}{2}\Big{(}\frac{v_{0p}^{2}}{H_{p}^{4}}\Big{)}_{p}=O(\tau_{*}^{2})\;\;\mbox{in
$Q$},$ $\displaystyle
B_{0}v_{1}+\frac{3}{2}\frac{v_{0p}^{2}}{H_{p}^{4}}=O(\tau_{*}^{2})\;\;\mbox{for
$p=1$},$ $\displaystyle v_{1}=0\;\;\mbox{for $p=0$}.$ (5.28)

We are looking for the solution in the form

$v_{1}(q,p)=a_{1}(p)\cos^{2}(\tau_{*}q).$

Then $a_{1}$ satisfies the equation

$a_{1p}=\frac{3}{2}\frac{\alpha_{0p}^{2}}{H_{p}}+H_{p}^{3}c_{1},$

where $c_{1}$ is a constant. Integrating this relation from $0$ to $p$ we get

$a_{1}(p)=\int_{0}^{p}\Big{(}\frac{3}{2}\frac{\alpha_{0p}^{2}}{H_{s}}+H_{s}^{3}c_{1}\Big{)}ds.$

From the boundry condition for $p=1$ we get

$a_{1}(1)-c_{1}=0.$

Therefore

$c_{1}\Big{(}1-\int_{0}^{1}H_{s}^{3}ds\Big{)}=\int_{0}^{1}\frac{3}{2}\frac{\alpha_{0p}^{2}}{H_{s}}ds.$

Since $F<1$ and the left hand side is equal to $c_{1}(1-F^{-2})$ the
coefficient $c_{1}$ is negative and

$c_{1}=(1-F^{-2})^{-1}\int_{0}^{1}\frac{3}{2}\frac{\alpha_{0p}^{2}}{H_{s}}ds.$
(5.29)

Integrating relation (5.3) with respect to $q$ and keeping terms containing
$c_{1}$ we get

$\lambda_{2}\tau_{*}^{2}\int_{0}^{1}\frac{\alpha_{0}^{2}}{H_{p}}dp=c_{1}\frac{9}{8}\Big{(}\int_{0}^{1}\Big{(}\frac{\alpha_{0p}^{2}}{H_{p}}+O(\tau_{*})\Big{)}dp\Big{)}+O(1).$
(5.30)

Since $F$ is less than $1$ and close to $1$ the number $c_{1}$ is negative and
large in the absolute value when $R_{1}$ is close to $R_{c}$. Therefore
$\lambda_{2}<0$, which proves Proposition 5.2.

##  6\. Proofs of Theorem 1.5 and Proposition 1.7

###  6.1. The main steps of the proof of Theorem 1.5

.

The assertions (i) and (iii) in Theorem 1.5 follow from Propositions 4.7 nd
4.3 respectively. Therefore it remains to prove the assertion (ii).

The proof of the assertion (ii) consists of three steps.

(i) In the paper [30], Sect.4, it was proved that the Frechet derivative at
the elements of the solitary branch (1.7) is invertible for small $t$. This
means that $\mu_{1}(t)\neq 0$ for small $t$.

(ii) Let us take a small $t$ and consider the solitary wave
$(\Psi,\xi,R)=(\Psi(t),\xi(t),R(t))$. Then the Bernoulli constant $R$ is close
to $R_{c}$ and we can construct the branch of Stokes waves (5.1) starting from
the uniform stream solution $(U_{+},d_{+})$. We prove that the eigenvalue
$\tilde{\mu}_{1}(\tilde{t})$ of the Frechet derivative is positive for all
values of $\tilde{t}$ on this branch of Stokes waves.

(iii) By Theorem 5.1 and by (i), (ii) the firs eigenvalue is positive also for
the Frechet derivative at the solitary solution $(\Psi,\xi,R)$.

Among these steps only the second step must be proved.

###  6.2. Solitary waves: the eigenvalue $\mu_{1}(t)$ for small $t$, proof of
(ii)

Consider the Frechet derivative on the Stokes branch (5.1). As it was proved
in [10] the lowest eigenvalue $\tilde{\mu}_{0}(\tilde{t})$ of the Frechet
derivative is always negative and according to [10] the eigenvalue
$\tilde{\mu}_{1}(\tilde{t})$ cannot change sign. Indeed if it changes then we
can consider the first bifurcation point and according to [10] there are
infinitely many subharmonic bifurcations near such point, which contradicts to
the uniqueness assertion in Theorem 1.5. Thus $\tilde{\mu}_{1}(\tilde{t})\geq
0$ for all $\tilde{t}\geq 0$. Assume that there exists a sequence
$\tilde{t}_{j}\to\infty$ as $j\to\infty$ such that
$\tilde{\mu}_{1}(\tilde{t_{j}})=0$ for all $j$. This implies that the second
eigenvalue $\mu_{1}(t)$ is zero, but this contradicts to (i). Hence the
eigenvalue $\tilde{\mu}_{1}(\tilde{t})>0$ for large $t$ and the same is true
for the second eigenvalue $\mu_{1}$. This proves (ii) and consequently Theorem
1.5.

###  6.3. Proof of Proposition 1.7

According to the assumption of Proposition 1.7 there exists a sequence
$\\{t_{j}\\}_{j01}^{\infty}$ such that $R(t_{j})\to R_{*}$ and

$\xi_{j}(0)\to R\;\;\mbox{as $j\to\infty$}.$

In this case the limit point $(0,\xi_{*}(0),R)_{*}$ is a stagnation point and
the limit configuration $(\Psi_{*},\eta_{*},F)$ satisfies: there is the angle
of $120^{\circ}$ at the stagnation point.

Reasoning as in Sect. 1.6 [11], we get

$|\nabla\Psi_{j}(X,Y)|\leq C(R_{*},\omega_{0}),\;\;\mbox{for
$(X,Y)\in\overline{{\mathcal{D}}_{\xi_{j}}}$},$ (6.1)

where $C$ depends only on $R_{*}$ and $\omega_{0}$, $\omega_{0}=\max_{0\leq
p\leq 1}\omega(p)$. One can choose a subsequence of $\\{\xi_{j}\\}$ which is
convergent in $C^{0,\alpha}_{pe}(\mathbb{R})$ for any $\alpha\in(0,1)$, to a
function $\xi_{*}\in C^{0,1}_{pe}(\mathbb{R})$. By (6.1) we can assume also
that the sequence $\\{\widetilde{\Psi}_{j}\\}$, where $\widetilde{\Psi}_{j}$
is the extension of $\Psi_{j}$ by $1$ for $Y>\xi(X)$ and by $0$ for $Y<0$, is
also convergent in $L^{\infty}(Q_{R_{*},a})$ to a function
$\widetilde{\Psi}_{*}$, where

$Q_{R,a}=\\{(X,Y)\,:\,X\in(-a,a),\;Y\in(0,R)\\}.$

Moreover

$\nabla\widetilde{\Psi}_{j}\;\mbox{weak}^{*}\;\nabla\widetilde{Psi_{*}}\;\mbox{in}\;L^{\infty}(Q_{R,a,b}).$

Then the limit functions $(\Psi_{*},\xi_{*},R_{*})$ satisfies (1) in a weak
sense, see [33]. One can verify that all conditions of Theorem 5.2, Varvaruca
[33], are satisfied and according to that theorem there are two options for
the limit function $\xi_{*}$:

$\lim_{X\to
0+}\frac{\xi_{*}(X)}{X}=-\frac{1}{\sqrt{3}}\;\;\mbox{or}\;\;\lim_{X\to
0}\frac{\xi_{*}(X)}{X}=0.$ (6.2)

It is assumed in the theorem that the first option holds, i.e. the limit
Stokes wave has an opening angle $120$ degree at the stagnation points.

######  Remark 6.1.

It is proved in [22] that the second option in (6.2) is imposible. So the
first option in (6.2) is always satisfied when we approach a stagnation point.
Therefore one can remove the text ”with angle $120^{\circ}$ at the crest” in
Proposition 1.7.

In the paper [11] it was shown that the number of negative eigenvalues of the
Frechet derivative tends to infinity if we consider the eigenvalue problem in
a bounded domain between $Y=\pm a$ with the homogeneous Neuman boundary
condition on the new boundary corresponding to the boundary. Since the
corresponding eigenfunctions decay outside a small neighborhood of the crest,
we can use these function to show that the same is true for the eigenvalue
problem in $D(t_{j})$, i.e. the number of eigenvalues for the Frechet
derivative in $D(t_{j})$ tends to $\infty$ as $j\to\infty$.

Denote by ${\mathcal{A}}_{j}$ the Frechet derivative at the point $t_{j}$. As
a result of the above arguments we get

######  Proposition 6.2.

Let $N_{j}$ be the number of negative eigenvalues of the operator
${\mathcal{A}}_{j}$. Then $N_{j}\to\infty$ as $j\to\infty$.

Now using Theorem 1.5 and the above proposition we obtain that the eigenvalue
$\mu_{1}(t)$ is non-negative on $(0,t_{*})$ for certain positive $t_{*}$ and
then negative for $t\in(t_{*},t_{*}+\epsilon)$ for a certain positive
$\epsilon$. Since $\mu_{1}(t_{*})=0$ is a simple eigenvalue and $\mu_{1}$ is
analytic in a neighborhood of $t_{*}$ we get that $\mu_{1}$ is positive for
$t<t_{*}$ and $t$ close to $t_{*}$. Therefore the crossing number at $t_{*}$
is $1$. This proves Proposition 1.7.

##  7\. Acknowledgments

This material is based upon work supported by the Swedish Research Council
under grant no. 2021-06594 while the author was in residence at Institut
Mittag-Leffler in Djursholm, Sweden during Oct 08- Oct 29, 2023.

##  8\. Data availability

No data was used for the research described in the article.

##  9\. References

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