数理解析研究所講究録 1436 巻 2005 年 1-21

1Global structure of solutions forthe 1-D Ginzburg-Landau equation
龍谷大学・理工学部 (日本学術振興会 特別研究員) 小杉 聡史 (Satoshi Kosugi)

Department of Applied Mathematics and Informations,Ryukoku University

1

Introduction.

This is a joint work with Prof. Y. Morita and Prof. S. Yotsutani (Ryukoku
University). In this article we are dealing with a simplified model of thesuperconductivity in a thin uniform superconducting ring. The energy functional in a one-dimensional form of such a model is given by$E( \psi):=]_{0}^{2\pi}\frac{1}{2}|D_{h}\psi|^{2}+\frac{\lambda}{4}(1-|\psi|^{2})^{2}dx$,$D_{h}:= \frac{d}{dx}-\mathrm{i}h(x))$

expresses the density ofwhere is a complex-valued order parameter (superconducting electrons) , is a positive parameter, and $h(x)$ is a periodic
function. Note that $h(x)$ is the projection of magnetic potential of an
applied magnetic field to the tangent direction of a parametrized ring (see[8], [9] . We consider this functional on a space of -periodic functions in. Then the Euler-Lagrange equation of this functional is given by$|\psi|^{2}$$\psi$$\lambda$$C^{1}$$2\pi$$)$$H_{1\mathrm{o}\mathrm{c}}^{1}(\mathbb{R})$’$\{\begin{array}{l}D_{h}^{2}\psi+\lambda(1-|\psi|^{2})\psi=0\psi(x+2\pi)=\psi(x)\end{array}$(1.1)$x\in \mathbb{R}x\in \mathbb{R},$which is the Ginzburg-Landau equation of this model. One feature of thisequation is that it is transformed into the equation’$\{\begin{array}{l}u_{xx}+\lambda(1-|u|^{2})u=0u(x+2\pi)\mathrm{e}\mathrm{x}\mathrm{p}(2\pi\mu i)=u(x)\end{array}$$x\in \mathbb{R}x\in \mathbb{R},$(1.2)

by the change of variable$\psi$$=u\exp$($\mathrm{i}[_{0}^{x}h(s)ds$

),

(1.3)

2
where
$\mu:=\frac{1}{2\pi}\int_{0}^{2\pi}h(s)ds$

(1.4)

.

. We wilIand
Our goal is to completely solve (1.2) for eachalso discuss the global structure of solutions to (1.2) for the parametersand .
, let be a constant
We here give a remark on in (1.2). For each, all
. Sincesuch that $\tilde{\mu}0\in[-1/2,1/2]$ and. We
are also all solutions to (1.2) for
the solutions to (1.2) for.
then realize that it suffices to solve (1.2) for $\mu\in[-1/2,1/2]$ instead ofin this paper for a simple expression of each
However we assume
solution to (1.1) which is given by (1.3). We also note that given solution$u(x)$ of (1.2) the symmetry of the equation allows $u(x)e^{ic}$ and $u(x+c)$ to be
R. However we will not mention about thissolutions for any constantfact explicitly unless we need to state clearly, we note that a complete global bifurcation
As for a specific casediagram for A is obtained in the previous paper [4]. We will extend thisstudy to the present case (see also [1] and [5]). However the bifurcation
structure exhibits more complex in the presence of an add itional parameter. Nonetheless we can see the global bifurcation structure by solving theand $\lambda>0$ . The approach developed in [4]equation (1.2) for any
fortunately works in the present situation so that a small modification of
the argument can provide an explicit expression of every solution. In consequence one can observe how the secondary bifurcating solution deforms as
varies until it disappears through another bifurcation.
To achieve it, we first classify all the solutions to (1.2) according to theirconfiguration. In what follows the idea of the classification is quite simple
but crucial for drawing the whole bifurcation diagram. Here we exclude thetrivial solution $u=0$ and modify the classification found in [4] a little forconvenience of dealing with the present problem. Thus all the nontrivial
solutions to (1.2) are classified into three types as$\mu\in \mathbb{R}$$\lambda>0$$\lambda$$\mu$$\tilde{\mu}\in \mathbb{R}$$\tilde{\mu}0$$\mu$$\tilde{\mu}-\tilde{\mu}_{0}\in \mathbb{Z}$$\exp(2\pi\tilde{\mu}_{0}\mathrm{i})=\exp(2\pi\tilde{\mu}\mathrm{i})$$\mu=\tilde{\mu}0$$\mu=\tilde{\mu}$$\mu\in \mathbb{R}$$\mu\in \mathbb{R}$$c\in$$\mu\in \mathbb{Z}$$\mu$$\mu\in \mathbb{R}$$\mu$(I) Solutions with zero.
(Ha) Solutions with constant amplitude.

(lib) Solutions with nonconstant and nonvanishing amplitude.
Note that this classification also works in (1.1).
We here characterize solutions in each class. As will be discussed in thenext section, the solution of Type (I) is written in the form$u(x)=\exp(\mathrm{i}c)\phi(x)$3where is a constant in and is a real-valued function. Thus the paramif solutions of TyPe (I) exist. In other words,
eter must satisfy
Z. More precisely we will prove
solutions of Type (I) do not exist if
the following. There exist solutions of Type (I) which have even zeros in$[0, 2\pi)$ if and only if
is even and $\lambda>1$ , otherwise, there exist solutions
is odd and $\lambda>1/4$ .
which have odd zeros in $[0, 2\pi)$ if and only if
Next we consider the solution of TyPe (Ha). It is easy to obtain thefollowing nontrivial (constant amplitude) solution to (1.2)$\mathbb{R}$$c$$\phi$$2\mu\in \mathbb{Z}$$\mu$$2\mu\not\in$$2\mu$$2\mu$(1.5)$u_{\lambda,\mu,m}^{\mathrm{c}}:=\sqrt{1-(m-\mu)^{2}/\lambda}\exp(\mathrm{i}(m-\mu)x)$

for each$m\in \mathbb{Z}$. This solution exists if and only if$\lambda>\lambda_{\mu,m}:=$

$($

rn

$-\mu)^{2}$$(\mu, \lambda)$

satisfies.

It gives a solution to (1.1) as$\psi_{m}:=u_{\lambda,\mu,m}^{\mathrm{c}}(x)\exp(\mathrm{i},\int_{0}^{x}h(s)ds)$

where pa is defined in (1.4). For each $m$ , this solution emerges from the. The study ofcrosses the curve A
trivial solution 0 when
[10] tells a local bifurcation structure of (1.2) by using a standard local
bifurcation analysis. As a result they showed a secondary bifurcation, thatis, bifurcations from the nontrivial solution take place at$=\lambda_{\mu,m}$$(\mu, \lambda)$

A $=\lambda_{\mu_{2}m,n}:=3(m-\mu)^{2}-n^{2}/2$ ,$(n\in \mathrm{N})$

.

(1.6)

Besides the local bifurcation structure, we are interested in a global one of(1.2). Among other things it is interesting to show how the configuration of
the secondary bifurcating solution changes as the parameters varies.
Finally we deal with solutions of Type (IIb). It is much more difficult
than the other case. We will discuss it in \S 3 and show that a Type (lib)solution arises through a secondary bifurcation which exists in regions$D_{m,n}^{-}:=\{(\mu, \lambda) :$D_{m,n}^{+}.=${ , ) : $\mu>m+n/2$ , A$(\mu$

,

(1.7)

},

(1.8)

\mu<m-n/2, \lambda>\lambda_{\mu,m,n}\}$$\lambda$$>\lambda_{\mu,m,n}$. For fixed $\mu<m-n/2$ (resp.
andfor arbitrarily given,$\mu>m+\mathrm{n}/2,$ , as A increase in a neighborhood of the curve A
a secondary bifurcating branch emanates from a branch of a Type (Ha). The secondary bifurcating branch is compose
solution at$m\in \mathbb{Z}$$n\in \mathrm{N}$$=\lambda_{\mu,m,n}$$\mathrm{d}$$\lambda=\lambda_{\mu,m,n}$

4of a Type (lib) solution. Similarly, for fixed A $>n^{2}/4$ , as increase in a
, a secondary bifurcating branch emanates fromneighborhood. A Type (lib)a branch of a Type (Ila) solution at the curveis the component of the secondary bifurcating
solution forbranch. As increase through $\mu=m-n/2$ , the Type (lib) solution changes
via a Type (I) solution.
into another Type (lib) solution for
The branch ends up by connecting itself with the branch of a Type (Ila).
solution at$\mu$$D_{m,n}^{-}\cup D_{m-n,n}^{+}$$\lambda=\lambda_{\mu\}m,n}$$(\mu, \lambda)\in D_{mn)}^{-}$$\mu$$(\mu, \lambda)\in D_{m-n,n}^{+}$$\lambda=\lambda_{\mu,m-n,n}$2Type (I) solutions.

In this section we treat the TyPe (I) solutions to (1.2). We will show that,, there exists a solution to (1.2) which has zero points onfor each$[0, 2\pi)$ if and only if$n$$n\in \mathrm{N}$$\lambda>n^{2}/4$,$\mu=m+n/2$$(\forall m\in \mathbb{Z})$

.

(2.1)

Each solution is written in the form
$u=u_{\lambda,n}^{\mathrm{s}}(x+\omega)\exp(\mathrm{i}c)$where$k\in(\mathrm{O}, 1)$,$u_{\lambda,n}^{\mathrm{s}}:=k\sqrt{2/(1+k^{2})}\mathrm{s}\mathrm{n}(nK(k)x/\pi, k)$

is a unique solution to$\sqrt{1+k^{2}}K(k)=\pi\sqrt{\lambda}/n$

and are arbitrary constant of
whose inverse is given by$c$$\mathbb{R}$$\omega$,$\mathrm{s}\mathrm{n}(x, k)$

,

(2.2)

is the Jacobi elliptic function$\mathrm{s}\mathrm{n}^{-1}(u, k)=\int_{0}^{u}\frac{1}{\sqrt{1-\tau^{2}}\sqrt{1-k^{2}\tau^{2}}}d\tau$,

and $K(k)$ is a complete elliptic integral$K(k)$ $:= \int_{0}^{1}\frac{1}{\sqrt{1-\tau^{2}}\sqrt{1-k^{2}\tau^{2}}}d\tau$

Recall thatto show$\phi$

is extended to with period $4K(k)$ and it is not difficult
is a solution to the real-valued equation$\mathbb{R}$$\mathrm{s}\mathrm{n}(x, k)$$=u_{\lambda,n}^{\mathrm{s}}$.

$\phi_{xx}+\lambda(1-\phi^{2})\phi=0$

.

(2.3)

To achieve the above results, we first show that if a nontrivial solution to(1.2) has a zero point, thenis an integer and the solution is a real-valued$2\mu$

5function multiplied by a complex constant. Consider a nontrivial solution$u(x)$ which vanishes at $x=x_{0}$ . Denote $u(x)=u_{1}(x)+\mathrm{i}u_{2}(x)$ . Then (1.2)
allows the expression’$\{\begin{array}{l}(u_{1})_{xx}+Q(x)u_{\mathrm{l}}=0(u_{2})_{xx}+Q(x)u_{2}=0\end{array}$$Q(x):=\lambda(1-|u(x)|^{2})$ .

$x\in \mathbb{R}x\in \mathbb{R},$Since $uj(x\mathrm{o})=uj(x0+2\pi)=0(j=1, 2)$ , each $uj$ is an eigenfunction of theoperator$L:= \frac{d^{2}}{dx^{2}}+Q(x)$,$D(L)=\{u\in H^{2}(x_{0}, x_{0}+2\pi) :u(x_{0})=u(x_{0}+2\pi)=0\}$. It thus follows that
corresponding to zero eigenvalue if
for some constants , $c_{2}\in \mathbb{R}((c1, c2)\neq(0,0))$ from the Sturm-Liouvilletheorem, which tells that the dimension of each eigenspace is one. Put.
). Then the solution is written in(or$c_{1}u_{1}=c_{2}u_{2}$$u\mathrm{i}\not\equiv \mathrm{O}$$c_{1}$$\phi:=\sqrt{1+c_{1}^{\mathit{2}}/c_{2}^{\mathit{2}}}.$$\sqrt{1+c_{2}^{2}/c_{1}^{2}}u_{2}$$u_{1}$$u$$u(x)=\phi(x)\exp(\mathrm{i}c)$

for a constant$c\in \mathbb{R}$. Thus the second condition of (1.2) implies$\phi(x+2\pi)\exp(2\pi\mu \mathrm{i})=\phi(x)$

.

(2.4)

must be an integer.
is real valued,
Since
We next verify that any solution of (2.3) with (2.4) is written in theup to translation and (2.1) is a necessary and sufficient
form
condition of existence. It follows from an elementary argument of ordinary
differential equations.
Let be a nontrivial solution to (2.3) with (2.4) which has zero points.
Without loss of generality we may assume $\phi(\mathrm{O})=\phi(2\pi)=0$ and $\phi_{x}(0)>0$
because (A is not the trivial solution. Thus there exists a point $x_{1}\in(0,2\pi)$such that$\phi_{x}(x)>0$.
$\phi_{x}(x_{1})=0$ ,$2\mu$$\phi(x)$$\phi$$\phi$$=u_{\lambda_{1}n}^{\mathrm{s}}$$\phi$$\forall x\in[0, x_{1})$Put a $=\phi(x_{1})$ . Then the equation (2.3) implies$\forall x\in[0, x_{1}]$$\frac{d\phi}{dx}=$

where$\alpha^{2}+\beta^{2}=2$

,

$0<$

a

$<\beta$.

(2.5)

$\mathrm{G}$Changing variable $\xi(x):=\phi(x)/\alpha$ , we have$\forall x\in[0, x_{1}]$$x= \frac{\sqrt{2}}{\beta\sqrt{\lambda}}\int_{0}^{\phi(x)/\alpha}$

and hence$x_{1}=\sqrt{2}K(k)/\beta^{\sqrt{\lambda}}$Therefore$\phi$,$k:=\alpha/\beta$

(2.6)

.

is written in the form$\phi(x)=\alpha$(2.7)

sn $(K(k)x/x_{1}, k)$

and this equality is satisfied on the whole R. Sincealso satisfies
Jacobi elliptic function$\phi$

satisfies (2.4) and the$\mathrm{s}\mathrm{n}(\cdot, k)$$\mathrm{s}\mathrm{n}(x+2K(k)n, k)=(-1)^{n}\mathrm{s}\mathrm{n}(\mathrm{x};k)$there exist$n\in \mathrm{N}$and$m\in \mathbb{Z}$,$x\in \mathbb{R}$,$\mu=m+n/2$ .

,On the other hand, since and satisfy (2.5) andand are written in the form$\alpha$$n\in \mathrm{N}$such that$x_{1}=\pi/n$$\alpha$,$\beta$(2.8)$k$is defined by$k=\alpha/\beta$,$\beta$$\alpha=k\sqrt{2/(1+k^{2})}$Thus (2.7) implies the expression of,$\beta=\sqrt{2/(1+k^{2})}$$u_{\lambda_{\mathrm{t}}n}^{\mathrm{s}}$.

and (2.6) changes into

.

(2.9)

$x_{1}=\sqrt{1+k^{2}}K(k)/\sqrt{\lambda}$

Since (2.8) and (2.9), must satisfy (2.2).
Therefore if the equation (2.3) with (2.4) has a nontrivial solution with(up to translation) and $k\in(0,1)$
zero points in $[0, 2\pi)$ , the solution issatisfies (2.2) and $\mu=m+n/2(m\in \mathbb{Z})$ . On the other hand, it is clear that,,solves the equation (2.3) with (2.4) if $k\in(0,1)$ satisfies
for each(2.2) and $\mu=m+n/2(m\in \mathbb{Z})$ .
Now let us consider (2.2). Since$\mathrm{k}$$n$$u_{\lambda,n}^{\mathrm{s}}$$n\in \mathrm{N}$$u_{\lambda,n}^{\mathrm{s}}$$K(0)= \frac{\pi}{2}$,$\frac{dK}{dk}>0$,$\lim_{karrow 1}K(k)=\infty$,. Accordingly we can assert(2.2) has a unique solution if and only ifthat any solution of (2.3) with (2.4) is written in the form
andthe solution exists if and only if (2.1) is satisfied$n^{2}<4\mathrm{A}$$\phi$$\phi$$=u_{\lambda,n}^{\mathrm{s}}$7Type (lib) solutions.

3

In this section we consider a solution with nonconstant amplitude but nonvanishing everywhere. The method developed in the previous paper [4] canstill work in this present 2-parameters case.
Since $|u(x)|>0$ , we can write $u=w(x)\exp(\mathrm{i}\theta(x))$ where $w(x)>0$ .
Putting it into the equation (1.2) yields$w_{xx}-\theta_{x}^{2}w+\lambda(1-w^{2})w=0$$x\in \mathbb{R}$,$(w^{2}\theta_{x})_{x}=0$$x\in$

R.

(3.1)

Then the periodic condition in (L2) is reduced to$\theta(x+2\pi)[perp]_{1}2\pi\mu=\theta(x)+2m\pi$

,

(3.2)

, we have thatfor an integer $m$ . Integrating the equation
R. Integrating this equality again and using (3.2),
for a constant
we obtain$(w^{2}\theta_{x})_{x}=0$$\theta_{x}=$$b\in$$b/w^{2}$$2(m- \mu)\pi=b[_{0}^{2\pi}\frac{1}{w(x)^{2}}dx$.

Thus the equation (3.1) is reduced to
’

(3.3)

$\ovalbox{\tt\small REJECT} bw_{xx}w(x+2\pi))=w-\frac{b^{2}}{w^{3}}+\lambda(,’ 0=2(m-\mu)\pi/\int_{(x)}^{1-w^{2}}0\frac{)w=dx}{w(x)^{2}}w(x)>02\pi$$x\in \mathbb{R}x\in \mathbb{R}x\in \mathbb{R},,$.

Then a solution of (1.1) is obtained by solving the above equation and itis written in the form$\psi$$\psi=w(x)\exp\{2(m-\mu)\pi \mathrm{i}]_{0}^{x}\frac{1}{w(s)^{2}}ds/f_{0}^{2\pi}\frac{1}{w(s)^{2}}ds+\mathrm{i}\int_{0}^{x}h(s)ds\}$

We note that (3.3) has a constant solution$w=\sqrt{1-(m-\mu)^{2}/\lambda}$

if A $>(m-\mu)^{2}$ . This gives a solution of Type (Ila). Since stands for theamplitude of a solution , we exclude this constant solution. Finding all thesolution of (3.3), we also give an attention to a solution of a higher mode,that is, a $2\pi/n$-periodic solution of (3.3) for $n\in$ N. Consequently all Type$w$$\psi$

8(lib) solutions are obtained by solving the following system of equations for$w(x)$ and 6:
’

(3.4)

$x\in \mathbb{R}x\in \mathbb{R},$$\{\begin{array}{l}w_{xx}-\frac{b^{2}}{w^{3}}+ \mathrm{A}(1-w^{2})w=0w(x)>0T_{w}=2\pi/n\end{array}$

and
(3.5)

$b=2(m- \mu)\pi/]_{0}^{2\pi}\frac{1}{w(x)^{2}}dx’$’

denotes the fundam entai, where,, and,for each
period of $w(x)$ .
The following result establishes not only the existence but also the configuration of every secondary bifurcating solution.
$m\in \mathbb{Z}$$n\in \mathrm{N}$$\mu\in \mathbb{R}$$\lambda>0$$T_{w}$, if and only if
andTHEOREM 3.1 For each
which is defined in (1.7) ancl (1.8), there ex\^i$m\in \mathbb{Z}$$n\in \mathrm{N}$$D_{m,n}^{+}\cup D_{m,n^{f}}^{-}$$u_{\lambda,\mu_{7}m,n}^{\mathrm{o}}:=w(x)\exp$

belongs toa solution$(\mu, \lambda)$$te$(i2 (x))$w(x):=\sqrt{\frac{2}{3}+\frac{2n^{2}K(k)^{2}}{\lambda\pi^{2}}\{k^{2}\mathrm{s}\mathrm{n}^{2}(\frac{nK(k)}{\pi}x,k)-\frac{k^{2}+1}{3}\}}$0(x)$:=2(m- \mu)\pi(]_{0}^{2\pi}\frac{dy}{w(y)^{2}})^{-1}\int_{0}^{x}\frac{dy}{w(y)^{2}}$

,

(3.6)

,

where $k\in$ $(0, 1)$ is a unique solution of$\{\begin{array}{l}2(m-\mu)^{2}K(k)^{2}-\lambda\gamma(k)\Pi(\mathcal{B}(k)/\alpha(k)-1, \mathrm{A})^{2}\beta(k)/\alpha(k)=0\alpha(k)>0\end{array}$a $nd$ ,$\alpha$$\beta$, , and$\gamma$$\Pi$

are defined as$\Pi(\nu, k):=\int_{0}^{1}\frac{1}{(1+\nu\tau^{2})\sqrt{1-\tau^{2}}\sqrt{1-k^{2}\tau^{2}}}d\tau$$\alpha(k):=\frac{2}{3}-\frac{2n^{2}K(k)^{2}(k^{2}+1)}{3\lambda\pi^{2}}$,$\beta(k);=\frac{2}{3}-\frac{2n^{2}K(k)^{2}(1-2k^{2})}{3\lambda\pi^{2}}$$\gamma(k):=\frac{2}{3}-\frac{2n^{2}K(k)^{2}(k^{2}-2)}{3\lambda\pi^{2}}$

.

,

(3.7)
(3.8)

,

(3.9)
(3.10)

8For$(\mu, \lambda)\in D_{m,n}^{+}\cup D_{m,n}^{-}$, the solutio$u_{\lambda,\mu,m,n}^{\mathrm{o}}-u_{\lambda,\mu,m}^{\mathrm{c}}arrow 0$$u_{\lambda,\mu,m,n}^{\mathrm{o}}-u_{\lambda,\mu,m}^{\mathrm{c}}arrow 0$

satisfies$nu_{\lambda,\mu,m,n}^{\mathrm{o}}$uniformly on$\mathbb{R}$asuniformly on$\mathbb{R}$

as $\muarrow m\pm$$\lambdaarrow\lambda_{\mu,m,n}$,(3.11)(3.12)

uniformly on$u_{\lambda,\mu,m,n}^{\mathrm{o}}\pm \mathrm{i}u_{\lambda_{l}n}^{\mathrm{s}}arrow 0$

Moreover, for given$u\equiv 0$
is given by one of$\mu\in \mathbb{R}$$e^{ic}u_{\lambda,n}^{\mathrm{s}}(x+\omega)$$e^{ic}u_{\lambda,\mu,m}^{\mathrm{c}}(x)$$e^{ic}u_{\lambda,\mu,m,n}^{\mathrm{o}}(x+\omega)$

where

$c$

and

$\omega$$\mathbb{R}$

as $\muarrow m\pm n/2$ .

(3.13)and A $>0$ , every solution of (1.2) except $/or$

such as $n^{2}/4<\lambda$ , $n/2-\mu\in \mathbb{Z}$ ,such as $(m-\mu)^{2}<\lambda$ ,for$n\in \mathrm{N}$for$m\in \mathbb{Z}$for$(m, n)\in \mathbb{Z}\mathrm{x}$$\mathrm{N}$such as $n^{2}/4<3(m-\mu)^{2}-n^{2}/2<\lambda$ ,are arbitrarily taken real numbers.

periodic
COROLLARY 3.1 Let A $>0$ . For givenfunction , defineas (1.4). Then each nontrivial solution to (1.1) is one of the following$2\pi$$u_{\lambda,n}^{s}(x+\omega)\exp(\mathrm{i}[_{0}^{x}h(s)ds+\mathrm{i}c)$for$n\in \mathrm{N}$$u_{\lambda,\mu,m}^{\mathrm{c}}(x) \exp(\mathrm{i}\int_{0}^{x}h(s)ds+\mathrm{i}c)$for$m\in \mathbb{Z}$$C^{1}$

$h$

$\mu$

such as $n^{2}/4<\lambda$ , $n/2-\mu\in \mathbb{Z}$ ,such as $(m-\mu)^{2}<\lambda$ ,and$u_{\lambda,\mu,m,n}^{\mathit{0}}(x+ \omega)\exp(\mathrm{i}\int_{\mathrm{f}l}^{x}h(s)ds+\mathrm{i}c)$

for
where

4

$c$

on

$(m, n)\in \mathbb{Z}\cross$$d\omega$$\mathrm{N}$such as $n^{2}/4<3(m-\mu)^{2}-n^{2}/2<\lambda$ ,are real numbers.

Appendix.

We will give a sketch of the proof of Theorem 3.1. The readers can refer to[4] for the detailed argument.
is a nonconstant
We solve (3.4) (without considering (3.5)). Sincesuch that $X1<x_{2}$ and, there exist ,periodic function in$w$$C^{2}$$w_{x}(x_{1})=w_{x}(x_{2})=0$$x_{1}$,$x_{2}\in \mathbb{R}$$w_{x}(x)>0$(Vr$\in(x_{1}$,$x_{2}$

)).

(4.1)

10
Multiplying$2w_{x}$to the equation in (3.4), we have$\frac{d}{dx}((w_{x})^{2}+\frac{b^{2}}{w^{2}}+\frac{\lambda}{2}(2w^{2}-w^{4}))=0$.

Thus$w_{x}(x)^{2}= \frac{\lambda\{w(x)^{2}-w(x_{1})^{2}\}}{2w(x)^{2}w(x_{1})^{2}}($$\{w(x)^{2}+w(x_{1})^{2}-2\}w(x)^{2}w(x_{1})^{2}+\frac{2b^{2}}{\lambda})$

.

(4.2)
Put

a $=w(x_{1})^{2}$ ,$\beta=w(x_{2})^{2}$,

and $x=x_{2}$ in (4.2). Then we obtain($\beta+$a -2)$\alpha\beta+2b^{2}/\lambda=0$,

which implies$\frac{2b^{2}}{\lambda}=\alpha\beta\gamma$,$\gamma:=2-\alpha-\beta$

.

(4.3)

Introducing the new variable$v(x):=w(x)^{2}$ ,and substituting (4.3) into (4.2), we can easily verify$(v_{x}(x))^{2}=2\lambda(v(x)-\alpha)(v(x)-\beta)(v(x)-\gamma)$,$\forall x\in \mathbb{R}$

.

Since
$0<\alpha<v(x)<\beta$ ,the ordering$\gamma\geq\beta$$v_{x}(x)>0$$(\forall x\in(x_{1}, x_{2})))$

holds. In the sequel(4.4)$\{\begin{array}{l}v_{x}(x)=\forall x\in[x_{1_{7}}x_{2}]0<\alpha<\beta\leq\gamma,\alpha+\beta+\gamma=2\end{array}$

Next we solve (4.4). By integration of (4.4)$\forall x\in[x_{1}, x_{2}]$$x-x_{1}= \frac{1}{\sqrt{2\lambda}}\int_{\alpha}^{v(x)}$

.

(4.5)

Changing the variable $y=\alpha+(\beta-\alpha)\tau^{2}$ in (4.5) an putting$k:=\sqrt{(\beta-\alpha)/(\gamma-\alpha)}$

,

(4.6)

11
we see.

$\frac{dy}{\sqrt{(y-\alpha)(y-\beta)(y-\gamma)}}=\frac{2d\tau}{\sqrt{(\gamma-\alpha)(1-\tau^{2})(1-k^{2}\tau^{2})}}$

Applying this to (4.5) yields$x-x_{1}=\sqrt{\frac{2}{\lambda(\gamma-\alpha)}}\mathrm{s}\mathrm{n}^{-1}$($\sqrt{\frac{v(x)-\alpha}{\beta-\alpha}}$

,

$k$

),

$\forall x\in[x_{1}, x_{2}]$

(4.7)

.

Thus on the interval $[x1, x2]$ it holds(4.8)$w(x)=\sqrt{v(x)}=$Since this $w(x)$ is defined over and periodic with period(sn 2( , ) has a period $2K(k)$ ), $T_{w}=2\pi/n$ implies$\mathbb{R}$$x$$2K(k)\sqrt{2}/\lambda(\gamma-\alpha)$$k$$2K(k) \sqrt{\frac{2}{\lambda(\gamma-\alpha)}}=\frac{2\pi}{n}$

.

(4.9)

Combining (4.3), (4.6), and (4.9), we obtain the expressions (3.8), (3.9), andand(3.10). In the sequel we obtained solutions of (3.4). In fact let
and
A $>0$ . Then $(w(x), \lambda, b)$ satisfies (3.4) if and only if there exist$k\in(0,1)$ such that $\alpha=\alpha(k)>0$ and$n\in \mathrm{N}$$x_{1}\in \mathbb{R}$$\sqrt{\alpha+(\beta-\alpha)\mathrm{s}\mathrm{n}^{2}(\frac{nK(k)}{\pi}x,k)}\backslash$,(4.10)$\{_{b^{2}=\frac{\lambda\alpha\beta\gamma}{2}}^{w(x+x_{1})=}$.
Now we take the condition (3.5) into account. Since $T_{w}=2\pi/n$ and asymmetry,$]_{0}^{2\pi} \frac{dx}{w(x)^{2}}=\int_{0}^{2\pi}\frac{dx}{w(x+x_{1})^{2}}=2n]_{0}^{\pi/n}\frac{dx}{w(x+x_{1})^{2}}$.

The similar argument used in the derivation of (4.5) and (4.7) leads us to$/n$$\underline{dx}$$]_{\mathrm{O}}\pi$$w$($x$ $+$ $x$ $1$)$\underline{1}$$=$$\int_{\alpha}^{\beta}$$2$$\sqrt{2\lambda}$$= \sqrt{\frac{2}{\lambda}}\frac{\Pi(\beta/\alpha-1,k)}{\alpha\sqrt{\gamma-\alpha}}$,(4.11)

12
where II is the complete elliptic integral defined in (3.7). Since $\alpha>0$ anda $=2n^{2}K(k)^{2}/\lambda\pi^{2}$ , the equation (3.5) is written as$\gamma-$$b=(m-\mu)\alpha K(k)/\Pi(\beta/\alpha-1, k)$.

Substituting this into the second equation in (4.10), we can reduce our problem to solving the equation(4.12)$2(m-\mu)^{2}K(k)^{2}-\lambda\gamma\Pi(\beta/\alpha-1, k)^{2}\beta/\alpha=0$

under the constraint $\alpha>0$ . To simplify the notation in the rest of thisfor eachpaper, we denote the left hand of the above equation by, that is, we put
and$\rho(k, \lambda, \mu)$$m\in \mathbb{Z}$$n\in \mathrm{N}$$\rho(k, \lambda, \mu):=2(m-\mu)^{2}K(k)^{2}-\lambda\gamma\Pi(\beta/\alpha-1, k)^{2}\beta/\alpha$(4.13).

Summarizing the above argument, we can assert that for each given, every nonconstant solution of, A $>0$ , and,,(3.4) with (3.5) is written as$n\in \mathrm{N}$$m\in \mathbb{Z}$$x_{1}\in \mathbb{R}$$\mu\in \mathbb{R}$$w(x+x_{1})=\sqrt{\alpha+(\beta-\alpha)\mathrm{s}\mathrm{n}^{2}(\frac{nK(k)}{\pi}x,k)}$,$b=\mathrm{s}\mathrm{g}\mathrm{n}(m-\mu)\sqrt{\frac{\lambda\alpha\beta\gamma}{2}}$,(4.14)

if the equation $\rho(k, \lambda, \mu)=0$ has a solution $(k, \lambda)\in A$ where$A:=\{(k, \lambda)\in(0,1)\mathrm{x} \mathbb{R}^{+} :\alpha(k)>0\}$.

The following proposition guarantees the unique existence of a solution
to $\rho=0$ .
PROPOSITION 4.1 Let$n\in \mathrm{N}$ancl$m\in \mathbb{Z}$.

(i) The equation (4.12) has a solution $(k, \lambda)=(k(\lambda, \mu),$$(\mu, \lambda)\in D_{m,n}^{-}\cup D_{m,n}^{+}$$\lambda)\in A$,

which are defined in (1.7) and (1.8), Moreover
each
.

if
(4.15)$k(\lambda, \mu)$

is unique for

as A $arrow 3(\mu-m)^{2}-n^{2}/2$ ,(4.16)

as $\muarrow m\pm\sqrt{\lambda}/3+n^{2}/6$ ,as $\muarrow m\pm n/2$ .

(4.17)$(\mu, \lambda)\in D_{m,n}^{-}\cup D_{m^{1},n}^{\lrcorner}\vee$(ii) Let$(\mu, \lambda)\in D_{m,n}^{-}\cup D_{m,n}^{+}$$k(\lambda, \mu)arrow 0$$k(\lambda, \mu)arrow 0$$\alpha(k(\lambda, \mu))arrow 0$. Then(4.12)

13(hi) There is no solution to (4.12) in $A$ if$(\mu, \lambda)\not\in\cup(D_{m,n}^{-}\cup D_{m,n}^{+})m\in \mathbb{Z},n\in \mathbb{N}$.

satisfy $\alpha(k)=0$ . It isis uniquely determined for each A $>n^{2}/4$ andProof of Proposition 4.1 (i). Leteasy to verify thatis written in the form

$k=k_{\alpha}(\lambda)$$k=k_{\alpha}(\lambda)\in(0,1)$

$A$

$A=$

{ , ):$(k$$\lambda$$0<k<k_{\alpha}(\lambda)$, A $>n^{2}/4$ }.

. By using the same manner as theif
We here remark that
proof of Proposition 3.1 (i) in [4], we can see$A=\emptyset$$\lambda\leq n^{2}/4$(4.19)$\rho(0, \lambda, \mu)=\frac{\pi^{2}}{6}(\frac{6(m-\mu)^{2}-n^{2}}{2}-\lambda)$an$\mathrm{n}\mathrm{d}$$\lim\rho(k, \lambda, \mu)=(4(m-\mu)^{2}-n^{2})\frac{K(k_{\alpha}(\lambda))^{2}}{2}$(4.20).

$k\uparrow k_{\alpha}(\lambda)$

Indeed, changing the variable $t=\sqrt{\nu+1}\tau/\sqrt{1-\tau^{2}}$ andwe have$\nu$$=\tilde{l/}-1$

into (3.7),(4.21)$\sqrt{\tilde{lJ}}\Pi(\tilde{\nu}-1, k)=[_{0}^{\infty}\frac{1}{1+t^{2}}\sqrt{\frac{\tilde{I/}+t^{2}}{\tilde{\nu}+(1-k^{2})t^{2}}}dt.$

Clearly the integral kernel satisfies$\underline{1}$$0$$\sqrt{\frac{\tilde{\nu}+t2}{\tilde{\nu}+(1-k\mathrm{z})\mathrm{f}2}}$

$<$
$1$

$+$

$\frac{1}{1+t2}$ $\sqrt{\frac{1}{1-k\alpha(\lambda)2}}$$\leq$

$2$

?

$t$

$(\forall k\in[0, k_{\alpha}(\lambda)], \forall\overline{\nu}\geq 0, \forall t\geq 0)$.

Thus it follows that$\sqrt{\overline{\iota/}}\Pi(\overline{\nu}-1, k)arrow\pi/2$

Since$\beta/\alphaarrow\infty$as$k\uparrow k_{\alpha}(\lambda)$,$\mathrm{w}\mathrm{e}$$(\tilde{\nu}arrow\infty)$.

obtain$\sqrt{\beta}/\alpha\Pi(\beta/\alpha-1, k)arrow\pi/2$(A$\uparrow k_{\alpha}(\lambda)$

and hence we obtain (4.20). Thus the function. Consequently
tinuous function on
is satisfied if and only if$\rho(\cdot, \cdot, \mu)$$\overline{A}\backslash \{\lambda=n^{2}/4\}$)(4.22)

is exten ded as a can, $\mu)<$$\rho(0, \lambda, \mu)\rho(k_{\alpha}(\lambda))\lambda$$0$$\frac{6(m-\mu)^{2}-n^{2}}{2}<\lambda$,$0<4(m-\mu)^{2}-n^{2}$ ,(4.23)

14
because the inequality $\lambda>n^{2}/4$ implies$\frac{6(m-\mu)^{2}-n^{2}}{2}-\lambda<\frac{3(4(m-\mu)^{2}-n^{2})}{4}$ .

that (4.12) has a solution
Therefore it follows from the continuity of$k=k(\lambda, \mu)$ if.
andsatisfies (4.15) for each
The following lemma implies that a solution $k=k(\lambda, \mu)$ to (4.12) is.
if it exists inunique for each$\rho(k, \lambda, \mu)$$(0, k_{\alpha}(\lambda))$$(\mu, \lambda)$

LEMMA 4.1 If$m\in \mathbb{Z}$$n\in \mathrm{N}$$(\mu, \lambda)$$k\in(0, k_{\alpha}(\lambda))$satisfiesthen$\rho(k, \lambda, \mu)=0_{f}$$\frac{\partial\rho}{\partial k}(k, \lambda, \mu)>0$.

The proof of Lemma 4.1 is performed literally in the same way as in [4],we omit it here (see the proof Lemma 3.5 in [4]).

$\mathrm{C}1$

Proof of Proposition 4.1 (ii). We first show (4.16). It is clear that, forandfixed which satisfies $(\mu-m)^{2}>n^{2}/4$ , the both
. Prom Lemma 4.1, it follows thatare strictly positive if$\rho(0, \lambda, \mu)$$\mu$$\rho(k_{\alpha}(\lambda), \lambda, \mu)$$\lambda\in(n^{2}/4, \lambda_{\mu,m,n})$$\rho(k, \lambda, \mu)>0$,$\forall k\in[0, k_{\alpha}(\lambda)]$,(4.24)$\forall\lambda\in(n^{2}/4, \lambda_{\mu,m,n})$and hence$\rho(k, \lambda_{\mu,m_{\}n}, \mu)=\mathrm{I}\mathrm{i}\mathrm{m}\rho(k, \lambda, \mu)\geq 0\lambda\uparrow\lambda_{\mu,m,n}$’$\forall k\in[0, k_{\alpha}(\lambda_{\mu,m,n})]$.

By using (4.19), (4.20), and Lemma 4.1 again, we can conclude that$\{\begin{array}{l}p(k,\lambda_{\mu,m,n},\mu)>0,\forall k\in(0,k_{\alpha}(\lambda_{\mu_{1}m,n})]\rho(0,\lambda_{\mu,m,n},\mu)=0\end{array}$(4.25). SinceLetbe any sequence satisfying
as ais bounded and is continuous, there exists a subsequencesuch that a limitofasexists inandEl Thus the limit
must be 0 by (4.25). This concludes
the proof of (4.16).
We next prove (4.17). That is similar to the above argument. Let$\lambda>n^{2}/4$ be fixed. If
satisfies pa $-m)^{2}>\lambda/3+n^{2}/6$$\{\lambda_{\sigma}\}$$arrow\infty$$\lambda_{\sigma}\downarrow\lambda_{\mu,m,n}$$k(\lambda, \mu)$$\{\lambda_{\sigma’}\}\subset\{\lambda_{\sigma}\}$$\rho$$k_{*}$$\sigma’arrow\infty$$k(\lambda_{\sigma}/, \mu)$$\rho(k_{*}, \lambda_{\mu,m,n}, \mu)=$$[0, k_{\alpha}(\lambda_{\mu_{r}m,n})]$$k_{*}$

$\mu$

$($

$\rho(k, \lambda, \mu)>0$,

VA$\in[0, k_{\alpha}(\lambda)]$.

15
Because the both
is applied. Thus$\rho(0, \lambda, \mu)$and$\rho(k_{\alpha}(\lambda), \lambda, \mu)$$p(k, \lambda, m\pm\sqrt{\lambda}/3+n^{2}/6)\geq 0$

are strictly positive and Lemma 4.1
VA,$\in[0, k_{\alpha}(\lambda)]$.

Combining (4.19), (4.20), Lemma 4.1, and the above inequality, we obtain(4.17) by a similar argument to the proof of (4.16).
Now we observe (4.18), that is, we will prove(4.26)$\lim_{\muarrow m\pm n/2}k(\lambda, \mu)=k_{\alpha}(\lambda)$for. Let A $>n^{2}/4$ be fixed. The bothare strictly negative if satisfies$(\mu, \lambda)\in D_{m,n}^{+}\cup D_{m,n}^{-}$$\rho(k_{\alpha}(\lambda), \lambda, \mu)$$\rho(0, \lambda, \mu)$

and

$\mu$

$m-n/2<\mu<m+n/2$ .
Lemm a 4.1 implies$\rho(k, \lambda, \mu)<0$,$\forall k\in[0, k_{\alpha}(\lambda)]$$\forall\mu\in(m-n/2, m+n/2)$ .

,Thus$\rho(k, \lambda, m\pm n/2)\leq 0$,$\forall k\in[0, k_{\alpha}(\lambda)]$.

(4.27)(4.28)

Combining (4.19), (4.20), (4.28), and Lemma 4.1, we obtain$\{\begin{array}{l}\rho(k,\lambda_{7}m\pm n/2)<0,\forall k\in[0,k_{\alpha}(\lambda))p(k_{\alpha}(\lambda),\lambda,m\pm n/2)=0\end{array}$

Therefore it follows thatargument to the proof of (4.16).

$k(\lambda, \mu)arrow k_{\alpha}(\lambda)$

as$\muarrow m\pm \mathrm{n}/2$(4.29)

from a similar$\square$be fixed. As. Thus it suffices toifmentioned in the proof of (i),: A $>n^{2}/4$ }
if
{
for
prove
. Since (4.24) and (4.25), it is clear that $\rho(k, \lambda, \mu)>0$ for. On the other hand, it follows fromif
if $m-n/2\leq\mu\leq\square$(4.27) and (4.29) that $p(k, \lambda, \mu)<0$ for$m+n/2$ and $\lambda>n^{2}/4$ . Therefore (iii) was proved.

Proof of Proposition 4.1 (iii).

Let$m\in \mathbb{Z}$$A=\emptyset$$\rho(k, \lambda, \mu)\neq 0$$\forall k\in(0, k_{\alpha}(\lambda))$and$n\in \mathrm{N}$$\lambda\leq n^{2}/4$$(\mu, \lambda)\in$$(\mu,$$\lambda)$$\backslash$$(D_{m,n}^{+}\cup D_{m,n}^{-})$$\forall k\in(0, k_{\alpha}(\lambda))$$n^{2}/4<\lambda\leq\lambda_{\mu\}m,n}$$\forall k\in(0, k_{\alpha}(\lambda))$Proof of Theorem 3.1. As mentioned above we proved that the noncon-

stant amplitude solution is written in the form (4.14) and Proposition 4.1
directly implies the existence condition and the nonexistence condition of

1$\mathrm{G}$the nonconstant amplitude solutions which are stated in Theorem 3.1. Substituting (3.8), (3.9), and $x_{1}=0$ into (4.14), we obtain (3.6). Now we verify(3.11), (3.12), and (3.13). In the rest of the proof, $w=w(x)$ denotes (4.14)
with $x_{1}=0$ for simplicity.
We first prove (3.11). It follows from (4.16) that if. Thus, thenand$(\mu, \lambda)\in D_{m,n}^{-}\cup D_{m,n}^{+}$$k(\lambda, \mu)arrow 0$$\lambdaarrow\lambda_{\mu,m,n}$as
as$cearrow 2/3-n^{2}/6\lambda_{\mu,m,n}=1-(\mu-m)^{2}/\lambda_{\mu,m,n}$$\beta-\alphaarrow 0$

For each$\ell\in \mathbb{Z}$$\lambdaarrow\lambda_{\mu,m,n}$$\lambdaarrow\lambda_{\mu,m,n}$,(4.30).

(4.31), if $x\in[2\pi\ell, 2\pi(\ell+1)]$ then$\theta(x)-(m-\mu)x=(m-\mu)\{2\pi]_{0}^{x}\frac{1}{w(y)^{2}}dy/\int_{0}^{2\pi}\frac{1}{w(y)^{2}}dy-x\}$$=(m- \mu)\{2\pi]_{0}^{x-2\pi l}\frac{1}{w(y)^{2}}dy/\int_{0}^{2\pi}\frac{1}{w(y)^{2}}dy+2\pi\ell$

Since$-x\}$ .

,$\sqrt{\alpha(k(\lambda,\mu))}\leq w(x)\leq\sqrt{\beta(k(\lambda,\mu))}$a simple calculation implies$\frac{\alpha-\beta}{\beta}(x-2\pi\ell)\leq 2\pi\int_{0}^{x-2\pi\ell}\frac{1}{w(y)^{2}}dy/I_{0}^{2\pi}\frac{1}{w(y)^{2}}dy+2\pi\ell-x\leq\frac{\beta-\alpha}{\alpha}(x-2\pi\ell)$and hence we obtain$\sup$$| \theta(x)-(m-\mu)x|\leq\frac{2\pi|m-\mu|(\beta-\alpha)}{\alpha}$,$\forall\ell\in \mathbb{Z}$.

$x\in[2\pi\ell,2\pi(l+1)]$

It is clear that$|u_{\lambda,\mu,m,n}^{\mathrm{o}}(x)-u_{\lambda,\mu,m}^{\mathrm{c}}(x)|=|w(x)\exp(\mathrm{i}(\theta(x)-(m-\mu)x))-\sqrt{1-(m-\mu)^{2}/\lambda}|$$\leq|w(x)||\exp(\mathrm{i}(\theta(x)-(m-\mu)x))-1|$$+|w(x)-\sqrt{1-(m-\mu)^{2}/\lambda}|$and$\sup_{x\in \mathbb{R}}|w(x)-\sqrt{1-(m-\mu)^{2}/\lambda}|arrow 0$

as$\lambdaarrow\lambda_{\mu,m,n}$17
for (4.30) and (4.31). It is also clear that$|\exp(\mathrm{i}(\theta(x)-(m-\mu)x))-1|^{2}=\{\cos(\theta(x)-(m-\mu)x)-1\}^{2}$$+\sin^{2}(\theta(x)-(m-\mu)x)$$\leq 4\sin^{4}(\pi|m-\mu|(\beta-\alpha)/\alpha)$$+\sin^{2}(2\pi|m-\mu|(\beta-\alpha)/\alpha)$

and henceas$\sup_{x\in \mathbb{R}}|\exp(\theta(x)-(m-\mu)x)-1|arrow 0$for (4.31). Therefore (3.11) follows.
Similarly, (3.12) follows from (4.17).
Next we prove (3.13). Let
limit of ut $=w(x)$ as $\muarrow m\pm n/2$ . Since
as $\muarrow m\pm n/2$ ,$(\mu, \lambda)\in D_{m,n}^{-}\cup D_{m,n}^{+}$$\lambdaarrow\lambda_{\mu,m,n}$. First, we consider theand (4.18) implies$\alpha(k_{\alpha}(\lambda))=0$$k(\lambda, \mu)arrow k_{\alpha}(\lambda)$$\beta-\mathrm{o}\mathrm{r}$$arrow 2k_{\alpha}(\lambda)^{2}/(k_{\alpha}(\lambda)^{2}+1)$$(\muarrow m\pm n/2)$

and hence$warrow k_{\alpha}(\lambda)\sqrt{2/(k_{\alpha}(\lambda)^{2}+1)}|\mathrm{s}\mathrm{n}$ $(nK(k_{\alpha}(\lambda))x/\pi, k_{\alpha}(\lambda))|$uniformly for(2.2), Thus$x$$u_{\lambda,n}^{\mathrm{s}}$as $\muarrow m\pm n/2$ . On the other hand,$k=k_{\alpha}(\lambda)$(4.32)

satisfies

is written in the form

$u_{\lambda,n}^{\mathrm{s}}(x)=$Therefore we obtain the following: For any
such that$\epsilon>0$, there exists$\sup_{x\in \mathbb{R}}|w(x)-|u_{\lambda,n}^{\mathrm{s}}(x)||\leq\in$$\delta_{0}=\delta_{0}(\epsilon)>0$(4.33)

for $\forall\mu\in(m-n/2-\delta_{0}, m-n/2)\cup(m+n/2, m+\mathrm{n}/2\cdot\delta_{0})$ . We here remark
can be defined independently of , however it is not a solution
thatis zero at $x=2\ell\pi/n(\ell\in \mathbb{Z}))$ we can verify. Sinceto (1.2) if
such that
and
that, for any $>0$ , there exist$\mu$$u_{\lambda,n}^{\mathrm{s}}(x)$$2\mu\not\in \mathbb{Z}$$\epsilon$$u_{\lambda,n}^{\mathrm{s}}(x)$$d_{1}=d_{1}(\epsilon)$$\delta_{1}=\delta_{1}(\epsilon)$$\max_{\mathbb{Z}|}\sup_{x-2\ell\pi/n|\leq d_{1}}|w(x)|\ell\in\leq\in$,$\max_{\mathbb{Z}|}\sup_{x-2l\pi/n|\leq d_{1}}|u_{\lambda,n}^{\mathrm{s}}(x)|\ell\in\leq\epsilon$(4.34)(4.35)

18
for &pa $\in(m-n/2-\delta_{1}, m-n/2)\cup(m+n/2, m+n/2+\delta_{1})$ .
Next we deal with 0. Combining (3.8), (3.10), and (4.11), we have$]_{0}^{2\pi} \frac{1}{w(x)^{2}}dx=\frac{2\pi\Pi(\beta/\alpha-1,k)}{\alpha K(k)}$We also obtain that for each$\ell\in \mathbb{Z}$.

if $x\in(2\pi\ell/n, 2\pi(\ell+1)/n)$ then$\int_{0}^{x}\frac{\mathrm{I}}{w(s)^{2}}ds=\frac{(2\ell+1)\pi\Pi(\beta/\alpha-1,k)}{n\alpha K(k)}+\int_{(2\ell+1)\pi/n}^{x}\frac{1}{w(s)^{2}}ds$,

else if $x=2\pi\ell/n$ then$]_{0}^{x} \frac{1}{w(s)^{2}}ds=\frac{2\ell\pi\Pi(\beta/\alpha-1,k)}{n\alpha K(k)}$

.

Thus

if $x\in(2\pi P/n, 2\pi(P+1)/n)$ ,$\theta(x)=\{\begin{array}{l}\frac{(m-\mu)(2l+\mathrm{l})\pi}{n}+\frac{(m-\mu)\alpha K(k)}{\square (\beta/\alpha-\mathrm{l},k)}]_{(2\ell+1)\pi/n}^{x}\frac{1}{w(s)^{2}}ds\frac{(m-\mu)\ell\pi}{n}\mathrm{i}\mathrm{f}x=2\pi\ell/n,(\ell\in \mathbb{Z})\end{array}$$(l\in \mathbb{Z})$,(4.36)The limit of 0 as $\muarrow m\pm n/2$ is characterized as follows. Since (4.22), it
holds that$\alpha/\Pi(\beta/\alpha-1, k)arrow 0$(4.37).
$(k\uparrow k_{\alpha}(\lambda))$Let $d_{2}>0$ . It is also clear that$|]_{(2p+1)\pi/n}^{x} \frac{1}{w(s)^{2}}ds$$\leq\frac{\pi}{n}\frac{1}{w(d_{2})^{2}}$,Vr $\in[2\pi\ell/n+d_{2},2\pi(\ell+1)/n-d_{2}]$ .

Thus it follows from (4.32), (4.36), (4.37), and the above inequality that forsuch that
any $>0$ there exists$\delta_{2}=\delta_{2}(\epsilon, d_{2})>0$$\epsilon$$x \in(2\pi l/n+d_{2},2\pi(l+1)/n-d_{2})\sup|\theta(x)-(2\ell+1)\pi/2|\leq\epsilon$$\forall\mu\in(m-n/2-\delta_{2}, m-n/2)$$\sup$,(4.38).

(4.36)$|\theta(x)+(2\ell+1)\pi/2|\leq\epsilon$$x\in(2\pi\ell/n+d_{2_{\gamma}}2\pi(\ell+1)/n-d_{2})$$\forall\mu\in(m+n/2, m+n/2+\delta_{2})$Ia
Now we estimate$|u_{\lambda,\mu,m,n}^{\mathrm{o}}(x)\pm \mathrm{i}u_{\lambda,n}^{\mathrm{s}}(x)|^{2}$$=w(x)^{2}\cos(\theta(x))^{2}+\{w(x)\sin(\theta(x))\pm u_{\lambda,n}^{\mathrm{s}}(x)\}^{2}$For$\epsilon$$(4,40)$$>0$ , put$\delta=\delta(\epsilon):=\min\{\delta_{0}(\epsilon), \delta_{1}(\epsilon), \delta_{2}(\epsilon, d_{1}(\epsilon))\}$.

(4.41)

Then it is clear that, if $\mu\in(m-n/2-\delta, m-n/2)\cup(m+n/2, m+n/2+\delta)$ ,$\sup_{x\in \mathbb{R}}w(x)^{2}\cos^{2}(\theta(x))\leq\max\sup w(x)^{2}\cos^{2}(\theta(x))l\in \mathbb{Z}|x-2l\pi/n|\leq d_{1}$$+ \max_{\mathbb{Z}x\in(2\pi l/}\sup_{n+d_{1},2\pi(\ell+1)/n-d_{1})}w(x)^{2}\cos^{2}(\theta(x))\ell\in$$\leq$.

$\epsilon^{2}+\max\beta\cos^{2}((2\ell+1)\pi/2+\epsilon)l\in \mathbb{Z}$$\underline{<}\epsilon^{2}+2\sin^{2}(\epsilon)$(4.42).

Next we estimate the second term of the right hand side of (4.40). Since(4.34) and (4.35), we have the following estimate in neighborhoods of zero:
points of$u_{\lambda,n}^{\mathrm{s}}$$\max\sup\{w(x)\sin(\theta(x))\pm u_{\lambda,n}^{\mathrm{s}}(x)\}^{2}\leq 4\epsilon^{2}l\in \mathbb{Z}|x-2\ell\pi/n|\leq d_{1}$

if $\mu\in(m-n/2-\delta, m-n/2)\cup(m+n/2, m+n/2+\delta)$ . In the complement, the second term of the rightto the neighborhoods of the zero points ofhand side of $(4,40)$ is estimated as follows: Let $\mu\in(m-n/2-\delta, m-n/2)$
and $x\in(2\pi\ell/n+d_{1},2\pi(\ell+1)/n-d_{1})$ . Then$u_{\lambda,n}^{\mathrm{s}}$$|w(x)\sin(\theta(x))-u_{\lambda,n}^{\mathrm{s}}(x)|$$\leq|w(x)-|u_{\lambda,n}^{\mathrm{s}}(x)||\sin(\theta(x))|+|u_{\lambda_{\}n}^{\mathrm{s}}(x)||\sin(\theta(x))-(-1)^{l}|$$-(-1)^{\ell}|$$\leq\in$$\leq\in$Since$\theta(x)$$+$has an estimate (4.38) and $\sin(\theta(x))-(-1)^{\ell}$ is estimated as$|\sin(\theta(x))-(-1)^{f}|=|(-1)^{l}\{\cos(\theta(x)-(2\ell+1)\pi/2)-1\}|$$=2\sin^{2}((\theta(x)-(2\ell+1)\pi/2)/2)$$\leq 2\sin^{2}(\epsilon/2)$,

20
we obtain$\sup$$|w(x)\sin(\theta(x))-u_{\lambda,n}^{\mathrm{s}}(x)|\leq\in+2\sqrt{2}\sin^{2}(\in/2)$$x\in(2\pi l/n+d_{1},2\pi(\ell+1)/n-d_{1})$.

(4.43)

Similarly, if $\mu\in(m+n/2, m+n/2+\delta)$ , it holds that$\sup$$|w(x)\sin(\theta(x))+u_{\lambda,n}^{\mathrm{s}}(x)|\leq\in$$+2\sqrt{2}\sin^{2}(\epsilon/2)$.

$x\in(2\pi l/n+d_{1},2\pi(l+1)/n-d_{1})$(4.44)
Combining (4.42) and (4.43), we obtain that there exists $C>0$ such thatfor any$\epsilon>0$,$\sup_{x\in \mathbb{R}}|u_{\lambda,\mu,m,n}^{\mathrm{o}}(x)-\mathrm{i}u_{\lambda,n}^{\mathrm{s}}(x)|\leq C\epsilon$$\forall\mu\in(m-n/2-\delta, m-n/2)$

On the other hand, (4.42) and (4.44) imply that, for any,$\sup_{x\in \mathbb{R}}|u_{\lambda,\mu,m,n}^{\mathrm{o}}(x)+\mathrm{i}u_{\lambda,n}^{\mathrm{s}}(x)|\leq C\epsilon$$\epsilon>0$,$\forall\mu\in(m+n/2, m+n/2+\delta)$

Therefore it completes the proof of (3.13).

.

.
$\square$Acknowledgments
The authors would like to express their thanks to Professor Toshiyuki Ogawa
for letting them know the paper by [10]. The author was supported in partby the Grant-in-Aid for JSPS Fellows, No. 9335.

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