94

数理解析研究所講究録
第2080巻 2018年 94-105

Eigenvalue problems associated with 1‐dimensional
scalar field equations
Tohru Wakasa

Kyushu Institute of Technology

1Introduction

Let us consider the boundary value problem\left{\begin{ar y}{l $\varepsilon$^{2}u_{x}+f(u)=0,\mathrm{i}\mathrm{n}(0,1)\ u_{x}(0)=u_{x}(1)=0,\ u>0,\mathrm{i}\mathrm{n}(0,1). \end{ar y}\right.

(NP)

Here $\varepsilon$ is a positive parameter andf(u)=-u+u^{p}with p> 1 . The problem (NP) is called the scalar field equation, which is motivated by
the parabolic PDEs modelling pattern formation and chemotaxis, and also by PDEs of

Klein‐Gordon/nonlinear Schrödinger type ([2] and [4]).
In the 1‐D problem (NP) a classical shooting argument solves the global bifurcation
structure of all solutions ([5]). In particular, one can observe the families of periodic
solutions: for any n , there exist $\varepsilon$_{n} > 0 such that there exist 1/n‐periodic solutions
u_{n, $\varepsilon$}^{\pm}(x) for $\varepsilon$\in(0, $\varepsilon$_{n}) ;u_{n, $\varepsilon$}^{+}(0)=\displaystyle \max_{x\in[0,1]}u_{n, $\varepsilon$}^{+}(x) , u_{n, $\varepsilon$}^{-}(0)=\min_{x\in[0,1]}u_{n, $\varepsilon$}^{-}(x) .

(1.1)

In view of pattern formation, we say u_{n, $\varepsilon$}^{+} is interior spike solution, and u_{\overline{n},$\varepsilon$} is boundary

spike solution.
We fix n\in \mathrm{N} arbitrarily and assume $\varepsilon$\in (0, $\varepsilon$_{n}) . Consider the linearized eigenvalue
problems associated with u_{n, $\varepsilon$}^{+} and u_{\overline{n}, $\varepsilon$}\left\{ begin{ar y}{l $\varepsilon$^{2}$\varphi$_{x }(x)+f_{u}( _{n,$\varepsilon$}^{\pm}(x) $\varphi$(x)+$\lambda\varphi$(x)=0&\mathrm{i}\mathrm{n}(0,1),\ $\varphi$_{x}(0)=$\varphi$_{x}(1)=0.& \end{ar y}\right.

(\mathrm{L}\mathrm{P}_{\pm})

For j\in \mathrm{N}\cup\{0\} we denote by $\lambda$_{j}^{\pm}=$\lambda$_{j, $\varepsilon$}^{\pm} and $\varphi$_{j}^{\pm}(x)=$\varphi$_{j, $\varepsilon$}^{\pm}(x) , the (j+1) ‐th eigenvalue
and the corresponding eigenfunction, respectively. By the Sturm‐Liouville theory showsthat$\lambda$_{0, $\varepsilon$}^{+}<$\lambda$_{1, $\varepsilon$}^{+}<\cdots<$\lambda$_{j, $\varepsilon$}^{+}<\cdots<+\infty,$\lambda$_{0, $\varepsilon$}^{-}<$\lambda$_{1, $\varepsilon$}^{-}<\cdots<$\lambda$_{j, $\varepsilon$}^{-}<\cdots<+\infty,
and the both $\varphi$_{j,$\varepsilon$}^{\pm} have exactly j zeros in (0,1) .

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We are interested in the $\epsilon$‐dependence of the both eigenpairs $\lambda$_{j,$\epsilon$}^{\pm} and $\varphi$_{j,$\varepsilon$}^{\pm} for (\mathrm{L}\mathrm{P}_{\pm}) ,and the qualitative difference between \{$\varphi$_{j, $\varepsilon$}^{+}\} and \{$\varphi$_{j, $\varepsilon$}^{-}\} . In the above problems we focus

on the case p=3 :f(u)=-u+u^{3}.

In this case we can give expressions of u_{n, $\varepsilon$}^{\pm} in terms of the Jacobi elliptic function. We notethat the similar expressions could be observed for the case p=2 . Moreover, the method ofrepresentation equation, which is proposed by the author and S. Yostsutani [7], [10] and[12] for analyzing the linearized eigenvalue problems to (NP) with bistable nonlinearities,
can be applied to show the precise information on eigenvalues and eigenfunctions of (\mathrm{L}\mathrm{P}_{\pm}) :\bulletrepresentation formulas on eigenfunctions,\bulletcharacterization of eigenvalues,\bulletasymptotic formulas of eigenpairs as $\varepsilon$\rightarrow 0.

More precisely, we obtain the three special eigenfunctions for both (\mathrm{L}\mathrm{P}_{+}) and (LP‐)

when p= 3 : $\varphi$_{0, $\varepsilon$}^{\pm}, $\varphi$_{0,$\varepsilon$}^{\pm} and $\varphi$_{2n, $\varepsilon$}^{\pm} . They are given explicitly in the algebraic expressionof elliptic function, and also possess fine symmetric and periodic properties. Moreover,

the other eigenfunctions can be given by the Liouvillian form ([3]). The corresponding
eigenvalues $\lambda$_{j,$\varepsilon$}^{\pm} of (\mathrm{L}\mathrm{P}_{\pm}) is determined by the characteristic equation,\displaystyle \mathcal{A}(k, $\mu$)=\frac{j $\pi$}{2n},where A is the characteristic function, which can be expressed by the complete elliptic

integral (of the third kind).
The asymptotic formulas of every eigenvalue are obtained by asymptotic formulas on
elliptic integrals, and the asymptotic profiles of every eigenfunction are obtained by the

Floquet / Bloch type argument with the asymptotic formulas of eigenvalues ([11] and [13]).
It should be noted that in the both problem (\mathrm{L}\mathrm{P}_{\pm}) for f(u)=-u+u^{3} the three special

eigenpairs play an important role for the limit classification; every eigenpair is classified
into the three classes, and roughly speaking, each class is provided by the corresponding
special eigenpair. The limit classification above is described by the asymptotic formulasof eigenvalues and the asymptotic profiles of eigenfunctions when $\epsilon$ is sufficiently small.
We will see that the limit classification on (\mathrm{L}\mathrm{P}_{+}) is essentially the same type as the limit

classifications for the cases of nonlinear bistable f ’s ([11] and [13]), which justify and gen‐eralize a conjecture by E. Yanagida on the asymptotic profiles of the first n‐eigenfunctions.

The limit classification for (LP‐) will show us that Yanagida’s conjecture also hold forthe first n‐eigenfunctions. However, it appears a different asymptotic characterizaton on
the eigenfunctions from the case (\mathrm{L}\mathrm{P}_{+}) . which is due to the difference of the special

eigenvalues between in (\mathrm{L}\mathrm{P}_{+}) and (LP‐). See Figs. 1 and 2.

In this article we obtain the representation formulas of eigenfunctions and derive thecharacteristic functions for (\mathrm{L}\mathrm{P}_{\pm}) . Also, we discuss the asymtotic results on the eigen‐values and eigenfunctions. The organization of the article is as follows. In Section 2 we
prepare the elliptic integrals and functions. In Section 3 we give the main results. In
Section 4 we introduce some key lemmas in the analysis and give a sketch of proofs.

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Figure 1: Profiles of $\varphi$_{j,$\varepsilon$}^{+} (j=0, \ldots, 8) for f(u)=-u+u^{3} with n=3.

Figure 2: Profiles of $\varphi$_{j,$\varepsilon$}^{-} (j=0,\ldots,97

2Preliminaries

We start with giving standard notations on the elliptic integrals and elliptic functions.

See for instance, the Handbook by Byrd and Fi iedman[1]. Let k \in (0,1) and let $\nu$ \in\mathrm{C}\backslash (-\infty, -1] . We denote by K(k) , E(k) and $\Pi$(\mathrm{v}, k) , the complete elliptic integrals of
the first, the second and the third kind, respectively. The Jacobi elliptic function \mathrm{s}\mathrm{n}(x, k)
is a periodic analytic function with the period 4K(k) as the function for the real domain,and is defined locally by

x=\displaystyle \int_{0}^{\mathrm{s}\mathrm{n}(x,k)}\frac{1}{\sqrt{(1-s^{2})(1-k^{2}s^{2}) }ds

for x \in [0, K(k)] . In a standard manner the elliptic functions cn (x, k) and dn (x, k) is
defined by\mathrm{s}\mathrm{n}(x, k)^{2}+\mathrm{c}\mathrm{n}(x, k)^{2}=1, \mathrm{s}\mathrm{n}(x, k)^{2}+k^{2}\mathrm{d}\mathrm{n}(x, k)^{2}=1

for x\in \mathrm{R} and k\in(0,1) .

Here we introduce the modified form of $\Pi$ (see [8] and [9])\displaystyle \mathcal{M}( $\nu$, k):= \sqrt{\frac{(1+ $\nu$)(k^{2}+ $\nu$)}{ $\nu$} \int_{0}^{1}\frac{1}{(1+ $\nu$ s^{2})\sqrt{(1-s^{2})(1-k^{2}s^{2}) }ds(= \sqrt{\frac{(1+ $\nu$)(k^{2}+ $\nu$)}{ $\nu$} $\Pi$( $\nu$, k)for ( $\nu$, k)\in \mathcal{D} , where\displaystyle \sqrt{z}:=\exp(\frac{1}{2}{\rm Log} z) , z\in \mathrm{C}\backslash (-\infty, 0],\mathcal{D}:=\{( $\nu$, k)\in \mathrm{C}\times \mathrm{R}| $\nu$\not\in(-\infty, -1]\cup[-k^{2},0Figure 3: A Graph of \mathcal{M}( $\nu$, k) for real \mathrm{v}(k=1/\sqrt{2}) .

(2.1)

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3

Main results-u+u^{3} . A standard shooting
Let us show the expressions of u_{n, $\varepsilon$}^{\pm} of (NP) for f(u)argument can be applied to (NP) (see Section 4 of [12]), and in a situation that $\epsilon$ is smallenough, one can obtain the monotone increasing and decreasing solutions of (NP), which
are symmetric with the line x=1/2 . In addition, one can describe the global bifurcation
structue of (NP) by using the two monotone solutions. In particular, we obtain the
1/n‐periodic solutions u_{n, $\varepsilon$}^{\pm} for each n\in \mathrm{N}.
=

Proposition 3.1. Fư n\in \mathrm{N} arbitrality and assume $\varepsilon$\in (0, \sqrt{2}/(n $\pi$)) . Let k_{ $\varepsilon$}=k_{n, $\varepsilon$} \in(0,1) be a solution of\displaystyle \sqrt{2-k^{2} K(k)=\frac{1}{n $\varepsilon$}(3.1)(note that it is uniquely determined). Then,u_{n,$\varepsilon$}^{+}(x)=\sqrt{\frac{2}{2-k_{$\varepsilon$}^{2} \mathrm{d}\mathrm{n}(K k_{$\varepsilon$})(1+2nx),k_{$\varepsilon$}),and

u_{n,$\varepsilon$}^{-}(x)=\sqrt{\frac{2} -k_{$\varepsilon$}^{2} \mathrm{d}\mathrm{n}(2nK(k_{$\varepsilon$})x,k_{$\varepsilon$}).

is the 1/n ‐periodic solutions of (NP) satisfying (1.1).

For a convenience, we introduce a notation of the two modified sn‐functions\mathrm{S}\mathrm{N}^{+}(x;k)=\mathrm{s}\mathrm{n}(K(k)(1+2nx), k) , \mathrm{S}\mathrm{N}^{-}(x;k)=\mathrm{s}\mathrm{n}(2nK(k)x, k).

Both functions have the same period 1/n for any k\in(0,1) . The functions \mathrm{C}\mathrm{N}^{\pm} and \mathrm{D}\mathrm{N}^{\pm}
are defined in a similar manner;

u_{n,$\varepsilon$}^{\pm}(x)=\sqrt{\frac{2} -k_{$\varepsilon$}^{2} \mathrm{D}\mathrm{N}^{\pm}(x;k_{$\varepsilon$}).

The representation formulas of eigenvalues and eigenfunctions are given by the follow‐
ing two theorems.

Theorem 1. The Linearized problem (\mathrm{L}\mathrm{P}_{+}) has the following pairs of eigenvalues andeigenfunctions:(i)$\lambda$_{0,$\varepsilon$}^{+}=-1\displayst le\frac{2\sqrt{1-k_{$\epsilon$}^{2}+k_{$\varepsilon$}^{4} {2-k_{$\varepsilon$}^{2},$\varphi$_{0, $\varepsilon$}^{+}(x)=1-(1+k_{ $\varepsilon$}^{2}-\sqrt{1-k_{ $\varepsilon$}^{2}+k_{ $\varepsilon$}^{4} )\mathrm{S}\mathrm{N}^{+}(x;k_{ $\varepsilon$})^{2},(ii)$\lambda$_{n,$\varepsilon$}^{+}=-\displaystyle\frac{3(1-k_{$\varepsilon$}^{2}){2-k_{$\varepsilon$}^{2}f$\varphi$_{n,$\varepsilon$}^{+}(x)=\mathrm{S}\mathrm{N}^{+}(x,k_{$\varepsilon$})\mathrm{D}\mathrm{N}^{+}(x,k_{$\varepsilon$}) ,

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(iii)

$\lambda$_{2n,$\varepsilon$}^{+}=-1+\displayst le\frac{2\sqrt{1-k_{$\varepsilon$}^{2}+k_{$\epsilon$:}^{4} {2-k_{$\varepsilon$}^{2},$\varphi$_{2n, $\varepsilon$}^{+}(x)=-1+(1+k_{ $\varepsilon$}^{2}+\sqrt{1-k_{ $\varepsilon$}^{2}+k_{ $\varepsilon$}^{4} )\mathrm{S}\mathrm{N}^{+}(x;k_{ $\varepsilon$})^{2},
where k_{$\varepsilon$} is the unique solution of (3.1).

Theorem 2. The Lineanized problem (LP‐) has the following pairs of eigenvalues andeigenfunctions:(i)$\lambda$_{0,$\varepsilon$}^{-}=1-\displayst le\frac{2\sqrt{1-k_{$\varepsilon$}^{2}+k_{$\varepsilon$}^{4} {2-k_{$\varepsilon$}^{2},$\varphi$_{0, $\varepsilon$}^{-}(x)=1-(1+k_{ $\varepsilon$}^{2}-\sqrt{1-k_{ $\varepsilon$}^{2}+k_{ $\varepsilon$}^{4} )\mathrm{S}\mathrm{N}^{-}(x;k_{ $\varepsilon$})^{2},(ii)$\lambda$_{n, $\varepsilon$2}-k^_{{-}$=\-v\aurenpsdileornl$in}^e{2{^3{f}} $\varphi$_{n, $\varepsilon$}^{-}(x)=\mathrm{C}\mathrm{N}(x, k_{ $\varepsilon$})\mathrm{D}\mathrm{N}(x, k_{ $\varepsilon$})(iii)$\lambda$_{2n,$\varepsilon$}^{-}=1+\displayst le\frac{2\sqrt{1-k_{$\varepsilon$}^{2}+k_{$\varepsilon$}^{4} {2-k_{$\varepsilon$}^{2},,$\varphi$_{2n, $\varepsilon$}^{-}(x)=-1+(1+k_{ $\varepsilon$}^{2}+\sqrt{1-k_{ $\varepsilon$}^{2}+k_{ $\varepsilon$}^{4} )\mathrm{S}\mathrm{N}^{-}(x;k_{ $\varepsilon$})^{2},
where k_{$\varepsilon$} is the unique solution of (3.1).
Here we introduce the following notations\left{bginary}l$\Lmbda_{0}(k):=-1\frc2sqt{k^}+42-{,\$Lambd_{1}^-(k):=\frac3{2^},\$Lambd_{1}^+(k):=-\frac31^{2})-k,\$Lambd_{2}(k):=-1+\fracsqt{-k^2}+4 {.\endary}ight.

(3.2)

Note that $\Lambda$_{0}(k)<$\Lambda$_{1}^{-}(k)<$\Lambda$_{1}^{+}(k)<0<$\Lambda$_{2}(k) for k\in(0,1) . In addition, the functions$\Lambda$_{0}, $\Lambda$_{1}^{-} and the functions $\Lambda$_{1}^{+}, $\Lambda$_{2} are monotone decreasing and increasing in k\in (0,1) ,0 and $\Lambda$_{0}(1)-3,respectively. Also, $\Lambda$_{1}(0)-2, $\Lambda$^{\pm}(0)-3/2, $\Lambda$_{2}(0)$\Lambda$_{1}^{-}(1)
=

=

=

=

=

$\Lambda$_{1}^{+}(1)=0, $\Lambda$_{2}(1)=1 . We see from Theorems 1 and 2 that$\lambda$_{0, $\varepsilon$}^{\pm}=$\Lambda$_{0}(k_{ $\varepsilon$}) , $\lambda$_{2n, $\varepsilon$}^{\pm}=$\Lambda$_{2}(k_{ $\varepsilon$})$\lambda$_{n, $\varepsilon$}^{-}=$\Lambda$_{1}^{-}(k_{ $\varepsilon$}) , $\lambda$_{n}^{+},\ovalbox{\t \smal REJECT}=$\Lambda$_{1}^{+}(k_{ $\varepsilon$}),,and in particular,$\lambda$_{0, $\varepsilon$}^{-}=$\lambda$_{0, $\varepsilon$}^{+}, $\lambda$_{+, $\varepsilon$}^{-}<$\lambda$_{n,e}^{+}, $\lambda$_{2n, $\varepsilon$}^{-}=$\lambda$_{2n, $\varepsilon$}^{+}.
For j\neq 0, n, 2n , let us show the representation formula of $\varphi$_{j,$\varepsilon$}^{\pm} . Set$\Sigma$_{1}^{$\Sigma$_{0} \mathrm{f}(k, $\mu$)(k, $\mu$)|k \in\in(0,1(0,1 $\mu$/(2 $\mu$/(2- k^{2})k^{2})\in\in ($\Lambda$_{1}^{+}(k),0)\}($\Lambda$_{0}(k),$\Lambda$_{1}^{-}(k) \} ,$\Sigma$_{2} :=\{(k, $\mu$)|k\in(0,1), $\mu$/(2-k^{2})\in($\Lambda$_{2}(k), +\infty)\}.

(3.3)

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and set$\Sigma$:=$\Sigma$_{0}\cup$\Sigma$_{1}\cup$\Sigma$_{2}.

For (k, $\mu$)\in $\Sigma$ , the characteristic function is defined by\mathcal{A}( $\mu$, k) :=|\mathcal{M}(\mathrm{v}_{-}(k, $\mu$), k)-\mathcal{M}($\nu$_{+}(k, $\mu$), k(3.4)

where \mathcal{M} is defined by (2.1),$\nu$_{\pm}(k, $\mu$):=\displaystyle \frac{3k^{2} {2}\cdot\frac{ $\mu$-3k^{2}\pm\sqrt{-3$\mu$^{2}+6(k^{2}-2) $\mu$+9k^{4} { $\mu$( $\mu$+3-3k^{2})

.

(3.5)

Theorem 3. Suppose j \neq 0, n, 2n . Let k_{$\varepsilon$} be the solution of (3.1) and let $\mu$_{j}(k) be theunique solution of\displaystyle \mathcal{A}(k, $\mu$)=\frac{j $\pi$}{n} ,(3.6)

where \mathcal{A} is given by (3.4). Then,$\lambda$_{j,$\varepsilon$}^{\pm}=\displaystle\frac{$\mu$_{j}(k_{e}) 2-k_{$\varepsilon$}^{2}.
Moreover,$\varphi$_{j, \epsilon$j}^{\pm}(x)=\displaystle\sqrt{|h(u_{n,$\varepsilon$}^{\pm}(x),$\lambda$_{j, \varepsilon$}^{\pm}(k_{$\varepsilon$});k_{$\varepsilon$}\cos(\frac{1}$\varepsilon$}\int_{0}^x\frac{\sqrt{$\rho$(\lambda$_{j \epsilon$j}^{\pm},k_{$\varepsilon$}){|h(u_{n,$\varepsilon$}^{\pm}($\xi),$\lambda$_{j, \varepsilon$}^{\pm}(k_{$\varepsilon$});k_{$\varepsilon$})|d$\xi)

,

where

and

h(u, $\lambda$, k) :=-[\displaystyle \frac{k^{2} {(1+k^{2})^{2} -\frac{1}{2}u^{2}+\frac{1}{4}u^{4}] +\frac{ $\lambda$}{6}(u^{2}-2)-\frac{$\lambda$^{2} {9}(3.7)$\rho$( $\lambda$, k)=\displaystyle \frac{1}{81}( $\lambda-\Lambda$_{0}(k) ( $\lambda-\Lambda$_{1}^{-}(k) ( $\lambda-\Lambda$_{1}^{+}(k) $\lambda$( $\lambda-\Lambda$_{2}(k) .

(3.8)

Combining Theorems 1‐3 we have all eigenvalues and eigenfunctions to the both (\mathrm{L}\mathrm{P}_{+})and (LP‐).

4

Key lemmas on characteristic function

Proofs of Theorems 1‐3 are done by an algorithm of the representation equation for the

linearized equation of (\mathrm{L}\mathrm{P}_{\pm}) (see [12]). We would like to describe whole procedure in theforth‐coming papers.
Here we only focus on the two key lemmas for justyfing existence and uniqueness ofsolution to (3.6).

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4.1

Fundamental properties on the modified elliptic integral

We prepare a fundamental properties on the modified elliptic integral \mathcal{M} ([8] and [9]).
Proposition 4.1. Let ( $\nu$, k)\in \mathcal{D} and let \mathcal{M} be defined by (2.1)

Then

(i)

\displaystyle \frac{\partial \mathcal{M} {\partial k}( $\nu$, k)=\sqrt{\frac{(1+ $\nu$)(k^{2}+ $\nu$)}{ $\nu$} \frac{kE(k)}{(k^{2}+ $\nu$)(1-k^{2}) ,(ii)\displaystyle \frac{\partial \mathcal{M} {\partial $\nu$}( $\nu$, k)=\sqrt{\frac{(1+ $\nu$)(k^{2}+ $\nu$)}{ $\nu$} [-\frac{K(k)}{2 $\nu$(1+ $\nu$)}+\frac{E(k)}{2(1+ $\nu$)(k^{2}+\mathrm{v}) ].

Remark 4.1. It is well known that (1-k^{2})K(k)<E(k)<K(k) for k\in(0,1) . Se we see

from (ii) of Proposition 4.1 that for each k\in(0,1) the real function \mathcal{M}(\cdot, k) is decreasing
for $\nu$\in(-1, k^{2})\cup(0, \infty) (see Fig. 3).
Proposition 4.2. Fix k \in (0,1) and consider the real function \mathcal{M} (. , k) : (-1, -k^{2})\cup(0, \infty)\rightarrow \mathrm{R} . Then the following formulas hold:4.2\displaystyle \lim_{ $\nu$\rightar ow-1}\mathcal{M}( $\nu$, k)=\frac{ $\pi$}{2} ,(4.1)\displaystyle \lim_{ $\nu$\rightar ow-k^{2} \mathcal{M}( $\nu$, k)=0 ,(4.2)\displaystyle \lim_{ $\nu$\rightar ow 0}\mathcal{M}( $\nu$, k)=\infty ,(4.3)\displaystyle\lim_{$\nu$\rightar ow\infty}\mathcal{M}($\nu$,k)=\frac{$\pi$}{2} .

(4.4)

Fundamental properties of \mathrm{v}\pm

Recall the characteristic function\mathcal{A}(k, $\mu$) :=|\mathcal{M}(\mathrm{v}_{+}(k, $\mu$), k)-\mathcal{M}($\nu$_{-}(k, $\mu$), k)|for (k, $\mu$)\in $\Sigma$ . We first remark that $\nu$_{\pm}(k, $\mu$) of (3.5) is characterized by$\nu$_{+}(k, $\mu$)+$\nu$_{-}(k, $\mu$)=\displaystyle \frac{3k^{2}( $\mu$-3k^{2}) { $\mu$( $\mu$-3+3k^{2})
and$\nu$_{+}(k, $\mu$)$\nu$_{-}(k, $\mu$)=\displaystyle \frac{9k^{4} { $\mu$( $\mu$-3+3k^{2}) .
Then an elementary algebra gives($\nu$_{+}(k, $\mu$)-$\nu$_{-}(k, $\mu$) ^{2}=\displaystyle \frac{-27k^{4}( $\mu$-(2-k^{2})$\Lambda$_{0}(k) ( $\mu$-(2-k^{2})$\Lambda$_{2}(k) }{$\mu$^{2}( $\mu$+3-3k^{2})^{2} .

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Moreover, we see\left{begin{ary}l \frac{1}$\nu_{+}(k,$\mu)}+\frac{1}$\nu_{-}(k,$\mu)}=\frac{$\mu-3k^{2}3k^{2},\ frac{1}$\nu_{+}(k,$\mu)}.\frac{1}$\nu_{-}(k,$\mu)}=\frac{$\mu($\mu+3-k^{2})9k^{4}, \end{ary}\ight.
\displaytle\{ }_1^{}+\frac{}\frac{1}$\nu_{+}(k,$\mu$),\nu$_{+}(k,$\mu$)1}\{+(1\frac{1}$\nu_{-1}(k,$\mu$,(k \mu$)}(1+\frac{+}$\nu_{-}\frac{=$\mu$( \mu$}{\frac{$\mu$+3k^{2}9k^{4}+3)k^{2}\displaystle\{ }_1^{}+\frac{}+\frac{k^2}{$\nu$_{+}(k,$\mu$),\nu$_{+}(k,$\mu$)k^{2}\+(1\frac{k^2}{$\nu$_{-}(k,$\mu$,(k \mu$)k^{2})(1+\frac{+}$\nu$_{-}\frac{=\frac{$\mu$+6-3k^{2}+3)($\mu$+3k^{2}($\mu$-3k^{2})9 ,,andrespectively.

(4.5)

(4.6)

(4.7)

The following proposition helps some calculations on the characteristic functions to‐gether with M.

Proposition 4.3. For $\nu$_{\pm}=$\nu$_{\pm}(k, $\mu$) of (3.5)\displaystyle \frac{1}{ $\nu$\pm}(1+\frac{1}{ $\nu$\pm}) (1+\frac{k^{2} { $\nu$\pm}) =\frac{ $\mu$( $\mu$-3)( $\mu$+3-3k^{2}) {27k^{4} .
4.3

Several limit for characteristic function

Lemma 4.1. Let k\in (0,1) be fixed and (k, $\mu$) \in $\Sigma$ . Then the following (i) and (ii) hold
true:

(i)

\displaystyle \lim_{ $\mu$\rightar ow(2-k^{2})$\Lambda$_{0}(k)}\mathcal{A}(k, $\mu$)=0, \displaystyle \lim_{ $\mu$\rightar ow-3}\mathcal{A}(k, $\mu$)=\frac{ $\pi$}{2}.

(ii)\displaystyle \lim_{ $\mu$\rightar ow-3(1-k^{2}) \mathcal{A}(k, $\mu$)=\frac{ $\pi$}{2}, \displaystyle\lim_{$\mu$\rightar ow0}\mathcal{A}(k, $\mu$)=$\pi$.

Proof of Lemma 4.1. (i) Suppose that (k, $\mu$)\in$\Sigma$_{0} . Then, it follows from (4.5)-(4.7) that

-1<$\nu$_{-}(k, $\mu$)<$\nu$_{+}(k, $\mu$)<-k^{2} and\bullet\bullet\displaystyle \lim_{ $\mu$\rightar ow(2-k^{2})$\Lambda$_{0}(k)}$\nu$_{\pm}(k, $\mu$)=\mathrm{v}_{*}(k)\in(-1, k^{2}),\displaystyle \lim_{ $\mu$\rightar ow-3}$\nu$_{-}(k, $\mu$)=-1, \displaystyle \lim_{ $\mu$\rightar ow-3}$\nu$_{+}(k, $\mu$)=-k^{2}.

Therefore, by using monotonicity of \mathcal{M}\displaystyle \lim_{ $\mu$\rightar ow(2-k^{2})$\Lambda$_{0}(k)}\mathcal{A}(k, $\mu$)= \lim_{ $\mu$\rightar ow(2-k^{2})$\Lambda$_{0}(k)}(\mathcal{M}($\nu$_{-}(k, $\mu$), k)-\mathcal{M}(\mathrm{v}_{+}(k, $\mu$), k)=\mathcal{M}(\mathrm{v}_{*}(k), k)-\mathcal{M}(\mathrm{v}_{*}(k), k)
=0.

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In the similar manner we see from (4.1) and (4.2) of Proposition 4.2 that\displaystyle \lim_{ $\mu$\rightar ow-3}\mathcal{A}(k, $\mu$)= \lim_{ $\mu$\rightar ow-3}(\mathcal{M}($\nu$_{-}(k, $\mu$), k)-\mathcal{M}($\nu$_{+}(k, $\mu$), k))=(\mathcal{M}(-1, k)-\mathcal{M}(-k^{2}, k))=^{\underline{ $\pi$}2Thus it complete a proof.
\square(ii) of the lemma is similarly done. So we omit it.

Previous results for the cases of bistable nonlinearities suggest that we will also have(iii)

\displaystyle \lim_{ $\mu$\rightar ow(2-k^{2})$\Lambda$_{2}(k)}\mathcal{A}(k, $\mu$)= $\pi$.

However, a different type of asymptotic formulas should be applied to show the above
claim. So we omit it.

4.4

Monotonicity of characteristic function

Lemma 4.2. Let (k, $\mu$)\in $\Sigma$ . For each

i=0 , 1, 2,\displayst le\frac{\partial\mathcal{A}{\partial$\mu$}(k,$\mu$)>0

in $\Sigma$_{i}.

Proof. Denote $\nu$_{\pm}:=\mathrm{v}_{\pm}(k, $\mu$) and set\displaystyle \mathcal{R}(k, $\mu$):=\frac{ $\mu$( $\mu$-3)( $\mu$+3-3k^{2})}{27k^{4} .
For simplicity we only consider the case (k, $\mu$)\in$\Sigma$_{0} . By using (ii) of Proposition 4.1 and
Proposition 4.3, we are led to\displayst le\frac{\parti l\mathcal{A}{\parti l$\mu$}=\frac{\parti l\mathcal{M}{\parti l$\nu$}( \nu$_{-},k)\cdot\frac{\parti l$\nu$_{-} \parti l$\mu$}-\frac{\parti l\mathcal{M}{\parti l$\nu$}( \nu$_{+},k)\cdot\frac{\parti l$\nu$_{+}{\parti l$\mu$}=\displaystyle \sqrt{\frac{(1+$\nu$_{-})(k^{2}+$\nu$_{-}) {$\nu$_{-} [-\frac{K(k)}{2$\nu$_{-}(1+$\nu$_{-}) +\frac{E(k)}{2(1+$\nu$_{-})(k^{2}+$\nu$_{-}) ] \displayte\frac{ptial$\nu_{-}\partil$\mu}-\displaystyle \sqrt{\frac{(1+$\nu$_{+})(k^{2}+$\nu$_{+}) { $\nu$+} [-\frac{K(k)}{2$\nu$_{+}(1+$\nu$_{+}) +\frac{E(k)}{2(1+$\nu$_{+})(k^{2}+$\nu$_{+}) ] \displayte\frac{prtial$\nu_{+}\partil$\mu}.

.

= \displaystyle \sqrt{\mathcal{R}(k, $\mu$)}(-$\nu$_{-}\frac{\partial$\nu$_{-} {\partial $\mu$}) [-\frac{K(k)}{2$\nu$_{-}(1+$\nu$_{-}) +\frac{E(k)}{2(1+$\nu$_{-})(k^{2}+$\nu$_{-}) ]-\displaystyle \sqrt{\mathcal{R}(k, $\mu$)}(-$\nu$_{+}\frac{\partial $\nu$+}{\partial $\mu$}) [-\frac{K(k)}{2$\nu$_{+}(1+$\nu$_{+}) +\frac{E(k)}{2(1+$\nu$_{+})(k^{2}+$\nu$_{+}) ]=\displaystyle\frac{\sqrt{\mathcal{R}(k,$\mu$)}{2}[(\frac{1}{1+$\nu$_{-}\frac{\partial$\nu$_{-}{\partial$\mu$}-\frac{1}{1+$\nu$+}\frac{\partial$\nu$+}{\partial$\mu$})K(k)-(\displaystyle\frac{$\nu$_{-}{(1+$\nu$_{-})(k^{2}+$\nu$_{-})\frac{\partial$\nu$_{-}{\partial$\mu$}-\frac{$\nu$+}{(1+$\nu$_{+})(k^{2}+$\nu$_{+})\frac{\partial$\nu$+}{\partial$\mu$})E(k)]104and furthermore,\displayst le\frac{\partial\mathcal{A}{\partial$\mu$}(k, $\mu$)=\frac{1}2\sqrt{\mathcal{R}(k,$\mu$)}[(\frac{k^2}+$\nu$_{-} $\nu$^{\underline{3} \frac{\partial$\nu$_{-} \partial$\mu$}-\frac{k^2}+$\nu$_{+}{$\nu$_{+}^{3}\frac{\partial$\nu$_{+}{\partial$\mu$})K(k)-(\displaystle\frac{1}$\nu$^{\underline{2} \frac{\partil$\nu$_{-} \partil$\mu$}-\frac{1}$\nu$_{+}^2}\frac{\partil$\nu$_{+} \partil$\mu$})E(k)]=\displaystyle\frac{1}{2\sqrt{\mathcal{R}(k,$\mu$)} [\frac{\partial}{\partial$\mu$}[(\frac{1}{$\nu$_{+} -\frac{1}{$\nu$_{-} )(1+\frac{k^{2} {2}(\frac{1}{$\nu$_{+} +\frac{1}{l\text{ノ_{-} ) ]K(k)-\displaystyle\frac{\partial}{\partial$\mu$}(\frac{1}{$\nu$_{+}-\frac{1}{$\nu$_{-})E(k)]=\displaystyle \frac{S_{1}(k,$\mu$)K(k)+S_{2}(k,$\mu$)E(k)}{6k^{2}\sqrt{\mathcal{R}(k,$\mu$)}\sqrt{-3$\mu$^{2}+6(k^{2}-2) $\mu$+9k^{4} ,where

S_{1} :=$\mu$^{2}+3(2-k^{2}) $\mu$+6(1-k^{2}) , S_{2} :=-3( $\mu$-k^{2}+2).

Finally, the claim of the lemma is proved by repeating argument in [12, Proof of Lemma\square6.2].

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