The method used to verify this type of stability is attributed to Grillakis, Shatah and Strauss [11], and we essentially apply a theorem presented therein to deal with it. For this purpose, we list the following assumptions:
-
(A1)
For every \(v_{0}\in H^{s}(R)\) (\(s > 3/2\)), there exists a solution \(v(t,x)\in C([0, T ); H^{s}(R))\cap C^{1}([0, T ); H^{s-1}(R))\) of (2.1) with \(v(0,x)=v_{0}\) for some \(T>0\). Furthermore, there exist functionals \(E(v)\) and \(F (v)\) such that they are conserved for solutions of (2.1).
-
(A2)
For every \(c\in(c_{1},c_{2})\), there exists a traveling wave solution \(\phi\in H^{2}(R)\) of (2.1), where \(\phi> 0\) and \(\phi_{x}\not\equiv0\). The mapping \(c\rightarrow\phi(x-ct)\) is \(C^{1}((c_{1},c_{2}); H^{2}(R))\). Moreover, ϕ satisfies \(cE'(\phi)- F'(\phi) = 0\), where \(E'\) and \(F'\) are the variational derivatives of E and F, respectively.
-
(A3)
For every \(c\in(c_{1},c_{2})\), the linearized Hamiltonian operator around ϕ defined by
$$ H_{c} : H^{1}(R)\rightarrow H^{-1}(R),\qquad H_{c} = cE''(\phi)-F''( \phi) $$
(4.1)
has exactly one negative simple eigenvalue, its kernel is spanned by \(\phi_{x}\) and the rest of its spectrum is positive and bounded away from zero.
Theorem 4.1
[11]
Under assumptions (A1), (A2) and (A3), a solitary wave solution
\(\phi(x-ct)\)
of (2.1) is stable if and only if the scalar function
\(d(c) = cE(\phi)-F(\phi)\)
is convex in a neighborhood of
c.
Firstly we verify that equation (2.1) satisfies assumptions (A1)-(A3). Assumption (A1) is guaranteed by Lemmas 2.1 and 2.2.
To prove assumptions (A2) and (A3), we need to calculate the variational derivatives of functionals \(E(v)\) and \(F(v)\).
$$\begin{aligned}& E'(v)=\bigl(1-\partial^{2}_{x} \bigr)v, \end{aligned}$$
(4.2)
$$\begin{aligned}& E''(v)=1-\partial^{2}_{x}, \end{aligned}$$
(4.3)
$$\begin{aligned}& F'(v)=p_{1}(k)v+\frac{p_{2}(k)}{2}v^{2}+ \frac {p_{3}(k)}{3}v^{3}+3v^{4}+p_{4}(k)v_{xx}+7v_{x}^{2}+14vv_{xx}, \end{aligned}$$
(4.4)
$$\begin{aligned}& F''(v)=p_{1}(k)+p_{2}(k)v+p_{3}(k)v^{2}+12v^{3}+p_{4}(k) \partial _{x}^{2}+14v_{xx}+14v\partial_{x}^{2}+14v_{x} \partial_{x}. \end{aligned}$$
(4.5)
From the above first order variational derivatives of \(E(v)\) and \(F(v)\), we know that equation (2.3) can be rewritten as
$$ cE'(\phi)-F'(\phi)=0. $$
(4.6)
Thus assumption (A2) is ensured.
By a direct calculation we obtain the following linearized operator \(H_{c}\):
$$\begin{aligned} H_{c} &=cE''( \phi)-F''(\phi) \\ &=-\partial_{x}\bigl(\bigl(c+p_{4}(k)+14\phi\bigr) \partial_{x}\bigr)+c-p_{1}(k)-p_{2}(k)\phi -p_{3}(k)\phi^{2}-12\phi^{3}-14\phi_{xx}. \end{aligned}$$
(4.7)
Since \(H_{c}\) is a second order linear differential operator, the corresponding spectrum equation \(H_{c}\nu=\lambda\nu\) can be written as the Sturm-Liouville problem
$$ -(p\nu_{x})_{x}+(q-\lambda)\nu=0, $$
(4.8)
where \(p=c+p_{4}(k)+14\phi\), \(q=c-p_{1}(k)-p_{2}(k)\phi-p_{3}(k)\phi ^{2}-12\phi^{3}-14\phi_{xx}\).
Review that a regular Sturm-Liouville system has an infinitely many real eigenvalues \(\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots\) with \(\lim_{n\rightarrow\infty}\lambda_{n}=\infty\) (see [16]). The eigenfunction \(\nu_{n}(x)\) corresponding to the eigenvalue \(\lambda_{n}\) is uniquely determined apart from the different constant factor and has exactly n zeros. Furthermore, via observation we know that \(H_{c}\) is a self-adjoint, second order differential operator, hence its eigenvalues λ are real and simple. By Weyl’s essential spectrum theorem, we have that its essential spectrum is expressed as \([c-p_{1}(k),\infty)\) owing to the fact that \(\lim_{x\rightarrow\infty} q(x)=c-p_{1}(k)\) (see [17]). It can be directly checked that (2.2) is equivalent to \(H_{c}(\phi_{x})=0\). From the properties of the solitary waves of (2.1), we know that \(\phi_{x}\) has exactly one zero on R, this indicates that 0 is the second eigenvalue of \(H_{c}\). The above analysis leads us to the conclusion that there is exactly one negative eigenvalue, and the rest of the spectrum is positive and bounded away from zero, which shows that assumption (A3) is satisfied.
Secondly, we prove that the scalar function \(d(c)\) is convex on a neighborhood of c. The following lemma is important in determining the sign of second order derivative of \(d(c)\).
Lemma 4.1
[18]
Set
\(\Omega=\mathbb {R}\)
and let
$$ G_{b}(x)=g_{n}(b)x^{n}+g_{n-1}(b)x^{n-1}+ \cdots+g_{1}(b)x+g_{0}(b) $$
(4.9)
be a family of real polynomials depending also polynomially on a real parameter
b. Assume that there exists an open interval
\(I\in \mathbb {R}\)
such that:
-
(i)
There is some
\(b_{0}\in I\)
such that
\(G_{b_{0}}(x)>0\)
on Ω.
-
(ii)
For all
\(b\in I\), the discriminant of
\(G_{b}\)
with respect to
x
is not equal to zero.
-
(iii)
For all
\(b\in I\), \(g_{n}(b)\neq0\).
Then, for all
\(b\in I\), \(G_{b}(x)>0\)
on Ω.
Set
$$ P_{1}(h)=18h^{5}+75h^{4}-835h^{2}-3 \text{,}000h-5\text{,}250 $$
(4.10)
and
$$ P_{2}(h)=13+10h+5h^{2}+6h^{3}/5. $$
(4.11)
We give our main result as follows.
Theorem 4.2
Let
\(h^{*}\)
be the only zero point of
\(P_{1}(h)\). If
\(k=1\)
and
\(13< c< P_{2}(h^{*})\), then the scalar function
\(d(c)\)
is convex in a neighborhood of
c. Therefore, the solitary waves with nonvanishing boundary are orbitally stable when wave speed
\(c\in (13,P_{2}(h^{*}))\).
Proof
Deriving \(d(c)\) with respect to c, we have
$$ d'(c)=\bigl(cE'(\phi)-F'(\phi), \phi_{c}\bigr)+E(\phi)=E(\phi). $$
(4.12)
Since \(\phi>0\) and \(\phi_{x}<0\) in \([0,+\infty)\), it is easy to obtain from (2.4) that
$$\begin{aligned} \begin{aligned}[b] \phi&=-\phi_{x}\sqrt{\frac{c+p_{4}(k)+14\phi}{c-p_{1}(k)-\frac {p_{2}(k)}{3}\phi-\frac{p_{3}(k)}{6}\phi^{2}-\frac{6}{5}\phi^{3}}} \quad \mbox{for } x\in(0,+ \infty) \\ &=-\phi_{x}\sqrt{\frac{f_{1}(c,k,\phi)}{f_{2}(c,k,\phi)}}, \end{aligned} \end{aligned}$$
(4.13)
where \(f_{1}(c,k,\phi)=c+p_{4}(k)+14\phi\), \(f_{2}(c,k,\phi)=c-p_{1}(k)-\frac {p_{2}(k)}{3}\phi-\frac{p_{3}(k)}{6}\phi^{2}-\frac{6}{5}\phi^{3}\).
Then we calculate the second derivative of \(d(c)\),
$$\begin{aligned} d''(c)&=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{R}\frac{1}{2} \bigl(\phi^{2}+ \phi^{2}_{x} \bigr)\, {\mathrm{d}}x \\ &=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{\infty}\phi^{2} \biggl(1+ \frac{f_{2}(c,k,\phi)}{f_{1}(c,k,\phi)} \biggr)\, {\mathrm{d}}x \\ &=-\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{\infty}\phi\phi _{x}\sqrt{ \frac{f_{1}(c,k,\phi)}{f_{2}(c,k,\phi)}} \biggl(1+\frac{f_{2}(c,k,\phi)}{f_{1}(c,k,\phi)} \biggr)\, {\mathrm{d}}x. \end{aligned}$$
(4.14)
Let \(h(c)\) denote the amplitude of solitary wave with wave speed c, then from (2.4) we know that h is the only positive real number satisfying
$$ c-p_{1}(k)-\frac{p_{2}(k)}{3}h-\frac{p_{3}(k)}{6}h^{2}- \frac{6}{5}h^{3}=0. $$
(4.15)
By using transformations \(\phi_{x}\, {\mathrm{d}}x={\mathrm{d}}y\) and \(y=hz\), (4.14) can be rewritten as
$$\begin{aligned} d''(c)&=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{h}y\sqrt{\frac {f_{1}(c,k,y)}{f_{2}(c,k,y)}} \biggl(1+ \frac{f_{2}(c,k,y)}{f_{1}(c,k,y)} \biggr)\, {\mathrm{d}}y \\ &=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{1}h^{2}z\sqrt{ \frac {f_{1}(c,k,hz)}{f_{2}(c,k,hz)}} \biggl(1+\frac{f_{2}(c,k,hz)}{f_{1}(c,k,hz)} \biggr)\, {\mathrm{d}}z. \end{aligned}$$
(4.16)
Further, substituting \(c=c(h)=p_{1}(k)+\frac{p_{2}(k)}{3}h+\frac {p_{3}(k)}{6}h^{2}+\frac{6}{5}h^{3}\)(see (4.15)) into the above equation, we have
$$\begin{aligned} d''(c)&=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{1}h^{2}z\sqrt{ \frac {f_{1}(c(h),k,hz)}{f_{2}(c(h),k,hz)}} \biggl(1+\frac{f_{2}(c(h),k,hz)}{f_{1}(c(h),k,hz)} \biggr)\, {\mathrm {d}}z \\ &=\frac{{\mathrm{d}}}{{\mathrm{d}}c} \int_{0} ^{1}h^{2}z\frac {f_{1}(c(h),k,hz)+f_{2}(c(h),k,hz)}{ \sqrt{f_{1}(c(h),k,hz)f_{2}(c(h),k,hz)}}\, { \mathrm{d}}z. \end{aligned}$$
(4.17)
Let \(F(h,k,z)\) denote the integrand in (4.17). For any interval \([h_{1},h_{2}]\) with \(h_{1}>0\) and \([k_{1}, k_{2}]\) with \(k_{1}>0\), \(\int_{0} ^{1}F(h,k,z)\, \mathrm{d}z\) can be regarded as an integral involving parameters. By a direct calculation, we have
$$ \partial_{h}F(h,k,z) =\frac{h^{2}z(1-z)P(h,k,z)}{ (f_{1}(c(h),k,hz)f_{2}(c(h),k,hz) )^{3/2}}, $$
(4.18)
where
$$\begin{aligned}& f_{1}\bigl(c(h),k,hz\bigr)=p_{1}(k)+p_{4}(k)+ \frac{p_{2}(k)}{3}h+\frac {p_{3}(k)}{6}h^{2}+\frac{6}{5}h^{3}+14hz, \end{aligned}$$
(4.19)
$$\begin{aligned}& f_{2}\bigl(c(h),k,hz\bigr)=h(1-z) \biggl(\frac{p_{2}(k)}{3}+ \frac {p_{3}(k)}{6}h(1+z)+\frac{6}{5}h^{2}\bigl(1+z+z^{2} \bigr) \biggr), \end{aligned}$$
(4.20)
and the expression of \(P(h,k,z)\) is given as
$$\begin{aligned} P(h,k,z) =&\frac {1}{2\text{,}700}\bigl(-186\text{,}624h^{8}-3\text{,}265 \text{,}920h^{6}z-186\text{,}624h^{8}z-6\text{,}350 \text{,}400h^{4}z^{2} \\ &{}-3\text{,}265\text{,}920h^{6}z^{2}-18\text{,}6624h^{8}z^{2}-6\text{,}350 \text{,}400h^{4}z^{3}-3\text{,}265\text{,}920h^{6}z^{3} \\ &{}+93 \text{,}312h^{8}z^{3}-6\text{,}350\text{,}400h^{4}z^{4}+1\text{,}632\text{,}960h^{6}z^{4}+93 \text{,}312h^{8}z^{4} \\ &{}+1\text{,}632\text{,}960h^{6}z^{5}+93 \text{,}312h^{8}z^{5}+1\text{,}632\text{,}960h^{6}z^{6}-233\text{,}280h^{5}p_{1} \\ &{}-680\text{,}400h^{3}zp_{1}-233 \text{,}280h^{5}zp_{1}-680\text{,}400h^{3}z^{2}p_{1}-233 \text{,}280h^{5}z^{2}p_{1} \\ &{}-680\text{,}400h^{3}z^{3}p_{1}+136 \text{,}080h^{5}z^{3}p_{1}+136\text{,}080h^{5}z^{4}p_{1}+136 \text{,}080h^{5}z^{5}p_{1} \\ &{}-16\text{,}200h^{2}p_{1}^{2}-16\text{,}200h^{2}zp_{1}^{2}-16 \text{,}200h^{2}z^{2}p_{1}^{2}-155 \text{,}520h^{6}p_{2} \\ &{}-1\text{,}814\text{,}400h^{4}zp_{2}-77 \text{,}760h^{6}zp_{2}-3\text{,}528\text{,}000h^{2}z^{2}p_{2}-302 \text{,}400h^{4}z^{2}p_{2} \\ &{}-77\text{,}760h^{6}z^{2}p_{2}-302 \text{,}400h^{4}z^{3}p_{2}+77\text{,}760h^{6}z^{3}p_{2}+756\text{,}000h^{4}z^{4}p_{2} \\ &{}+38 \text{,}880h^{6}z^{4}p_{2}+38\text{,}880h^{6}z^{5}p_{2}-129 \text{,}600h^{3}p_{1}p_{2}-441\text{,}000hzp_{1}p_{2} \\ &{}-16\text{,}200h^{3}zp_{1}p_{2}-16 \text{,}200h^{3}z^{2}p_{1}p_{2}+64 \text{,}800h^{3}z^{3}p_{1}p_{2}-13 \text{,}500p_{1}^{2}p_{2} \\ &{}-43\text{,}200h^{4}p_{2}^{2}-252\text{,}000h^{2}zp_{2}^{2}+84 \text{,}000h^{2}z^{2}p_{2}^{2}-3 \text{,}600h^{4}z^{2}p_{2}^{2} \\ &{}+18 \text{,}000h^{4}z^{3}p_{2}^{2}-18 \text{,}000hp_{1}p_{2}^{2}+7\text{,}500hzp_{1}p_{2}^{2}-4 \text{,}000h^{2}p_{2}^{3}+2\text{,}000h^{2}zp_{2}^{3} \\ &{}-77 \text{,}760h^{7}p_{3}-907\text{,}200h^{5}zp_{3}-77\text{,}760h^{7}zp_{3}-1\text{,}323 \text{,}000h^{3}z^{2}p_{3} \\ &{}-907\text{,}200h^{5}z^{2}p_{3}-38 \text{,}880h^{7}z^{2}p_{3}-1\text{,}323 \text{,}000h^{3}z^{3}p_{3}-189\text{,}000h^{5}z^{3}p_{3} \\ &{}+38 \text{,}880h^{7}z^{3}p_{3}+415\text{,}800h^{5}z^{4}p_{3}+38 \text{,}880h^{7}z^{4}p_{3}+415\text{,}800h^{5}z^{5}p_{3} \\ &{}+16\text{,}200h^{7}z^{5}p_{3}-64 \text{,}800h^{4}p_{1}p_{3}-157\text{,}500h^{2}zp_{1}p_{3}-64 \text{,}800h^{4}zp_{1}p_{3} \\ &{}-157\text{,}500h^{2}z^{2}p_{1}p_{3}-10\text{,}800h^{4}z^{2}p_{1}p_{3}+35 \text{,}100h^{4}z^{3}p_{1}p_{3}+35 \text{,}100h^{4}z^{4}p_{1}p_{3} \\ &{}-4 \text{,}500hp_{1}^{2}p_{3}-4\text{,}500hzp_{1}^{2}p_{3}-43\text{,}200h^{5}p_{2}p_{3}-252 \text{,}000h^{3}zp_{2}p_{3} \\ &{}-21\text{,}600h^{5}zp_{2}p_{3}-63 \text{,}000h^{3}z^{2}p_{2}p_{3}+94 \text{,}500h^{3}z^{3}p_{2}p_{3}+17\text{,}100h^{5}z^{3}p_{2}p_{3} \\ &{}+9 \text{,}900h^{5}z^{4}p_{2}p_{3}-18 \text{,}000h^{2}p_{1}p_{2}p_{3}-3 \text{,}750h^{2}zp_{1}p_{2}p_{3}+8 \text{,}250h^{2}z^{2}p_{1}p_{2}p_{3} \\ &{}-6\text{,}000h^{3}p_{2}^{2}p_{3}+2 \text{,}250h^{3}z^{2}p_{2}^{2}p_{3}-10 \text{,}800h^{6}p_{3}^{2}-63\text{,}000h^{4}zp_{3}^{2} \\ &{}-10 \text{,}800h^{6}zp_{3}^{2}-63\text{,}000h^{4}z^{2}p_{3}^{2}+26 \text{,}250h^{4}z^{3}p_{3}^{2}+5 \text{,}400h^{6}z^{3}p_{3}^{2} \\ &{}+26 \text{,}250h^{4}z^{4}p_{3}^{2}+4 \text{,}050h^{6}z^{4}p_{3}^{2}-4\text{,}500h^{3}p_{1}p_{3}^{2}-4 \text{,}500h^{3}zp_{1}p_{3}^{2} \\ &{}+2 \text{,}250h^{3}z^{2}p_{1}p_{3}^{2}+2 \text{,}250h^{3}z^{3}p_{1}p_{3}^{2}-3 \text{,}000h^{4}p_{2}p_{3}^{2}-1\text{,}500h^{4}zp_{2}p_{3}^{2} \\ &{}+1 \text{,}500h^{4}z^{2}p_{2}p_{3}^{2}+625h^{4}z^{3}p_{2}p_{3}^{2}-500h^{5}p_{3}^{3}-500h^{5}zp_{3}^{3}+250h^{5}z^{2}p_{3}^{3} \\ &{}+250h^{5}z^{3}p_{3}^{3}-233 \text{,}280h^{5}p_{4}-680\text{,}400h^{3}zp_{4}-233 \text{,}280h^{5}zp_{4} \\ &{}-680\text{,}400h^{3}z^{2}p_{4}-233\text{,}280h^{5}z^{2}p_{4}-680 \text{,}400h^{3}z^{3}p_{4}+136\text{,}080h^{5}z^{3}p_{4} \\ &{}+136 \text{,}080h^{5}z^{4}p_{4}+136\text{,}080h^{5}z^{5}p_{4}-32\text{,}400h^{2}p_{1}p_{4}-32 \text{,}400h^{2}zp_{1}p_{4} \\ &{}-32\text{,}400h^{2}z^{2}p_{1}p_{4}-129 \text{,}600h^{3}p_{2}p_{4}-441\text{,}000hzp_{2}p_{4}-16\text{,}200h^{3}zp_{2}p_{4} \\ &{}-16 \text{,}200h^{3}z^{2}p_{2}p_{4}+64 \text{,}800h^{3}z^{3}p_{2}p_{4}-27 \text{,}000p_{1}p_{2}p_{4}-18\text{,}000hp_{2}^{2}p_{4} \\ &{}+7\text{,}500hzp_{2}^{2}p_{4}-64 \text{,}800h^{4}p_{3}p_{4}-157\text{,}500h^{2}zp_{3}p_{4}-64 \text{,}800h^{4}zp_{3}p_{4} \\ &{}-157\text{,}500h^{2}z^{2}p_{3}p_{4}-10\text{,}800h^{4}z^{2}p_{3}p_{4}+35 \text{,}100h^{4}z^{3}p_{3}p_{4} \\ &{}+35 \text{,}100h^{4}z^{4}p_{3}p_{4}-9 \text{,}000hp_{1}p_{3}p_{4}-9\text{,}000hzp_{1}p_{3}p_{4}-18\text{,}000h^{2}p_{2}p_{3}p_{4} \\ &{}-3 \text{,}750h^{2}zp_{2}p_{3}p_{4}+8 \text{,}250h^{2}z^{2}p_{2}p_{3}p_{4}-4 \text{,}500h^{3}p_{3}^{2}p_{4}-4 \text{,}500h^{3}zp_{3}^{2}p_{4} \\ &{}+2\text{,}250h^{3}z^{2}p_{3}^{2}p_{4}+2 \text{,}250h^{3}z^{3}p_{3}^{2}p_{4}-16 \text{,}200h^{2}p_{4}^{2}-16\text{,}200h^{2}zp_{4}^{2} \\ &{}-16 \text{,}200h^{2}z^{2}p_{4}^{2}-13\text{,}500p_{2}p_{4}^{2}-4 \text{,}500hp_{3}p_{4}^{2}-4\text{,}500hzp_{3}p_{4}^{2} \bigr), \end{aligned}$$
where \(p_{i}=p_{i}(k)\) (\(i=1,2,3,4\)).
From (4.18) we know that there exists a positive constant K related to \([h_{1},h_{2}]\) and \([k_{1},k_{2}]\) such that
$$\bigl\vert \partial_{h}F(h,k,z) \bigr\vert \leq K(1-z)^{-\frac{1}{2}} \quad \mbox{for all } (h,k,z)\in[h_{1},h_{2}] \times[k_{1},k_{2}]\times(0,1). $$
Set \(g(z)=K(1-z)^{-1/2}\), then \(g(z)\in L^{1}(0,1)\). By the dominated convergence theorem, we have
$$\partial_{h} \int_{0} ^{1}F(h,k,z)\, \mathrm{d}z= \int_{0} ^{1}\partial_{h}F(h,k,z)\, \mathrm{d}z $$
for all \(h\in[h_{1},h_{2}]\) and \(k\in[k_{1},k_{2}]\). Therefore we can extend this result to all \(h\in(0,+\infty)\) and \(k\in(0,+\infty)\), with the monotonic increase of amplitude h with wave speed c, we have
$$ d''(c)= \int_{0} ^{1}\partial_{h}F(h,k,z)\, \mathrm{d}z \cdot h'(c), $$
(4.21)
where \(h'(c)>0\) denotes the derivative of amplitude h with respect to wave speed c.
It is observed from (4.18) and (4.21) that the sign of \(d''(c)\) is determined by \(P(h,k,z)\), but it is difficult to judge the sign of it. So we take a step back to consider the special case \(k=1\), then (4.18) is simplified as
$$ \partial_{h}F(h,z) =\frac{h^{2}z(1-z)P(h,z)}{ (f_{1}(c(h),hz)f_{2}(c(h),hz) )^{3/2}}, $$
(4.22)
where
$$\begin{aligned}& f_{1}\bigl(c(h),hz\bigr)=28+10h+5h^{2}+\frac{6}{5}h^{3}+14hz, \end{aligned}$$
(4.23)
$$\begin{aligned}& f_{2}\bigl(c(h),hz\bigr)=h(1-z) \biggl(10+5h(1+z)+ \frac{6}{5}h^{2}\bigl(1+z+z^{2}\bigr) \biggr), \end{aligned}$$
(4.24)
and
$$\begin{aligned} P(h,z) =&\frac{1}{125}\bigl(1\text{,}470\text{,}000 + 840 \text{,}000h + 233\text{,}800h^{2} - 21\text{,}000h^{4} -5 \text{,}040h^{5} \\ &{}+ 3\text{,}080\text{,}000hz+1\text{,}708\text{,}800h^{2}z + 601\text{,}200h^{3}z +126\text{,}000h^{4}z +12\text{,}960h^{5}z \\ &{}+ 2\text{,}473 \text{,}800h^{2}z^{2}+1\text{,}141\text{,}200h^{3}z^{2}+ 359\text{,}650h^{4}z^{2} + 74\text{,}210h^{5}z^{2} +11\text{,}250h^{6}z^{2} \\ &{}+ 900h^{7}z^{2} + 1\text{,}180\text{,}200h^{3}z^{3} + 470 \text{,}400h^{4}z^{3} + 146\text{,}390h^{5}z^{3} + 33\text{,}300h^{6}z^{3} \\ &{} + 5\text{,}400h^{7}z^{3} + 432h^{8}z^{3} + 380\text{,}275h^{4}z^{4} + 116\text{,}640h^{5}z^{4}+ 29\text{,}835h^{6}z^{4} \\ &{}+ 5\text{,}400h^{7}z^{4} + 432h^{8}z^{4} + 75\text{,}390h^{5}z^{5} + 12\text{,}960h^{6}z^{5} + 2\text{,}250h^{7}z^{5} \\ &{} + 432h^{8}z^{5} + 7\text{,}560h^{6}z^{6} \bigr), \end{aligned}$$
(4.25)
where \((h,z)\in(0,\infty)\times(0,1)\). The transformation \(z=\frac {x^{2}}{1+x^{2}}\) can be used to map the variable z from \((0,1)\) to the whole real line \(\mathbb {R}\), then we use Lemma 4.1 to determine the sign of polynomial \(P(h,z)\) with parameter h.
Substituting \(z=\frac{x^{2}}{1+x^{2}}\) into (4.25), we have
$$\begin{aligned} P_{h}(x) =&\frac {1}{125(1+x^{2})^{6}}\bigl(1\text{,}470\text{,}000+840 \text{,}000h+233\text{,}800h^{2}-21\text{,}000h^{4}-5 \text{,}040h^{5} \\ &{}+\bigl(8\text{,}820\text{,}000+8\text{,}120\text{,}000h+3\text{,}111 \text{,}600h^{2}+601\text{,}200h^{3}-17\text{,}280h^{5} \bigr)x^{2} \\ &{}+ \bigl(22\text{,}050\text{,}000+28\text{,}000\text{,}000h+14\text{,}524 \text{,}800h^{2}+4\text{,}147\text{,}200h^{3}+674 \text{,}650h^{4} \\ &{}+63\text{,}410h^{5}+11\text{,}250h^{6}+900h^{7}\bigr)x^{4} \\ &{}+ \bigl(29\text{,}400\text{,}000+47\text{,}600\text{,}000h+31\text{,}659 \text{,}200h^{2}+11\text{,}757\text{,}000h^{3} \\ &{}+2\text{,}749 \text{,}000h^{4}+472\text{,}030h^{5}+78\text{,}300h^{6}+9\text{,}000h^{7}+432h^{8} \bigr)x^{6} \\ &{}+ \bigl(22\text{,}050\text{,}000+43\text{,}400\text{,}000h+35\text{,}437 \text{,}800h^{2}+16\text{,}399\text{,}800h^{3} \\ &{}+4\text{,}894 \text{,}375h^{4}+1\text{,}055\text{,}070h^{5}+197\text{,}235h^{6}+27\text{,}000h^{7}+1 \text{,}728h^{8}\bigr)x^{8} \\ &{}+ \bigl(8\text{,}820\text{,}000+20\text{,}440\text{,}000h+19\text{,}842 \text{,}000h^{2}+11\text{,}111\text{,}400h^{3}+4\text{,}114 \text{,}350h^{4} \\ &{}+1\text{,}079\text{,}240h^{5}+217\text{,}530h^{6}+32\text{,}850h^{7}+2 \text{,}592h^{8}\bigr)x^{10} \\ &{}+ \bigl(1\text{,}470\text{,}000+3\text{,}920\text{,}000h+4\text{,}416 \text{,}400h^{2}+2\text{,}922\text{,}600h^{3}+1\text{,}315 \text{,}325h^{4} \\ &{}+420\text{,}550h^{5}+94\text{,}905h^{6}+13\text{,}950h^{7}+1 \text{,}296h^{8}\bigr)x^{12}\bigr). \end{aligned}$$
(4.26)
Set \(P_{h}(x)=\frac{N_{h}(x)}{D(x)}\), obviously the denominator \(D(x)\) in \(P_{h}(x)\) is absolutely greater than zero. We need to prove that the numerator \(N_{h}(x)\) is over zero, which can be treated as a one-parametric family of polynomials with parameter \(h\in(0,\infty)\) and can be handled by Lemma 4.1. With the help of Maple or Mathematica, the discriminant of \(N_{h}(x)\) can be obtained as
$$\begin{aligned} \Delta =&-974\text{,}098\text{,}582\text{,}732\text{,}800 \text{,}000h^{60}\bigl(-5\text{,}250-3\text{,}000h-835h^{2}+75h^{4}+18h^{5} \bigr) \\ &{}\times\bigl(58\text{,}800+30\text{,}100h+10\text{,}825h^{2}+2 \text{,}040h^{3}+2\text{,}415h^{4}+900h^{5}+108h^{6} \bigr)^{4} \\ &{}\times\bigl(1\text{,}470\text{,}000+3\text{,}920\text{,}000h+4\text{,}416 \text{,}400h^{2}+2\text{,}922\text{,}600h^{3} \\ &{}+1\text{,}315\text{,}325h^{4}+420\text{,}550h^{5}+94 \text{,}905h^{6}+13\text{,}950h^{7}+1\text{,}296h^{8} \bigr) \\ &{}\times\bigl(10\text{,}527\text{,}552\text{,}696\text{,}056\text{,}300 \text{,}836\text{,}353\text{,}072\text{,}000\text{,}000\text{,}000\text{,}000 \\ &{}+39 \text{,}306\text{,}760\text{,}598\text{,}215\text{,}044\text{,}821\text{,}831 \text{,}490\text{,}000\text{,}000\text{,}000\text{,}000h \\ &{}+75\text{,}333\text{,}718\text{,}338\text{,}168\text{,}614\text{,}240 \text{,}153\text{,}258\text{,}910\text{,}000\text{,}000\text{,}000h^{2} \\ &{}+99 \text{,}618\text{,}621\text{,}855\text{,}054\text{,}289\text{,}426\text{,}838 \text{,}010\text{,}255\text{,}000\text{,}000\text{,}000h^{3} \\ &{}+102\text{,}779\text{,}684\text{,}628\text{,}263\text{,}794\text{,}779 \text{,}502\text{,}999\text{,}420\text{,}625\text{,}000\text{,}000h^{4} \\ &{}+88 \text{,}249\text{,}916\text{,}544\text{,}459\text{,}627\text{,}952\text{,}310 \text{,}998\text{,}095\text{,}312\text{,}500\text{,}000h^{5} \\ &{}+65\text{,}303\text{,}345\text{,}107\text{,}623\text{,}498\text{,}315 \text{,}297\text{,}763\text{,}904\text{,}460\text{,}937\text{,}500h^{6} \\ &{}+42 \text{,}548\text{,}078\text{,}554\text{,}608\text{,}716\text{,}720\text{,}415 \text{,}000\text{,}636\text{,}792\text{,}968\text{,}750h^{7} \\ &{}+24\text{,}860\text{,}972\text{,}299\text{,}032\text{,}069\text{,}597 \text{,}285\text{,}337\text{,}353\text{,}435\text{,}937\text{,}500h^{8} \\ &{}+13 \text{,}275\text{,}168\text{,}601\text{,}838\text{,}580\text{,}408\text{,}774 \text{,}342\text{,}743\text{,}598\text{,}046\text{,}875 h^{9} \\ &{}+6\text{,}584\text{,}544\text{,}306\text{,}510\text{,}110\text{,}821 \text{,}769\text{,}951\text{,}522\text{,}397\text{,}265\text{,}625h^{10} \\ &{}+3 \text{,}061\text{,}087\text{,}654\text{,}026\text{,}717\text{,}393\text{,}086 \text{,}768\text{,}379\text{,}532\text{,}656\text{,}250h^{11} \\ &{}+1\text{,}335\text{,}781\text{,}177\text{,}106\text{,}125\text{,}623 \text{,}220\text{,}777\text{,}826\text{,}553\text{,}984\text{,}375h^{12} \\ &{}+546 \text{,}581\text{,}319\text{,}914\text{,}479\text{,}884\text{,}670\text{,}825 \text{,}019\text{,}152\text{,}734\text{,}375h^{13} \\ &{}+209\text{,}902\text{,}282\text{,}786\text{,}161\text{,}951\text{,}852 \text{,}852\text{,}247\text{,}181\text{,}906\text{,}250h^{14} \\ &{}+75 \text{,}835\text{,}942\text{,}043\text{,}197\text{,}348\text{,}513\text{,}832 \text{,}530\text{,}299\text{,}890\text{,}625h^{15} \\ &{}+25\text{,}790\text{,}280\text{,}220\text{,}863\text{,}758\text{,}433 \text{,}223\text{,}640\text{,}939\text{,}343\text{,}750h^{16} \\ &{}+8 \text{,}230\text{,}241\text{,}786\text{,}176\text{,}630\text{,}163\text{,}097 \text{,}372\text{,}845\text{,}306\text{,}250h^{17} \\ &{}+2\text{,}452\text{,}965\text{,}279\text{,}312\text{,}714\text{,}362 \text{,}680\text{,}952\text{,}709\text{,}421\text{,}875h^{18} \\ &{}+680 \text{,}524\text{,}665\text{,}454\text{,}209\text{,}261\text{,}845\text{,}696 \text{,}176\text{,}096\text{,}875h^{19} \\ &{}+175\text{,}481\text{,}686\text{,}500\text{,}525\text{,}607\text{,}025 \text{,}019\text{,}806\text{,}094\text{,}375h^{20} \\ &{}+41\text{,}967 \text{,}547\text{,}117\text{,}323\text{,}743\text{,}984\text{,}292\text{,}464 \text{,}355\text{,}000h^{21} \\ &{}+9\text{,}253\text{,}162\text{,}584\text{,}359\text{,}279\text{,}323 \text{,}694\text{,}970\text{,}356\text{,}250h^{22} \\ &{}+1\text{,}860 \text{,}515\text{,}408\text{,}068\text{,}033\text{,}078\text{,}135\text{,}476 \text{,}990\text{,}000h^{23} \\ &{}+336\text{,}138\text{,}021\text{,}290\text{,}024\text{,}276\text{,}087 \text{,}312\text{,}149\text{,}500h^{24} \\ &{}+53\text{,}620\text{,}961 \text{,}897\text{,}451\text{,}577\text{,}435\text{,}040\text{,}772 \text{,}000h^{25} \\ &{}+7\text{,}399\text{,}996\text{,}680\text{,}369\text{,}897\text{,}120 \text{,}613\text{,}849\text{,}200h^{26} \\ &{}+861\text{,}847\text{,}511 \text{,}676\text{,}410\text{,}348\text{,}045\text{,}222\text{,}400h^{27} \\ &{}+82\text{,}038\text{,}790\text{,}520\text{,}521\text{,}056\text{,}364 \text{,}723\text{,}200h^{28} \\ &{}+6\text{,}107\text{,}780\text{,}869 \text{,}054\text{,}049\text{,}060\text{,}536\text{,}320h^{29} \\ &{}+332\text{,}825\text{,}048\text{,}384\text{,}446\text{,}232\text{,}248 \text{,}320h^{30} \\ &{}+11\text{,}808\text{,}121\text{,}415\text{,}612 \text{,}085\text{,}043\text{,}200h^{31} \\ &{}+205\text{,}521\text{,}196\text{,}952\text{,}300\text{,}027 \text{,}904h^{32}\bigr)^{2}. \end{aligned}$$
(4.27)
We take \(h_{0}=1\) to get that
$$\begin{aligned} N_{1}(x) =&2\text{,}517\text{,}760+20\text{,}635\text{,}520x^{2}+69 \text{,}472\text{,}210x^{4}+123\text{,}724\text{,}962x^{6}+123 \text{,}463\text{,}008x^{8} \\ &{} +65\text{,}659\text{,}962x^{10}+14\text{,}575 \text{,}026x^{12}>0, \end{aligned}$$
(4.28)
which ensures the assumption (i) in Lemma 4.1. We write (4.27) simply as
$$ \Delta=-P_{1}(h)P^{+}(h), $$
(4.29)
where \(P_{1}(h)\) see (4.10) and \(P^{+}(h)>0\) consists of the remainder of Δ. From the expression of \(P_{1}(h)\) we know that it has only one real zero \(h^{*}\) which lies in the interval \((3,4)\) and \(P_{1}(h)<0\) as \(h\in(0,h^{*})\), namely the discriminant (4.29) of \(N_{h}(x)\) is positive, which ensures that the assumption (ii) in Lemma 4.1 is satisfied.
Moreover, for all \(h\in(0,h^{*})\), the coefficient of the most order term \(x^{12}\) in \(N_{h}(x)\) is positive, which guarantees the assumption (iii) in Lemma 4.1.
When wave amplitude \(h\in(0,h^{*})\), the corresponding wave speed \(c\in (13, P_{2}(h^{*}))\), where \(P_{2}(h)\) see (4.11). Therefore, in summary, we have \(d''(c)>0\), this completes the proof of Theorem 4.2. □