数理解析研究所講究録
913 巻 1995 年 17-30

17

STABILITY OF SOLITARY WAVESFOR THE ZAKHAROV EQUATIONS東京大学大学院数理科学研究科
太

雅人 (Masahito OHTA)

1. INTRODUCTION AND RESULTIn the present paper we consider the stability of solitary waves for the

Zakharov equations:$i \frac{\partial}{\partial t}u+\frac{\partial^{2}}{\partial x^{2}}u=nu$$\frac{\partial}{\partial t}n+\frac{\partial}{\partial x}v=0$,,$\frac{\partial}{\partial t}v+\frac{\partial}{\partial x}n=-\frac{\partial}{\partial x}|u|^{2}$

where $u,$$\mathbb{C},$$\mathbb{R}$

and

$n$

$\mathbb{R}$

and

$v$

,

$t>0$ ,

$x\in \mathbb{R}$

,

(1.1)

$t>0$ ,

$x\in \mathbb{R}$

,

(1.2)

$t>0$ ,

$x\in \mathbb{R}$

,

(1.3)

are functions on the time-space$\mathbb{R}\cross \mathbb{R}$with values in, respectively. From (1.2) and (1.3), we have$\frac{\partial^{2}}{\partial t^{2}}n-\frac{\partial^{2}}{\partial x^{2}}n=\frac{\partial^{2}}{\partial x^{2}}|u|^{2}$

.

(1.4)

The system of equations (1.1) and (1.4) was first obtained by Zakharov[20] as a model which describes the propagation of Langmuir turbulence

18

in a plasma. In this system,
and

$n$

$u$

denotes the envelope of the electric fieldis the deviation of the ion density from its equilibrium. On theother hand, $(1.1)-(1.3)$ was given by Gibbons, Thornhill, Wardrop andter Haar [4] from a Lagrangian formalism.

It is well known that $(1.1)-(1.3)$ has a two parameter family of solitary

wave solutions:$u_{\omega,c}(t, x)=\sqrt{2\omega(1-C^{2})}$sech$\sqrt{\omega}(x-Ct)\cdot\exp i(\frac{c}{2}x-\frac{c^{2}}{4}t+\omega t)$$n_{\omega,c}(t,x)=-2\omega$sech2$\sqrt{\omega}(x-Ct)$$v_{\omega,c}(t, x)=-2c\omega$sech2$\sqrt{\omega}(x-ct)$,, (1.5)
(1.6)

,

(1.7)

where $\omega>0$ and-l $<c<1$ . Our purpose in this paper is to show thestability of the solitary wave solution given by $(1.5)-(1.7)$ of $(1.1)-(1.3)$

for any $\omega>0$ and-l $<c<1$ .

There are a large amount of paper concerning the stability and in$s$

stability of solitary waves for the nonlinear Schr\"odinger equations (see,e.g., [2, 3, 7, 14, 15, 16, 18, 19] . However, to our knowledge, there are$)$

only a few results concerning the stability of solitary waves for coupledsystems of Schr\"odinger equations and other wave equations, except theabstract theory by Grillakis, Shatah and Strauss [8] and our recent re-

19

$s$

ults for the coupled nonlinear Schr\"odinger equations [10] and for the

coupled-Schr\"odinger equations [11].

$\mathrm{K}\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{n}-\mathrm{G}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{o}\mathrm{n}$We now state our main result.
Theorem 1.1. For any $\omega>0$ and $-1<c<1$ , the solitary wave
solu tion$(u_{\omega,c}(t), n_{\omega,C}(t),$sense: for any$\epsilon>0$$v_{\omega,C}(t))$

of$(\mathit{1}.\mathit{1})-(\mathit{1}.\mathit{3})$is stable in the following

there exists a $\delta>0$ such th at if $(u_{0}, n_{0,0}v)\in X$

verifies,$||(u_{0,0}n, v\mathrm{o})-(u_{\omega,c}(\mathrm{o}), n\omega,c(\mathrm{o}), v_{\omega,C}(\mathrm{o}))||_{X}<\delta$then th solu tion $(u(t), n(t),$ $v(t))$ of$e$$(u_{0}, n_{0,0}v)$

with$(u(\mathrm{O}), n(\mathrm{o}),$ $v(\mathrm{O}))=$

satisfies$\inf_{\theta,y\in \mathrm{R}}||(u(t), n(t),$$v(t))-(e^{i\theta}u_{\omega,c}(t, \cdot+y),$ $n_{\omega},c(t, \cdot+y),$ $v\omega,c(t, \cdot+y))||\mathrm{x}<\in$for any $t\geq 0$ , where$X=H^{1}(\mathbb{R})\cross L^{2}(\mathbb{R})\cross L^{2}(\mathbb{R})$Remark 1.2. For any ( $u0,$ no,$(u(\cdot), n(\cdot),$ $v(\cdot))\in L^{\infty}([0, \infty);x)$$(u_{0}, n_{0}, v0)$$(\mathit{1}.\mathit{1})^{-}(\mathit{1}.\mathit{3})$(see C. Sulem and$v_{0}$.

) $\in X$ , there exists a weak solutionof $(1.1)-(1.3)$ with $(u(0), n(\mathrm{o}),$$\mathrm{P}.\mathrm{L}$$v(\mathrm{O}))=$. Sulem [17]). We do not necessarily

have the uniqueness and the energy identity. However, by using themethod in Ginibre and Velo [5], we can find a weak solution satisfying$H(u(t), n(t),$ $v(t))\leq H(u_{0}, n_{0,0}v)$ ,$t\geq 0$

,

(1.8)

20

$N(u(t))=N(u_{0})$ ,$t\geq 0$$P(u(t), n(t),$ $v(t))=P$ ( $u0,$ no,$v_{0}$

),

,

(1.9)
$t\geq 0$,(1.10)

where$H(u, n, v)= \int_{-\infty}^{\infty}(|\frac{\partial}{\partial x}u|^{2}+n|u|^{2}+\frac{1}{2}n^{2}+\frac{1}{2}v^{2})dx$$N(u)= \int_{-\infty}^{\infty}|u|2d_{X}$,,$P(u, n, v)= \int_{-\infty}^{\infty}(i\overline{u}\frac{\partial}{\partial x}u-nv)d_{X}$.

For the Cauchy problem of the Zakharov equations, see also [1], [12] and[13].
Remark 1.3. Recently, Glangetas and Merle [6] proved the strong in-stability (instability by blow-up) of standing waves of the Zakharov equa-tions in two space dimensions.
In the next section, we give the proof of Theorem 1.1. We apply thevariational method introduced by Cazenave and Lions [3] to the coupledsystem of the Schr\"odinger equation and the wave equations as well as

in our previous papers [10] and [11]. In [3] they proved the stability ofstanding waves for some nonlinear Schr\"odinger equations. By a simple
inequality in Lemma 2.3 below, we reduce our problem for the Zakharov

equations to the case of the single nonlinear Schr\"odinger equation.

21

2. PROOF OF THEOREM 1.1
In what follows, we fix the parameter $c\in(-1,1)$ . First, we briefly re-call the proof by Cazenave and Lions [3] for the stability of standing wave$s$olution $u(t, x)=e^{i\omega t}\varphi\omega,C(x)$ of the nonlinear Schr\"odinger equation:$t>0$ ,,

$i \frac{\partial}{\partial t}u+\frac{\partial^{2}}{\partial x^{2}}u+\frac{1}{1-c^{2}}|u|^{2}u=0$

where$\varphi_{\omega,c}(x)=\sqrt{2\omega(1-C^{2})}$sech$\sqrt{\omega}x$$x\in \mathbb{R}$

,

(2.1)

. We consider the minimization

problem:u\in H^{1}(\mathbb{R}), N(u)=\mu\}$,$E^{1}(u)= \int_{-\infty}^{\infty}(|\frac{\partial}{\partial x}u|2-\frac{\mathrm{I}}{2(1-C^{2})}|u|^{4)dX}$,$I^{1}( \mu)=\inf\{E^{1}(u) :$\Sigma^{1}(\mu)=\{u\in H^{1}(\mathbb{R}) :E^{1}(u)=I^{1}(\mu), N(u)=\mu\}$

We note that $E^{1}(u)$ and $N(u)$ are the conserved

(2.2)

.

$\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\tilde{\mathrm{t}}\mathrm{i}\mathrm{e}\mathrm{s}$of (2.1).

The following two lemmas are crucial parts to prove the stability of thestanding wave of (2.1). We use them in the proof of Theorem 1.1 later.

Lemma 2.1. For any $\omega>0$ , we A

$av\mathrm{e}$$\Sigma^{1}(\mu(\omega))=\{e^{i\theta}\varphi\omega,C(\cdot+y) :where$\varphi_{\omega,c}(x)=\sqrt{2\omega(1-C^{2})}$sech$\mu(\omega)=N(\varphi_{\omega,C})=4(1-C^{2})\sqrt{\omega}$.

$\sqrt{\omega}x$\theta, y\in \mathbb{R}\}$

and

,

22

Lemma 2.2. Let $\mu>0$ . If$\{u_{j}\}\subset H^{1}(\mathbb{R})$and $N(u_{j})arrow\mu$ , then there exis
relatively compact in$H^{1}(\mathbb{R})$$\mathrm{t}s\{y_{j}\}\subset \mathbb{R}$sa tisfies $E^{1}(u_{j})arrow I^{1}(\mu)$such that$\{u_{j}(\cdot+y_{j})\}$

is.

Lemma 2.2 is proved by using the concentration compactness method
introduced by Lions [9]. For the proofs of Lemmas 2.1 and 2.2, see [3].

Rom the conservation laws of (2.1) and the compactness of any mini-

mizing sequence of (2.2), Lemma 2.2, one can easily show the stabilityof the set of minimizers$\Sigma^{1}(\mu)$for any $\mu>0$ . Moreover, the character-ization of the set of minimizers, Lemma 2.1, concludes the stability ofthe standing wave of (2.1) (for details, see [3]).

Following Cazenave and Lions [3], we consider the following minimization problem:$I( \mu)=\inf\{E(u, n, v) :(u, n, v)\in X, N(u)=\mu\}$ ,(2.3)$E(u, n, v)= \int_{-\infty}^{\infty}(|\frac{\partial}{\partial x}u|^{2}+n|u|^{2}+\frac{1}{2}n^{2}+\frac{1}{2}v^{2}-cnv)dx$,$\Sigma(\mu)=\{(u, n, v)\in X : E(u, n, v)=I(\mu), N(u)=\mu\}$ ,where$X=H^{1}(\mathbb{R})\cross L^{2}(\mathbb{R})\cross L^{2}(\mathbb{R})$. We note that$E(e^{-icx}u, n, v)/2=H(u, n, v)+cP(u, n, v)+ \frac{c^{2}}{4}N(u)$

.

(2.4)

23

The following lenuna plays an essential role in the proof of Theorem

1.1.
Lemma 2.3. For any $(u, n, v)\in X$ , we have $E^{1}(u)\leq E(u, n, v)$ . Moreover, th equality holds if and on $ly$ if $n=-(1/(1-c^{2}))|u|2$ and $v=cn$ .
$\mathrm{e}$Proof. Since$0\leq||u|^{2}+(1-C^{2})n|^{2}=|u|^{4}+2(1-c^{2})n|u|^{2}+(1-c^{2})^{2}n2$ ,(2.5)

we have$E(u, n, v) \geq\int_{-\infty}^{\infty}(|\frac{\partial}{\partial x}u|^{2}-\frac{1}{2(1-C^{2})}|u|4\frac{c^{2}}{2}n^{2}+\frac{1}{2}v^{2}-Cn+v)d_{X}$$= \int_{-\infty}^{\infty}(|\frac{\partial}{\partial x}u|^{2}-\frac{1}{2(1-C^{2})}|u|^{4}+\frac{1}{2}(cn-v)^{2)dx}$$\geq E^{1}(u)$

.

(2.6)

From (2.5) and (2.6), we see that the equality holds if and only if$n=-(1/(1-c^{2}))|u|2$ and $v=cn$ .

$\square$

The following lemma follows immediately from Lemma 2.3.
Lemma 2.4. For any $\mu>0$ , we $\mathrm{h}aveI(\mu)=I^{1}(\mu)$ and$\Sigma(\mu)=\mathrm{t}(u, n, v)$:$u\in\Sigma^{1}(\mu),$$n=- \frac{1}{1-c^{2}}|u|2,cv=n\}$.

24

Proof. We set$\Sigma^{0}(\mu)=\{(u, n, v)$

For$u\in\Sigma^{1}(\mu)$:$u\in\Sigma^{1}(\mu),$ $n=- \frac{1}{1-c^{2}}|u|^{2},$$v=cn\}$ .

, we have from Lemma 2.3$I(\mu)\leq E(u,$ $- \frac{1}{1-c^{2}}|u|^{2},$ $- \frac{c}{1-c^{2}}|u|2)=E^{1}(u)=I^{1}(\mu)\leq I(\mu)$Thus, we have $I(\mu)=I^{1}(\mu)$ and$\Sigma^{0}(\mu)\subset\Sigma(\mu)$.

.

Moreover, for $(u, n, v)\in\Sigma(\mu)$ , we have$I(\mu)---I^{1}(\mu)\leq E^{1}(u)\leq E(u, n, v)=I(\mu)$which implies that$u\in\Sigma^{1}(\mu)$from Lemma 2.3 that,and $E(u, n, v)=E^{1}(u)$ . Thus, it follows$\Sigma(\mu)\subset\Sigma^{0}(\mu)$. Hence, we have$\Sigma(\mu)=\Sigma^{0}(\mu)$.

$\square$

We note that from $(1.5)-(1.7)$ and Lemma 2.1, we have$e^{-icx/2}u\omega,c(t)\in\Sigma^{1}(\mu(\omega))$$n_{\omega,c}(t)=- \frac{1}{1-c^{2}}|u_{\omega,C}(t)|^{2}$for any$t\in \mathbb{R}$,,$v_{\omega,c}(t)=cn_{\omega,c}(t)$. Therefore, from Lemma 2.4, in order to show Theorem

1.1, we have only to prove the following proposition.

25

Proposition 2.5. For any $\mu>0$ , the set$A=\{(e^{i_{C}x/}u, n, v)2 :is$s\mathrm{t}\mathrm{a}ble$(u, n, v)\in\Sigma(\mu)\}$in the following sense: for any

that if $(u_{0}, n_{0}, v_{0})\in X$ verifies dist (tion $(u(t), n(t),$ $v(t))$ of$(\mathit{1}.\mathit{1})-(\mathit{1}.\mathit{3})$$\epsilon>0$$(u_{0},$

with

no,

there exis$v\mathrm{o}),$$A$)$(u(\mathrm{O}), n(\mathrm{o}),$$<\delta$$\mathrm{t}s$a $\delta>0$ such

, then the $sol$ u-$v(\mathrm{O}))=(u_{0}, n_{0}, v0)$

satisfies dist $((u(t), n(t),$ $v(t)),$ $A)<\epsilon$ for any $t\geq 0$ , where

dist $((u, n, v), A)= \inf\{||(u, n, v)-(u^{0}, n^{00}, v)||_{X} : (u^{0}, n^{0}, v^{0})\in A\}$ .

$\ln$order to prove Proposition 2.5, we need one lemma concerning thecompactness of any minimizing sequence of (2.3).
Lemma 2.6. Let $\mu>0$ . If
$I(\mu)$$\{(u_{j}, n_{j}, v_{j})\}\subset X$

and $N(u_{j})arrow\mu$ , then there exists$\{(u_{j}(\cdot+y_{j}), n_{j}(\cdot+y_{j}), v_{j}(\cdot+y_{j}))\}$

sa tisfies $E(u_{j}, n_{j}, v_{j})arrow$$\{y_{j}\}\subset \mathbb{R}$such th atis rela tively compact in $X$ .

Proof. From Lemma 2.3 and our assumption, we have $E^{1}(u_{j})arrow I(\mu)=$$I^{1}(\mu)$. Thus, from Lemma 2.2, there exists$\{u_{j}(\cdot+y_{j})\}$

is relatively compact in$\{y_{j}\}\subset \mathbb{R}$$H^{1}(\mathbb{R})$$u_{j}(\cdot+y_{j}),$ $n_{j}^{0}=n_{j}(\cdot+y_{j}),$ $v_{j}^{0}=v_{j}(\cdot+y_{j})$such that. Moreover, if we put, then$\{(u_{j}, n_{j}, vj)000\}$$u_{j}^{0}=$is bounded

in $X$ . Therefore, for some subsequence (still denoted by the same letter),26

we have$(u_{j’ j}^{0}n^{00}, v_{j})-(u^{0}, n^{0}, v^{0})$in$u_{j}^{0}arrow u^{0}$

weakly in $X$ ,$H^{1}(\mathbb{R})$.

Since $n^{2}+v^{2}-2cnv=(1-|c|)(n^{2}+v^{2})+|c|(n-(c/|c|)v)^{2}$ and $|c|<1$ ,we obtain$I( \mu)\leq E(u^{0}, n^{0}, v^{0})\leq\lim_{jarrow}\inf_{\infty}E(u_{j’ j}n^{0}, v_{j}^{0})0=I(\mu)$,from which it follows that

in $X$ ,$(u_{j}^{0}, n_{j}^{0}, v_{j}^{0})arrow(u^{0}, n^{0}, v^{0})$and$(u^{0}, n^{0}, v^{0})\in\Sigma(\mu)$.

$\square$

Proof of Proposition 2.5.

$\ln$what follows, we often extract subse-

quences without explicitly mentioning this fact. We prove by contradiction. If $A$ is not stable, then there exi

sequences $\{(u_{0j}, n_{0j}, v0j)\}\subset X$ and$s\mathrm{t}$a positive constant$\{t_{j}\}\subset \mathbb{R}$andsuch that(2.7)

dist $((u_{0j}, n0j, v_{0}j), A)arrow \mathrm{O}$ ,dist $((uj(tj), nj(t_{j}),$ $vj(tj)),$$\epsilon_{0}$$A)\geq\Xi_{0}$

,

(2.8)

27

where $(u_{j}(t), n_{j}(t),$$(u_{j}(\mathrm{o}), n_{j}(\mathrm{o}),$$v_{j(t)})$is a solution of $(1.1)-(1.3)$ with$v_{j}(0))=(u_{0j,0j,j}nv0)$. From the conservation laws $(1.8)-$(1.10), (2.4) and (2.7), we have$E(e^{-i/}ucx2j(tj), n_{j}(tj),$ $vj(t_{j}))\leq E(e^{-i/}cx2u0j, n0j, v0j)arrow I(\mu)$$N(e^{-icx/}u2j(tj))=N(u_{j}(t_{j}))=N(u_{0j})=N(e^{-i}cx/2u0j)arrow\mu$, (2.9). (2.10)

From (2.9), (2.10) and the definition of $I(\mu)$ , we have$E(e^{-i_{Cx}/2}uj(tj), n_{j}(tj),$ $vj(t_{j}))arrow I(\mu)$If we put$u_{j}^{1}(x)=e^{-i\mathrm{c}x/}u_{j}(2t_{j}, x),$$n_{j}^{1}(x)=n_{j}(tj, x),$then from (2.10), (2.11) and Lemma 2.6, there exists$(u_{j}^{1}(.

for some+y_{j}), n_{j}^{1} (.

+y_{j}),$ $v_{j}^{1}(.

$(u^{1}, n^{1}, v^{1})\in\Sigma(\mu)$.

(2.11)$v_{j}^{1}(x)=v_{j}(tj, x)$$\{y_{j}\}\subset \mathbb{R}$in $X$+y_{j}))arrow(u^{1}, n^{1}, v^{1})$. Since we have$u_{j}^{1}(X+yj)=e-icx/2e^{-}u_{j}j/iCy2(tj, X+y_{j})$it follows from (2.12) thatdist $((uj(tj), nj(t_{j}),$ $vj(tj)),$$A)arrow \mathrm{O}$which contradicts (2.8).
Hence, $A$ is stable. This completes the proof.

$\square$,,,such that(2.12)

28

3.
$1_{-\emptyset}^{\vee}\mathrm{f}\mathrm{f}\mathrm{l}_{J\mathrm{L}}^{g}\mathrm{b}\ovalbox{\tt\small REJECT}^{\mathrm{A}}=T-\ovalbox{\tt\small REJECT}\Phi \mathrm{b}\gamma’.k^{\sqrt}\prime \mathrm{b}$$/\mathrm{J}^{\rangle}\mathfrak{F}$

ffl%4 $l0$ Zakharov$[]^{\vee}$$1_{\overline{Q}}$$\langle$$\iota\tau_{\backslash }\mathrm{f}\mathrm{b}\Phi \mathrm{l}\underline{\mathrm{g}}\mathrm{X}\prime^{1}\neq*\Phi^{\prime b\pm}11\neq \mathfrak{o}\beta\ovalbox{\tt\small REJECT}\not\cong\ovalbox{\tt\small REJECT}\subsetneqq\emptyset$$\mathrm{A}^{\mathrm{o}}$$\Phi_{\mathrm{n}}^{=}\ovalbox{\tt\small REJECT}\epsilon$

クトノ R\Re \epsilon イ\acute J-

Zakharov$\iota\backslash \backslash$$\mathrm{b}T\ovalbox{\tt\small REJECT}\{1\prime x\emptyset \mathrm{p}\mathfrak{o}$$O\ovalbox{\tt\small REJECT} \mathfrak{F}\mathrm{g}\mathrm{X}\iota T\mathrm{t}\backslash \mathrm{g}_{\mathrm{t}}--\ \not\in\ovalbox{\tt\small REJECT}\check{\lambda}\tau$$arrowarrow\uparrow\vee\vee\vee.--\vec{\mathfrak{o}}\mathrm{E}-\iota\tau \mathrm{Y}/\rfloor\backslash \ovalbox{\tt\small REJECT}*\mathrm{g}[]^{arrow \mathrm{B}^{-\ovalbox{\tt\small REJECT}}}.-\Re\wedge f\backslash .,\iota\ovalbox{\tt\small REJECT}\tau 0$

(7) ス

. PH

$\mathfrak{B}\mathrm{E}\Re\circ?\mathrm{m}\backslash \mathrm{z}ae\mathbb{R}0\overline{\mathrm{r}}\not\in\prime \mathrm{E}[]arrow$. Y. Wu [21] . \d ag er$\mathfrak{F}\delta\backslash \mathrm{T}\backslash ^{\backslash }T\uparrow^{arrow}$Gf$1^{\backslash }\underline{\mathrm{g}}_{\hat{\mathrm{D}}}^{-}--\mathrm{E}$Eg.

$\ovalbox{\tt\small REJECT} \mathrm{b}f_{0}’$$x\ \ \mathrm{b}\tau \mathfrak{l}\mathfrak{x}_{\backslash }’\{,\mathrm{p}\# 4\mathrm{b}4Tffl\ovalbox{\tt\small REJECT}$

Grillakis, Shatah and Strauss$[8]\uparrow_{\iota}^{arrow}-\iota 6\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}\ovalbox{\tt\small REJECT}\infty\backslash$$X\mathrm{E}\Re[]arrow.\otimes_{I}\ulcorner^{\backslash }\llcorner\backslash \mathrm{b}\tau\ovalbox{\tt\small REJECT}\tau 6_{0}\pi\overline{\overline{-}}\mathrm{B}f\mathrm{g}1:\mathrm{b}\Phi T6\ ^{\mathrm{A}}\urcorner$$\mathrm{b}f^{\vee}.\mathrm{A}\cdot\check{2}r_{\tilde{\mathrm{A}}}\mathfrak{F}\#\#\backslash \mathrm{o}X\ \emptyset Xp\backslash ^{\mathrm{s}}\mathrm{E}\llcorner \mathrm{g}\mathrm{f}\mathrm{f}\mathrm{l}\backslash T\backslash b0_{\backslash }\mathrm{f}\ovalbox{\tt\small REJECT} \mathrm{R}\tau b6\mathrm{A}\cdot\check{2}[]arrow.\mathrm{f}\mathrm{f}\mathrm{l}*l\mathrm{L}\backslash \backslash )\mathrm{n}\epsilon 0$

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