数理解析研究所講究録
933 巻 1995 年 141-152141

長波短波相互作用方程式のパンルベ特性と
川積分性
Painlev\’e properties and integrability of thelong- and short-wave interaction equation
阪大基礎工吉永隆夫

Takao Yoshinaga
Department of Mechanical Engineering
Faculty of Engineering Science
University, Toyonaka, Osaka 560, Japan$\mathrm{O}_{\mathrm{S}}\mathrm{a}1\backslash \prime \mathrm{a}$AbstractThe
in the sense of Painlev\’e property is examinedin theand short- wave interaction equation. The equation described in a coupled form of the NLS equation with the K-dV equation
has only two parameters in theform. When the equationis reduced to the ODE through the traveling wave transformation. it
is shown to pass the Painlev\’e test for three cases of the parameters.
On the other hand, for these parameters, when the test is directly applied to the original PDE, it is found that two cases except for one do
not pass the test without any restrictions. However, the test is found. Furthermore, thenot to be successful in the nearly integrable
possibility of ’finite time
is discussed for a special caseof the parameters.
$i\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{t}\supset}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$1\mathrm{o}\mathrm{n}_{\Leftrightarrow}\sigma-$$\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}x_{\mathrm{z}\mathrm{e}\mathrm{d}}$$\mathrm{r}\mathrm{e}_{b}^{\sigma}\mathrm{i}\mathrm{o}\mathrm{n}$’$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\Leftrightarrow}^{\sigma}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$1Introduction.

In dispersive media, wave interactions play an important role in energy exchange
different two or more wave modes, if resonance conditions$\mathrm{a}\mathrm{m}\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$

142with respect to wave frequencies (and wave numbers) or wave velocities are
satisfied in these wave modes. The
and short-wave interaction is oneof such interactions and thought of as a special case of the three-wave interaction. [1] That is to say,awave
and twoshort waves, from the resonance condition of thethree-wave interaction given as$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma-$$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\min_{\mathrm{o}}\sigma$$(\triangle \mathrm{k}, \omega(\triangle k))$$\sin_{\mathrm{O}}\sigma 1\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}_{\circ}\sigma$$(\mathrm{k}\pm\triangle \mathrm{k}/2,\omega(k\pm\triangle k/2))$$\omega(\triangle k)=\omega(k+\triangle k/2)-\omega(k-\triangle k/2)$(1),

we can approximately obtain the following resonance condition between longand short waves:,(2)$\Delta/\mathrm{k}\cdot(\partial\omega/\partial \mathrm{k})|_{\mathrm{k}}\simeq\omega(/\Delta k)$

where
and
is assumed forfound to be equivalent to$\omega\ll 1$$k=|\mathrm{k}|$$\mathrm{v}_{p}\cdot \mathrm{v}_{g}\simeq v_{p}^{2}$or$\triangle/k(\ll k).\cdot$

The above condition is

,

(.3)

$v_{g}\cos\psi\simeq v_{p}$where the phase velocity of the
wave is given byvelocity of the short wave by
and the. Therefore,
if the above condition is satisfied. the interaction is possible between thetwo waves propagating in the different direction by . In particular, thiscondition is simplified towhen both waves propagate in the same
direction $(\psi=0)$ .
Such a resonance condition can be satisfied in water waves, plasma waves
and others in dispersive media. $[1]-[6]$ Although several nonlinear interaction
equations have been proposed for these waves, in this article, we deal with the, which is represented in a coupled form of the Nonlinear
following(NLS) equation with the Korteweg-de Vries (K-dV) equation: [7]$\mathrm{v}_{p}=\omega(\triangle k)\triangle \mathrm{k}/\Delta/k^{2}$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{v}_{g}=(\partial\omega/\partial \mathrm{k})|_{\mathrm{k}}$$\mathrm{o}\mathrm{P}\sigma \mathrm{r}\mathrm{o}\mathrm{u}$$\psi$$v_{g}\sim v_{p}$$\mathrm{e}\mathrm{q}_{\mathfrak{U}}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{r}\ddot{\mathrm{o}}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{r}$$\mathrm{i}S_{t}\pm S_{xx}=SL$,$L_{t}+\alpha LL_{x}+\beta L_{xxx}=|S|_{x}^{2}$,(4)

denote, respectively, the real
where
and
wave and the complex
amplitude of the envelope of the short wave, while and are spatial andtemporal coordinates in a frame of referencewith the phase velocityof the long wave or the group velocity of the short wave.
In the above equation, which is expressed in the normalized form with
only two parameters and , the parameters and the alternative of thesigns in front ofdepend upon the individual properties of the waves and
$L$

$S$

$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$

$x$

$t$

$\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{n}_{\circ}\sigma$$\alpha$$S_{xx}$$\beta$

$\pm$

143

the media concerned: [7] the gravity and capillary waves in a single layer fluid
and
, the
waves in a two-layer fluid (the ion acoustic and electron plasma waves
can be formally obtained if ,and so on. However, since the case, we will consider only theare replaced
and inin the followings.
caseand matheDepending upon the parameters and , physicalasfollows:
Whenboth andmatical properties of this equation can be said
represents the case when the magnitude of theare equal to zero,
wave is much less than that of the short wave $(|L|\ll|S|)$ . For this case,or to have the -soliton solution by
the equation is proved to betransform (IST) method. $[8, 9]$ On the othermeans of the inverse
hand, when both and have finite values, the equation represents the caseof theand short waves are of the same order
for which the$(|L|\sim|S|)$ . In this case, not only analytic solitary wave (one-soliton) solutions, but also a variety of numerical solitary wave solutions including ones
with oscillatory damped tails are found. [10] It is expected, however, that thetime asymptotic wave behavior may become chaotic for general initialwaves or soliton interactions, since the equation for $\beta=1$ is shown to be
IST [11]. Additionally, in the Hirota bilinear form
. $[11,12]$
for $a=-6\beta$ , the -soliton solution has not been found for
Nevertheless, for the nearlycase in the vicinity of $\alpha=\beta=0$ , it isnumerically shown that the wave behavior is regular orupon initial conditions and values of the parameters. [10]is shown to be non-integrable for the
As is seen in the above, though,
the integrability has not yet been analytically surveyed
particular and.
for all values of the parameters, in particular, in the nearly
Therefore, in this article, theofis examined in theparameter space by means of the Painlev\’e test, which is known as one oftheand practical techniques to test the integrability despite somedrawbacks. $[13, 14]$of this article is as follows: In section ‘2, the results
Theof the test are shown for the reduced ordinary differential equation (ODE)a variable transformation (Painlev\’e ODE test). In addition, they
the surface of section for particular parameters.
are confirmed by
In section 3, for the cases which pass the ODE test, the original partial
differential equation (PDE) is directly tested (Painlev\’e PDE test). And$(\alpha, \beta\leq 0\mathrm{a}\mathrm{n}\mathrm{d}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n})$$\beta\leq 0$$\circ\sigma \mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$-$(\alpha\geq 0, \beta\leq 0\mathrm{a}\mathrm{n}\mathrm{d}+\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n})$$\mathrm{s}\mathrm{i}\sigma \mathrm{n})\circ$’$t$$\mathrm{o}\mathrm{f}-\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n}$$L$$\mathrm{b}.\mathrm{v}-t,$$\mathrm{e}\mathrm{q}.(4)$$\beta$$-L\mathrm{a}\mathrm{n}\mathrm{d}-\grave{\beta}$$\mathrm{o}\mathrm{f}+\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n}$$\alpha$$\beta$$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{n}_{\circ}\sigma \mathrm{S}$$\alpha$$\mathrm{e}\mathrm{q}.(4)$$\beta$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{n}$$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}_{\circ}\sigma$$\alpha$$\beta$$\mathrm{m}\mathrm{a}_{\mathrm{o}}\sigma \mathrm{n}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}\mathrm{S}$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\circ}\sigma_{\Gamma \mathrm{a}\mathrm{b}}1\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}_{\circ}\sigma \mathrm{h}$$\mathrm{n}$$\alpha,$$\beta\neq 0$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}^{\Gamma \mathrm{a}\mathrm{b}}}\sigma 1\mathrm{e}$$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\Gamma \mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{e}\mathrm{q}.(4)$$\alpha$$\beta$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$\mathrm{u}\mathrm{S}\mathrm{e}\mathrm{f}\mathfrak{U}1$$\iota$$\mathrm{o}\mathrm{r}_{\mathrm{o}}\sigma \mathrm{a}\mathrm{n}\mathrm{i}_{\mathrm{Z}}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}$$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}\dot{\mathrm{u}}\mathrm{n}\mathrm{i}\mathrm{n}_{\circ}\sigma$$\mathrm{e}\mathrm{q}.(4)$$(\alpha, \beta)$

144

finally, in section 4, we remark the validity of the test in the nearly integrable
region and the possibility of the ’finite time integrability’.

Painlev\’e ODE test.

2

For the Painlev\’e ODE test, we first reducewave transformation:

to the ODE$\mathrm{e}\mathrm{q}.(4)$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{o}\sigma \mathrm{h}$the$\mathrm{f}\mathrm{o}11\mathrm{o}\mathrm{W}\mathrm{i}\mathrm{n}_{\mathrm{o}\mathrm{O}}\sigma \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\sigma$$S=f(\zeta)\exp[\mathrm{i}(\lambda/2)(x-Vt)]$,$L=g(\zeta)$,

where andare constants. Substituting (5) into
andwith respect to , we can easily obtain the reduced ODE$\lambda$(5)$(\zeta=x-\lambda t)$$V$$\mathrm{e}\mathrm{q}.(4)$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}_{\circ}\sigma_{\mathit{9}}$$\zeta$$f_{(\zeta}+(\lambda/2)(V-\lambda/2)f=fg$,$\beta g_{\zeta\zeta}+(\alpha/2)g^{2}-\lambda g=f2-c^{2}$

where we have imposed the boundary conditions: $farrow C$ (const.),, and $\lambda=2V$ for $C\neq 0$ .
as
use of the fol owin\mbox{\boldmath $\sigma$}o variable transformation into:

$g,$ $g_{\zeta},g_{\dot{\mathrm{t}}\zeta}arrow 0$(6),$f_{\zeta},$$f_{\zeta(}$,$|\zeta|arrow\infty$$\mathrm{e}\mathrm{q}.(6)$$\mathrm{M}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{O}}\sigma$$garrow(2/\beta)^{1/2}g$,$(arrow(\beta/2)^{1/4}\zeta,$$(7)$

we can show that our system has H\’enon-Heiles Hamiltonian$H=(1/2)[f_{\zeta}2g_{\zeta}^{2}+]+I(f, g)$

(8)

,

where
.

$I=(\beta/2)^{1/2}(\lambda/4)(V-\lambda/2)f^{2}-(2/\beta)^{1/2}(\lambda/4)g^{2}-(f^{2}-c^{2})g/2+\alpha g^{3}/(6\beta)$

Since the Painlev\’e properties ( -properties) in the above system have been
examined by
et al. [15] for $\beta>0$ and $C=0$ , it is expected that
our ODE has similar
structures. In fact, it is found thathas$\mathrm{P}$$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{e}\mathrm{q}.(6)$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$similar -properties. [10]According to the procedure of the test by Ablowitz et $al,$ $[16]$ the solutionsofare expanded in theLaurent series:$\mathrm{P}$$\mathrm{e}\mathfrak{c}_{1}.(6)$$\mathrm{f}\mathrm{o}\mathrm{l}1_{0}\mathrm{W}\mathrm{i}\mathrm{n}_{\circ}\circ$$.f=( \zeta-\zeta_{0}.)^{-}a\sum_{=j0}\infty fj(\zeta-\zeta 0)^{j}$,$g=( \zeta-(0)-b\sum_{j=0}gj(\zeta-(.0\mathrm{n}\urcorner)^{j}.$

(9)

145

Table 1: Painlev\’e ODE Testdenotes an arbitrary movable singularity depending upon initialwhereand equating coefconditions. Substituting the above expression into
ficients of powers of , we can obtain the leading orders and for $j=0$ , andfor $j\geq 1$ . From the recurand
the recursion relations with respect tobecome arbitrarysion relations, we can see that the coefficientsfor particular values of $j=r$ , which is called resonances. The resonances forto the arbitrariness of and$r=-1$ and are, respectively,
resonances for $r<-1$ are ignored. [18] For(go), though
and are required, at least, to be integers, which
the -property, theseshould be of the pole type or the single-valued.
solutionsmeans that thethe candidates for the -property are limited toThen, Table I showsand . It is found in this table that the case
of
a
cases
three
$\alpha=\beta=0$ has only
solution, while the other cases have both gensolution
solutions in pairs. In these solutions, theeral andlneans that the equation has equal arbitrary parameters to the order of thesolution means that the solution has less arbiequation, while thetrariness than the order of the equation. However, in order for these threecandidates to have the -property, the self-consistency of the resonance must$\zeta_{0}$$\mathrm{e}\mathrm{q}.(6)$$a$$\zeta$$f_{j}$$b$$g_{j}$$f_{i}\mathrm{o}\mathrm{r}/\mathrm{a}\mathrm{n}\mathrm{d}gj$$0$$f_{0}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{P}$$\zeta_{0}$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\sigma 0$$\mathrm{n}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{a}\mathrm{t}\mathrm{i}\mathrm{V}\mathrm{e}$

$a,$

$b$

$r$

$\mathrm{P}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}_{\wedge}$$\beta$$\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{t}$$\circ\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{P}$146

be checked in the recursion relations. Resultin from this, it is finally foundthat the Case I for $\alpha=-\beta$ has the -property under the restrictions thateither $V-\lambda/2+2/\beta=0$ for $C=0$ or $V=\lambda=0$ for. while the othercases have -property without any restrictions.
The results of the ODE test can be confirmed by examining the surfacesof section for the H\’enon-Heiles system (8) when $\beta>0$ and $C=0$ .
phase trajectories in this system move through the four-dimensional phasespace $(f, f_{(,g,g\zeta})$ , we can construct the two- dimensional surface of section, by
the phase space at $f=0$ andthe trajectories withfor the fixed total. In the, typical examplesof the surfaces of section are shown for bothandcases:
shows the sections for the integrable case $\alpha=-6\beta$ , where$\alpha=-2,$ $\beta=1/3,$ $\lambda=2$ and $V=1/2$ are taken.
and (b),respectively, show the surfaces whenand 0.262. We can see thatthe closed smooth curves areon the surface even if the energy increases,
which means that the trajectories move on the tori in thephasespace even in the nonlinear. The situation on the
motion of
the trajectories is similar when
, if the condition $V-\lambda/2+2/\beta=0$
is satisfied. Figure 2 shows this case, where we take $\alpha=-4,$ $\beta=4,$ $\lambda=2$
and $V=1/2$ . As is seen in bothforand (b) for,even if the energy increases, the smooth curves are retained on the surface,which means that the motion of the trajectories is. However, if thiscondition is not satisfied, the motion of the trajectories arein the$\lambda=2$
nonlinear. This is shown in Fig.3, where $\alpha=-1/\cdot 3,$$V=1/‘
2$
and. It is found from both Figs.(a) forand (b) forthat the smooth curves are partly replaced byscattered points, whentheincreases. Furthermore, in Fig.4 for, we can illustrate oneof the examples which do not pass the test and show the large regions ofchaotic motion, where we take $\alpha=\beta=1/\cdot 3,$
and $V=1/2$ .
closed smooth curves areon the surface for sufficiently small, when the energy increases to, the curves
become vague due to scattering points
them. Finally, when, allsmooth curves disappear and random spread of points are found all over thesurface within the maximum energy shell.
\mbox{\boldmath $\sigma$}$\circ$$\mathrm{P}$$C\neq 0_{\text{ノ}}$$\mathrm{P}$$\mathrm{A}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}$$(g, g_{\zeta})$$\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{C}\mathrm{i}\mathrm{n}\mathrm{o}\sigma$$\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$f_{\zeta}>0$$\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}_{\mathrm{o}}\sigma \mathrm{y}E(=H)$$\mathrm{f}\mathrm{o}\mathrm{l}1_{\mathrm{o}\mathrm{W}}\mathrm{i}\mathrm{n}\circ\sigma \mathrm{s}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{e}$$\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}1$$\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}\mathrm{e}\mathrm{s}(\mathrm{a})$$\mathrm{E}=0.0072$$1.\mathrm{v}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{o}\mathrm{r}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{n}\mathrm{a}1$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{m}\mathrm{e}$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\alpha=-\beta$$\mathrm{E}=0.0.36$$\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{s}.(\mathrm{a})$$\mathrm{E}=0.216$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\beta=1/\cdot 3,$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{m}\mathrm{e}$$\mathrm{E}=3.79$$\mathrm{E}=7.79$$\mathrm{v}\mathrm{a}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{e}\mathrm{l}\mathrm{y}$$\alpha=\beta$$\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}_{\mathrm{O}}\sigma \mathrm{y}$$\lambda=2$$\mathrm{A}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}_{\mathrm{O}}\sigma \mathrm{h}$$1\mathrm{y}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}_{\mathrm{o}}\sigma \mathrm{y}$$\mathrm{E}=0.05(\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma.(\mathrm{a}))$$\mathrm{E}=0.2(\mathrm{F}\mathrm{i}\mathrm{g}.(\mathrm{b}))$$\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$(\mathrm{F}\mathrm{i}_{\mathrm{o}}\sigma.(\mathrm{C}))$$\mathrm{E}=0.5$

147

(b)

(a)
$\mathrm{F}\mathrm{i}_{\mathrm{o}}^{\sigma \mathrm{u}\mathrm{r}\mathrm{e}}1$: Surface of section for the integrable case $\alpha=-6\beta,$$-2,$ $\beta=1/3,$$\lambda=2$

and $V=1/2:(\mathrm{a})\mathrm{E}=0.0072,$$(\mathrm{b})\mathrm{E}=0.262$)$\mathrm{w}\mathrm{l}$$\mathrm{e}\mathrm{r}\mathrm{e}\alpha=$

.

(b)

(a)

Figure 2: Surface of section for the integrable case $\alpha=-\beta$ when $V-$,$\lambda/2+2/\beta=0$ , where $\alpha=-4,$ $\beta=4,$
and $V=1/2$ : (a)$\lambda=2$$(\mathrm{b})\mathrm{E}=0.216$.

$\mathrm{E}=0.0036$

148

(a)

(b)

Figure 3: Surface of section for the non-integrable case $\alpha=-\beta$ when $V-$$\lambda/2+2/\beta\neq 0$ , where $\alpha=-1/3,\beta=1/3,$
and $V=1/2:(\mathrm{a})\mathrm{E}=.3.79$ ,.
$\lambda=2$$(\mathrm{b})\mathrm{E}=7.79$

(a)

(b)

(c)

Figure 4: Surface of section for the non-integrable case$\beta=1/3,$
and $V=1/2:(\mathrm{a})\mathrm{E}=0.005,$$\lambda=2$$(\mathrm{b})\mathrm{E}=0.2,$, where$\alpha=\beta$$(\mathrm{c})\mathrm{E}=0.5$.

$\alpha=$149

Painlev\’e PDE test.

3

It is known that the test in the reduced ODE gives only necessary conditions. $[16]$ In other words, a
PDE to be completely
for thewhen the ODE reduced from the PDE does
PDE is not completel.Vof theTherefore,
in this section, the-property.
thenot haveoriginal PDE is directly examined for the three cases that pass the ODE test
section.
in theLet us apply the Painlev\’e PDE test, whose direct procedure was introduced by Weiss et al. [17] In this test, a given partial differential equation is-valued in the neighsaid to have the P- property if the solutions are
manifold. Sinceborhood of the arbitrary and analytic (movable)with respectthemanifold for the ODE reduces to thevariable, the PDE test may be considered as a
to aextention of the ODE test with similar procedure. For convenience,in the fol owin\mbox{\boldmath $\sigma$}o form:$0\sigma \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{o}\mathrm{r}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{n}\mathrm{a}1$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{P}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{C}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{n}_{\circ}\sigma$$\sin\sigma \mathrm{l}\mathrm{e}\mathrm{o}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\sin_{\mathrm{O}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{h}\mathrm{t}\mathrm{f}_{\mathrm{o}\mathrm{r}}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{d}$$\sin\sigma \mathrm{l}\mathrm{e}\mathrm{o}$$\mathrm{r}\mathrm{e}\mathrm{w}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{e}\mathrm{q}.(4)$$\mathrm{i}n_{t}+u_{xx}=u\omega$,$-\mathrm{i}v_{t}+u_{xx}=vw$,$\omega_{t}+\alpha ww_{x}+\beta w_{xxx}=(uv)_{x}$

(10)

,

the solutions are set as$u= \phi^{-a}\sum_{j=0}u\infty j\varphi^{1j}$,$v= \phi^{-b}\sum_{j=0}v_{j}\phi^{j}\infty$

,

(11)

$w=( \delta^{-\mathrm{c}}\sum w_{j}\phi^{j}j=\infty 0^{\cdot}$

and
order
, we can deterlnine theuse of (11) into
and the resonances like in the ODE test, whose values are integers for thesame three cases of and as in Table I. The results of the PDE test are
shown in Table II, where the case $\alpha=\beta=0$ have only general solution, while.
andthe other two cases have both
it is finallyresonances,
the recursion relations for the self-consistency of thefound that the case of $\alpha=\beta=0$ and the Case II for $\alpha=-\beta$ hold the-property without any restrictions. The latter case, however, is excluded in
to vanish closely
the present context, since the solutions and are$\phi=0$solution isnear themanifold. Consequently, theonly which isbut that of the K-dV equation, where the resonances
occur for $r=-1,4,6$ . On the other hand, the other cases have the P-property
through thewave transformation like $\phi=x-ct$ ( :const.), that is
to say, the -property is conditional. Thus, only the case of a $=\beta=0$ is$\mathrm{e}\mathrm{q}.(10)$$\mathrm{M}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{O}}\sigma$$a,$$1\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

$b$

$r$

$c$

$\alpha$$\beta$$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}_{\mathrm{S}}\mathrm{o}\mathrm{I}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$[19]\mathrm{C}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{P}$$u$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$

$w$

$v$

$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{t}$$\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}_{\circ}\sigma$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\circ\circ$$\mathrm{P}$

$c$

150

completely

4$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$, which is consistent with the result of IST method. $[8, 9]$Concluding remarks.

We can see in Table II that the leading orders and some coefficients in the
expansions are coincident or adjustable between the completely
case $\alpha=\beta=0$ and the case for $\alpha=-6\beta$ (Case II). Although thisthat these two cases are closely related to each other, the test is found not tobe successful in the nearly
for $\alpha=-6\beta$ , since the
manifold expansions become non-uniformly valid when tends tozero. This non-unifo,rmity may be due to the small parameter in theorder derivative term in. Additionally, since there exists one-soliton$[10]$
solutions which are uniformly valid forthe usualmanifold expansions (9) and (11) is not appropriate to
examine thein this region.
On the other hand, in thesolution for $\alpha=-6\beta$ (Case II), we)should remark that thecondition that pernlits the P-property$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\circ\cdot \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$\mathrm{s}\mathrm{u}\mathrm{g}_{\mathit{0}}\sigma \mathrm{e}\mathrm{S}\mathrm{t}_{\mathrm{S}}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{o}\mathrm{n}\alpha,$$\beta\sim 0$$\beta$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\beta$$\mathrm{h}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{h}\mathrm{e}\mathrm{S}\mathrm{t}$$\mathrm{e}\mathrm{q}.(4)$$\alpha=-6\beta \mathrm{i}\mathrm{n}\mathrm{C}\mathrm{l}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma\alpha=\beta=0,$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$$\circ\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$$\mathrm{c}\mathrm{o}\mathrm{n}$$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$151

is found to be relaxed considerably for a finite time. That is to say, since thecompatibility condition for the -property is written as$\mathrm{P}$$\mathrm{s}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{n}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{t}$$\theta_{t}-\theta\theta_{x}=0$. the$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}\theta=\phi_{t}/c)\mathcal{I}_{\text{ノ}}$$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$

(12)

,

solution of the above wave equ.ation$\theta=(x \dagger t\theta)$

.

(13)

deis analytic for a finite time depending upon initial conditions, where
notes an arbitrary function. Therefore, for a certain class of which is given, the compatibility condition (12) can be satisfiedby (13)
which the solution (13) is analytic and arbitrary.
for a finite time
This means that the equation holds the -property for the finite time and is
expected to have multi-soliton solutions for the time. As a special case, it iseasil.v seen that the condition (12) is identically satisfied for an infinitely
wave transformation $\mathit{0}’=x-ct$ , which is confirmed
time under theone
by the existence of one-soliton solution. [10] Thus, for $\alpha=-6\beta,$
time., the soliton interactions due
soliton state is valid for an infinitely
to multi-soliton state
be elastic for the finite time, that is to say, thepossibility of ’finite timeis expected.
$\ominus$$\acute{\varphi}$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}\theta=\phi_{t}/\Phi_{x}’$$\mathrm{d}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{P}$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{Y}^{-}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}$$1\mathrm{o}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{n}\dot{\mathrm{u}}_{\mathrm{o}}\sigma \mathrm{h}\mathrm{t}$’$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$References[1] D.J. Benney, Stud. Appl.Math.55 (1976)93.

[2] A.D.D.Craik. PVaue interactions and fiuid fiows (Press, 1985).

$\mathrm{C}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{d}_{\mathrm{o}}\sigma \mathrm{e}$University[3] G.D.Crapper. Introduction to water waves (Ellis Horwood, 1984).

[4] V.E.Zakharov, Sov.Phys.35(1972)908, and see also Sov. J. Eksp.Theor.
Phys.62(1972)1745.
[5]$\perp l\mathrm{I}.\mathrm{V}$$[6].\mathrm{J}.\mathrm{D}$. Goldman, Rev. Mod.Phys.56(1984)709.

. Gibbon, Phil.Trans.R. Soc.Lond.

$\mathrm{A}315(19\mathrm{s}5).335$

.

152

[7]

, M.Wakamiya and T.Kakutani, Phys.Fluids $\mathrm{A}3(1991)8.3$ ,and see also
and T.Kakutani, J.Phys.Soc..Jpn. 63 (1994)445
and the references therein.
$\mathrm{T}.\mathrm{Y}_{\mathrm{o}\mathrm{S}}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{o}\sigma \mathrm{a}$$\mathrm{T}.\mathrm{Y}_{0}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{a}\sigma \mathrm{a}\mathrm{o}$[8] N.Yajima and M.Oikawa,$\mathrm{P}\mathrm{r}\mathrm{o}^{\sigma}.\mathrm{T}\mathrm{o}\mathrm{y}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}.\mathrm{P}\mathrm{h}\mathrm{s}.56(1976)1719$.

[9] Y-C.Ma, Stud. in Appl.Math.59(1978)201.

[10]

and T.Kakutani, J.Phys.Soc.Jpn.63(1994)445.

$\mathrm{T}.\mathrm{Y}^{r}\mathrm{o}\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{a}\sigma \mathrm{a}\circ$[11] E.S.Benilov and S.P.Burtsev, Phys.Lett. $\mathrm{A}98(198.3)256$ .
[12]$\mathrm{T}.\mathrm{Y}_{\mathrm{o}\mathrm{S}}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{a}_{\mathrm{o}}\sigma \mathrm{a}$, Proc.Estonian Acad.Sci.Phys.Math.,44 (1995) 96.

[1.3] M.Tabor, Chaos and integrability in nonlinear dynamics (John Wiley&
Sons, New York, 1989).

[14] M.J.Ablowitz and P.A.Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (London VlIathematical Society Lecture
Univ.Press,
Note Series, 1991).
$149,\mathrm{C}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{d}_{\mathrm{o}}\sigma \mathrm{e}$[15]$\mathrm{Y}.\mathrm{F}.\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{C}^{\mathrm{t}}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{d}\mathrm{o}\sigma \mathrm{e}$, M.Tabor and J.Weiss, J.Math.Phys.23(1982)531.

[16] M.J.Ablowitz, A.Ramani and$[17].].\mathrm{t}\mathrm{V}\mathrm{e}\mathrm{i}_{\mathrm{S}\mathrm{s}}$, J.Math.Phys.21 (1980) 715.

, M.Tabor and G.Carnevale, J.Math.Phys.24(1983) 522.

[18] R.Conte, A.P.Fordy and A.
references therein.
[19] T.

$\mathrm{H}.\mathrm{S}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{u}\mathrm{r}$$\mathrm{Y}\mathrm{o}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{a}\sigma \mathrm{a}\mathrm{o}$’, Physica D69 (1993)33 and the$\mathrm{P}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$

to appear in J.Phys.Soc.Jpn.64(1995).