数理解析研究所研究录

$\frac{\partial}{\partial T}v(A_{x}^{r}=0, I\iota_{y}’’=0, T)=0$
$v(It_{x}^{r}=0, I1_{y}’r=0_{\rangle}T)=v(I\zeta_{x}=0, I\iota_{y}^{\nearrow}=0, T=0)$
[2] (
Kad$\circ$mtsev-Petviashvili $(I<P)$
)
$KdV$
3
_{X}-6Qu^{2}u_{X}+u_{XXX})_{X}+ \frac{1}{9}u_{YY}=0$
(2)
Modified $K$ adomtsev-Petviashvili(MKP)
(4)
2
Tl 2\mbox{\boldmath $\pi$}
$x=(2\pi/L_{X})X,$ $y=(2\pi/L_{Y})Y$ (
$x,$ $y$
2
)
$x,$ $y$
( 2)
69
2
)
(4)
3
2:
–
$\kappa_{Y}$
128
(
$X,$ $Y$
$\theta=2\tan^{-1}(L_{X}/L_{Y})$
830 1993 65-74
65
– (Hidekazu Tsuji ) (Masayuki Oikawa)
1
Modified $KdV(MI<dV)$
Korteweg-de vries $(KdV)$ 3
[11
$u_{T}+6Puu_{X}-6Qu^{2}rx_{X}+u_{XXX}=0$
( $P,Q$
$\gamma=\gamma$
$\xi=x-t,$
$+\epsilon\beta(\gamma_{c}\equiv\sqrt{}\sigma\gamma$
$\tau=\epsilon^{2}t$ $\eta=\epsilon y,$
$\delta\simeq\epsilon$
\mbox{\boldmath $\zeta$} $=\zeta^{(0)}+\epsilon\zeta^{(1)}+\cdots$
$I\iota_{x}’’=0,$ $I\iota_{y}’’\neq 0$
$v=0$
y X
$v(I\backslash _{x}^{\nearrow}=0, I\acute{1}_{y}^{r}\neq 0, T)\equiv 0$
(5)
(2) (5)
70
3: $MKP$
$(\theta=143.1^{o})$
7: $MKP$
$(\theta=36.9^{o})$
$P=1.5,$ $Q=0.1,$ $\kappa=0.25,$ $L_{X}=100,$ $L_{Y}=$ 100*3, $T=100_{o}$
74
[1] T.Kakutani &N.Yamasaki: Solitary Waves on a Two-Layer Fluid,
$\eta=Y$
(2)
$P>0$
$P= \frac{3p_{c}}{2q_{c}}$
$Q= \frac{3r_{c}}{q_{c}}(>0)$
$P<0$
$uarrow-u$
$P>0$
(2)
3
)
wineberg [8] , Y $X$
$MKP$
Crank-Nicolson
(
$u_{1},$ $u_{2}$
$\triangle\equiv\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}$
,
$\nabla\equiv(\frac{\partial}{\partial x’}\frac{\partial}{\partial y})$
$u=u’+ \frac{P}{2Q}\text{
}$
)
(1), (2)
2
1
$y$
$z$
$z=-h_{2}$
(
1
2
$z=((x, y,t)$
$x$
$z=$ hl
$z=0$
) \rho 1, \rho 2
\mbox{\boldmath $\phi$}1, $\phi_{2}$
67
$h_{1}$ $\frac{a}{h_{1}}\equiv\epsilon\ll 1$
$\frac{\partial\zeta}{\partial t}+\triangle f_{2}+\frac{\epsilon}{\gamma}\nabla\cdot(\zeta\nabla f_{2})-\frac{\delta^{2}\gamma^{2}}{6}\triangle^{2}f_{2}+O(\epsilon\delta^{2})=0$
o
$(\theta=36.9^{\circ})$
( 4)
y
–
(
)
( $x$
)
y
$L_{Y}$
2
\theta
$KP$
$KdV$
$(i=1,2)$ $u_{1}=k_{Xi}^{2}sech^{2}(k_{Xi}X+k_{Yi}Y-\omega_{i}T)$
$MKParrow KP$
$MKdV$
(3)
$KdV$
$KP$
$( \zeta_{\tau}^{(0)}+\frac{3p_{c}}{2}\zeta^{(0)}(_{\xi}^{(0)}-3r_{c}(^{(0)2}\zeta_{\xi}^{(0)}+\frac{q_{c}}{6}\zeta_{\xi\xi\xi}^{(0)})_{\xi}+\frac{1}{2}\zeta_{\eta\eta}^{(0)}=0$
$\frac{\partial\zeta}{\partial t}-\triangle f_{1}+\epsilon\nabla\cdot(\zeta\nabla f_{1})+\frac{\delta^{2}}{6}\triangle^{2}f_{1}+O(\epsilon\delta^{2})=0$
)–
( 6) ““ MKP
(
$Y$
( 5) MKP MKP
1
3
\theta
$Q=0.1$ $Q=2.5$
$Q=2.5$
$KP$
MKP
MKP
( 7)
$Y$
3
73
6: $KP$
$(\theta=90.0^{o})$
$k_{X1}=k_{X2}=1/\sqrt{2},$ $k_{Y1}=-k_{Y2}=1/\sqrt{2},$ $L_{X}=50,$ $L_{Y}=50,$ $T=4.0$
$\Omega_{i}=-4\kappa_{Xi}^{3}-\kappa_{Yi}^{2}/4\kappa_{Xi}$
$\delta_{i}=-\frac{1}{2}\log\frac{P-2\kappa_{Xi}\sqrt{Q}}{P+2\kappa_{X\mathfrak{i}}\sqrt{Q}}$
$P=1.5,$ $Q=2.5,$ $\kappa=0.25,$ $L_{X}=100,$ $L_{Y}=$ 100/3, $T=6$
$I\iota_{x}^{\nearrow}=I\{_{y}’=0$
v
H|j
$\frac{\partial}{\partial T}\int\int udxdy=0$
\alpha
$\frac{h_{1}}{l}\equiv\delta\ll 1$
l
$h_{1}\simeq h_{2}$
\mbox{\boldmath $\phi$}1, $\phi_{2}$ \mbox{\boldmath $\delta$}2
\epsilon \mbox{\boldmath $\delta$}
J. Phys. Soc. Jpn.,45(1978) 674-679.
$\frac{\partial f_{1}}{\partial t}-\frac{\sigma}{\gamma}\frac{\partial f_{2}}{\partial t}+\frac{\epsilon}{2}(\nabla f_{1})^{2}-\frac{\epsilon\sigma}{2\gamma^{2}}(\nabla f_{2})^{2}$ $- \frac{\delta^{2}}{2}\frac{\partial}{\partial t}(\triangle f_{1})+\frac{\delta^{2}\gamma\sigma}{2}\frac{\partial}{\partial t}(\triangle f_{2})+O(\epsilon\delta^{2})=(1+\frac{\sigma}{\gamma})\zeta$
$L_{X},$ $L_{Y},$ $\kappa x$
\mbox{\boldmath $\kappa$}x \mbox{\boldmath $\kappa$}
$u$
$u \equiv I=-I\iota’+1\sum_{\mathfrak{i}_{xcut}’}^{It_{cut}’}\sum_{ycut}^{It_{cut}’}v(I1_{x}^{J}I\backslash ’=-I\zeta+17I_{1_{y}^{J}}rT)\cdot\exp\{i(I1_{x}^{y}Xr+I_{1_{y}^{\nearrow}}y)\}$