Title 非線形シュレディンガー方程式に対する3 次分散項の効果 ( 流体における波動現象の数理とその応用 ) Author(s) 及川 正行 Citation 数理解析研究所講究録 (1993) 830: 244-253 Issue Date 1993-04 URL http://hdlhandlenet/2433/83338 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
830 1993 244-253 244 3 (Masayuki Oikawa) 1 (NLS) $i \frac{\partial A}{\partial\tau}+\frac{1}{2}\omega (k_{0})\frac{\partial^{2}a}{\partial\chi^{2}}+\alpha A ^{2}A=0$ (1) 1) $\epsilon A(\chi \tau)e^{i(k_{0}x-\omega(k_{0})t)}$ $+cc$ $\chi=\epsilon(x-\omega (k_{0})t)$ $\tau=\epsilon^{2}t$ $\omega=\omega(k)$ a $\epsilon=ak_{0}\ll 1$ $k=k_{0}$ $k_{c}$ $\omega (k_{0})$ 3 $X=\epsilon^{2/3}(x-\omega (k_{0})t)$ $T=\epsilon^{2}t$ (1) $i \frac{\partial A}{\partial T}-\beta\frac{\partial^{2}A}{\partial X^{2}}- A ^{2}A=\dot{\iota}\gamma\frac{\partial^{3}A}{\partial X^{3}}$ (2) $\alpha$ $\beta=-\frac{\omega^{u}(k_{0})}{2\epsilon^{2/3}}$ $\gamma=\frac{1}{6}\omega^{u/}(k_{0})$ (3) $A$ 1 (2) MNLS (2) Akylas $\ Kung^{2)}$ Wai Chen Lee3) $\gamma$ MNLS ( ) NLS ( $\gamma$ \langle ) MNLS $\gamma$ 2 $A_{0}=a_{0}\exp(-ia_{0^{2}}T+ib_{0})$ ( $a_{0}>0$ $b_{0}$ : ) (4) MNLS $\hat{a}(x T)\hat{\theta}(X T)$ $A=(a_{0}+\hat{a})\exp(-ia_{0^{2}}T+ib_{0}+i\hat{\theta})$ $\hat{a}\hat{\theta}$ $(\hat{a}\hat{\theta})\propto e^{i(\kappa X-\Omega T)}$ (2) $\Omega=\gamma\kappa^{3}\pm\sqrt{\kappa^{2}(\beta^{2}\kappa^{2}-2\beta a_{0^{2}})}$ (5)
: $\gamma$ 245 (4) (i) $\beta<0$ $(\omega^{n}(k_{0})>0)$ $\Rightarrow$ (ii) $\beta>0$ $(\omega (k_{0})<0)$ $\Rightarrow$ $0<\kappa<\kappa_{c}\equiv\sqrt{2/\beta}a_{0}$ $\gamma$ ( $\gamma>0$ NLS $\gamma\neq 0$ ) 3 $\beta$ $\beta=01817\cdots$ $00936\cdots$ $\gamma=001$ $\gamma=0005$ 4 $\gamma=0$ $\gamma=$ $A(X 0)=1-01\cos(\kappa X)$ $0\leq X\leq L\equiv 2\pi/\kappa$ $L$ $\kappa$ $\kappa=s\kappa_{c}(s=07035$ 027 022 018) 5 $s=07035027022018$ $n=12345$ Yuen $\ Ferguson^{4)}$ NLS $0<\kappa<\kappa_{c}$ 5) 3 $I_{1}= \int_{0}^{l} A ^{2}dX$ $I_{2}= \dot{\iota}\int_{0}^{l}(a^{*}\frac{\partial A}{\partial X}-A\frac{\partial A^{*}}{\partial X})dX$ $I_{3}= \int_{0}^{l}\{\beta\frac{\partial A}{\partial X}\frac{\partial A^{*}}{\partial X}+\frac{i\gamma}{2}(\frac{\partial A^{*}\partial^{2}A}{\partial X\partial X^{2}}-\frac{\partial A}{\partial X}\frac{\partial^{2}A^{*}}{\partial X^{2}}I-\frac{1}{2} A ^{4}\}dX$ $H=iI_{3}$ MNLS MNLS $\frac{\partial A}{\partial T}=\frac{\delta H}{\delta A^{*}}$ $\frac{\partial A^{*}}{\partial T}=-\frac{\delta H}{\delta A}$ ( $\delta$ ) 4 1 2 3 4 $\gamma=0$ (NLS) $\gamma=00936\cdots$ $\gamma=001$ $\gamma=$ 0005 $ A $ ( ) $A$ ( ) $\beta=01817\cdots$ $L$ $2\pi$ $\xi=(2\pi/l)x$ XI-AXIS N-AXIS
246 ( ) NLS MNLS $(\ddagger\supset)$ NLS $(J\backslash )$ $(\gamma=0)$ $ A $ NLS $\gamma=00936\cdots$ NLS ( ) 6) ( ) $\gamma=00936\cdots$ $s=07$ $ A $ NLS $s=07$ $(+)$ $\gamma=001$ $s=07$ $ A $ $\gamma=00936\cdots$ $s=07035027022018$ $n=817222835$ ( ) $\gamma=0005$ $ A $ NLS $(+)$ $s=$ 07 035 027 022 018 $n=1631414961$ 5 Wai 3) MNLS $i \frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial^{2}u}{\partial x^{2}}+ u ^{2}u=\dot{\iota}\sigma\frac{\partial^{3}u}{\partial x^{3}}$ (6) $\sigma$ NLS 1 $1/(2\sigma)$ $(\dagger\backslash )$ ( ) $\xi$ MNLS (2) $k_{r}$ $k_{f}= \beta\sqrt{\frac{\beta}{2}}\frac{1}{\gamma s}=\frac{005477}{\gamma s}$ ( $\beta=01817\cdots$ ) (7) 1 $(+)$ ( )
247 1 (7) 1 $\sim 4$ $\overline{a}(x T)$ (2) (2) $A$ $\overline{a}(x T)+B$ $B$ $B$ $i \frac{\partial B}{\partial T}-\beta\frac{\partial^{2}B}{\partial X^{2}}-2 \overline{a} ^{2}B-\overline{A}^{2}B^{*}=i\gamma\frac{\partial^{3}B}{\partial X^{3}}$ (8) $T=0$ $B(X 0)$ (8) $d(t)= B(X T) $ ( $B(X T)$ $ B(X T) $ ) To $\lambda_{t}=\frac{1}{t-t_{0}}\log\frac{d(t)}{d(t_{0})}$ $Tarrow\infty$ $\lambda=\lim_{tarrow\infty}\lambda_{t}$ 7) $B(X 0)$ $d(t)$ $B(X T)$ $\lambda>0$ $\lambda$ $\lambda$ $\lambda=0$ 6 $\lambda=0$ 7 $\lambda>0$ NLS $\lambda=0$ 6 $\lambda=0$ $\lambda$ 2 $0$ $\lambda=0$ $\lambda>0$ 2
248 Wei $\gammaarrow 0$ $0$ NLS $\gammaarrow 0$ MNLS(2) 1) TTaniuti and NYajima :J math Phys 10(1969) 1369 2) TRAkylas and T-JKung : J Fluid Mech 214(1990) 489 3) PKAWai HHChen and YCLee : Phys Rev A41(1990) 426 4) HCYuen and WEFerguson Jr : Phys Fluids 21(1978) 1275 5) :FORTRAN 77 ( 1987) 6) MJAblowitz and BM erbst : SIAM J Appl Math 50(1990) 339 7) AJLichtenberg and MALieberman : Regular and Chaotic Dynamics 2nd ed (Springer 1991)
$\frac{\omega}{\aleph}\frac{<_{1}}{l}-$ 249 1 $ A $ $A$ $(\gamma=0)$
$- \backslash 5 ) \ovalbox{\tt\small REJECT}$ $c_{\lrcorner}$ $1^{\backslash } $ $\backslash _{\bigwedge_{\lrcorner }^{\wedge}\vee^{-}}\nwarrow\underline{l}\vee^{-}\vee$ $c\ulcorner$ $=$ } $ _{-}$ $ = $ $\cuparrow$ $- A^{1}-\frac{\infty}{\aleph}<$ 250 $r]$ $=$ $c= $ 1 $*^{\prime^{\backslash }}\cdot 0$ $o$ $0_{O}$ 2 $ A $ $A$ $(\gamma=00936\cdots)$
$-\wedge-<_{1}$ 251 3 $ A $ $A$ $(\gamma=001)$
$-\llcorner$ $\frac{\frac{\omega}{\aleph}\prec_{\}}}{-"}$ 252 $<$ 4 $ A $ $A$ $(\gamma=0005)$
253 5 $T=550$ $ A $ $(\gamma=0005 s=018)$ 6 $\lambda_{t}$ $\lambda$ ( $\lambda=0$ ) 7 $\lambda_{t}$ $\lambda$ ( ) $\lambda>0$