数理解析研究所講究録
第 1989 巻 2016 年 170-178

170

MODIFIED KDV AND ASYMPTOTICS OF SOLUTIONSNAKAO HAYASHI AND PAVEL I. NAUMKIN

1. INTRODUCTION

We survey the large time asymptotics of solutions to the Cauchy problem for themodified Korteweg-de Vries $(KdV)$ equation(1.1)$\{\begin{array}{l}\partial_{t}u-\frac{1}{3}\partial_{x}^{3}u=\lambda\partial_{x}(u^{3}) , t>0, x\in R,u(O, x)=u_{O}(x) , x\in R\end{array}$with mass condition $\int_{R}u_{O}(x)dx\neq 0,\lambda\in R$
Large time asymptotics of solutions to the generalized Korteweg-de Vries equation$\partial_{t}u-\frac{1}{3}u_{xxx}=\partial_{x}(|u|^{\rho-1}u)$

was studied by W. A. Strauss [16], [17], S. Klainerman [11], S. Klainerman and G.
Ponce[12], J. Shatah [15] G. Ponce and L. Vega [14], F.M. Christ and M.I. Weinstein[2], and ours [5] for diﬀerent values of in the super critical region $\rho>3$ . Stabilityof solutions in the neighborhood of solitary waves was shown by T. Mizumachi [13].
For the $KdV$ equation and the modified $KdV$ equation (1.1), the Cauchy problem
was solved by the Inverse Scattering bansform method and thus the large time
asymptotic behavior of solutions was studied (see [1], [3]). This method depends
on the nonlinearity in the equation and note that if we replace nonlinear term by$a(t)\partial_{x}(u^{3})$ with $|a(t)|\leq C$ it
does not work. Therefore it is important to develop
alternative methods for studying the large time asymptotics of solutions to the
Cauchy problem (1.1).
To state our results precisely we introduce Notation and FUnction Spaces. The
weighted Sobolev space is$\rho$$H^{k,s}=\{\varphi\in S’;||\phi||_{H^{k,s}}=\Vert\langle x\rangle^{s}\langle i\partial_{x}\rangle^{k}\phi\Vert_{L^{2}}<欧禾\infty\},$$k,$ $\mathcal{S}\in R,$ $1\leq p\leq\infty,$$H^{k}=H^{k,O}$$\langle x\rangle=\sqrt{1+x^{2}},$$\langle i\partial_{x}\rangle=\sqrt{1-\partial_{x}^{2}}$. We also use the notation

shortly.

2. ZERO TOTAL MASS CASE
In [6], we showed the large time asymptotics of solutions to (1.1) in the case ofsmall real-valued initial data $u_{O}\in H^{1,1}$ with zero total mass assumption $\int_{R}u_{O}(x)dx=$
O. We have the asymptotics$u(t, x) = \sqrt{2\pi}t^{-*}R\epsilon Ai(xt^{-{\}})\hat{u_{+}}(x)\exp(-3i\pi|\hat{u_{+}}(x)|^{2}\log t)$(2.1)$+O(t^{-*-\lambda})$171
N. HAYASHI AND P. NAUMKIN

for large time , where $0< \lambda<\frac{1}{21},$ $x=(x/t)^{l}1$ ,, is such that $\hat{u_{+}}(O)=0$ , andthe data$t$$\hat{u_{+}}\in L^{\infty}$

is uniquely defined by$u_{O}$$Ai(x)= \frac{1}{\pi}\int_{0}^{\infty}e^{ix\xi-\xi\xi^{3}}d\xi$

is the Airy function.
It is known that the following asymptotics for the Airy function$Ai( \eta)=C|\eta|^{-\geq}\exp(\frac{2}{3}i|\eta|^{\frac{\delta}{2}}+i\frac{\pi}{4})+O(|\eta|^{-7/4})$

is valid.

as$\eta=xt^{-\doteqdot}arrow\infty$

Airy function oscillates rapidly and decays slowly as $xarrow\infty$ , Whendecays exponentially as$xarrow-\infty,$ $Ai(\eta)$$Ai(\eta)=C|\eta|^{-g_{e^{-g|\eta|}}\S}2+O(|\eta|^{-7/4}e^{-\frac{2}{s}|\eta|}\S)$a$s$$\eta=xt^{-\xi}arrow-\infty.$

These asymptotics are obtained by the stationary method (see [4]).

3. STABILITY OF THE SELF SIMILAR SOLUTION$K$

In [7] we showed
Proposition 3.1. Assume hat the initial data $u_{0}\in H^{1,1}$ are real valued functions
with suﬃciently small norm
. Then there exists a unique global solution$u\in C(|O, \infty);H^{1,1})$ of the Cauchy problem for (1.1) such that-$||u_{O}||_{IS^{1,1}}=\epsilon$$\langle t\rangle^{\xi-}\Leftrightarrow||u(t)||_{L^{p}}\leq Ce$

for all $t\in R$ , where $4<\beta\leq\infty.$
We denote by$v_{ln}(t, x\rangle=t^{-\xi}f_{\tau n}(xt^{-\xi})$

the self similar solution of (1.1). Note that if the function
Peinleve equation$f_{\tau n}(\eta)$

satisfies the second$\frac{d^{2}}{d\eta^{2}}f_{m}+\eta f_{rn}-3f_{ln}^{3}=0,$

thensatisfies (1.1).
The next result from [7] says the asymptotic stability of solutions in the neighborhood of the self similar solution.
$v_{\tau n}$

Proposition 3.2. Let $u\in C([O, \infty);H^{1,1})$ be the solution of (1.1) constructed in
Proposition 3.1 and $ﬀ_{m}(x)dx= \int u_{O}(x)dx$ . Then for any $u_{O}\in H^{1,1}$ , there exist
unique functions
and $B_{j}\in L^{\infty}$ (are real-valued), $j=1$ , 2, such that thefollowing asymptotic formula is valid for large time $t\geq 1$$If_{j}$$B_{j}$$u(t, x) = t^{-\doteqdot}f_{m}(xt^{-\S})$$+\sqrt{2\pi}t^{-\not\in ReAi}(xt^{-\xi})(H_{1}(x)\exp(iB_{1}(x)\log|x|t^{-\xi})$$+H_{2}\langle\beta f)\exp(iB_{2}(x)\log|x|t^{-1}\S))$

(3.1)

where

$+O(\epsilon t^{4\gamma-\#}(1+|x|t^{-\xi})^{-x/4})$$\gamma\in(0, \frac{1}{so})$and $x=(x/t)^{4}$ ,,172
MODIFIED KORTEWEG DE VRIES EQUATION

Since
in the second term of the right-hand side of (3.1) are not necessarily
zero at the or\^igin, and asymptotic property of solutions to the second Peinleve$H_{j}$

equation is not stated explicitly in [7], therefore it is not determined which one
is the leading term $f_{rn}(\eta)$ or $Ai(\eta)$ from the previous work. In the recent work[8], we proved that the leading term of $f_{rn}(\eta)$ as $\eta=xt^{-\xi}arrow\infty$ is similar to theleading term of $Ai(\eta)$ for $\eta>0$ . Thus the previous work says that the main term
consists of the first and the second terms of the right-hand side of (3.1). In [8], wedeveloped the factorization technique to obtain the sharp time decay estimate ofsolutions and make an improvement of the previous result from [7].

4. STABILITY OF THE SELF SIMILAR SOLUTION II

We are now in a position to state our first result from [8].

Theorem 4.1. Assume that the initial data$u_{O} \in H^{s}\cap H^{O,1}, s>\frac{3}{4}\backslash$

are real-valued with a suﬃciently small norm$||u_{O}||rr\epsilon\cap H^{0,1}\leq\epsilon.$

Then there exists a unique global solution$\mathcal{F}e^{-\tau^{e_{\partial_{x}^{3}}}}u:\in C([0, \infty) ; L^{\infty}\capH^{0,1})$of the Cauchy problem (1.1). Fbrthermore the estimate$\sup_{t>0}(\Vert \mathcal{F}e^{-\yen}\partial_{x}^{3}u(t)\Vert_{L\infty}+\langle t\rangle^{-\frac{1}{6}}\Vert xe^{-\S\partial_{x}^{3}}u(t)\Vert_{L^{2}}+\langle t\rangle^{\frac{1}{\theta}(1-\frac{1}{p})}||u(t)||_{L^{p}})\leq C\epsilon$

is true, where $p>4.$

In order to state the stability of global solutions in the neighborhood of the selfsimilar solution$v_{7n}(t, x)=t^{-\S}f_{rn}(xt^{-\doteqdot)},$we needTheorem 4.2. Assume that $m$ is suﬃciently small real number. Then there exists
a unique real-valued solution of the Cauchy problem (1.1) in the form $v_{rn}(t, x)=$$t^{-A}3f_{\tau n}(xt^{-1}3)$ , such that$\int f_{7n}(x)dx=m,$$\mathcal{F}e^{-\S^{e_{-\partial_{x}^{3}}}}v_{rn}\in C([1, \infty) ; L^{\infty})\wedge xe^{-\not\in\partial_{x}^{3}}v_{\gamma\iota}\in C([1, \infty) ; L^{2})$.

Fbrthervnore the estimates$\sup_{t>1}(\Vert \mathcal{F}e^{-\yen^{a_{x}^{3}}}v_{rn}(t)\Vert_{L\infty}+t^{-1}6\Vert xe^{-\frac{e}{3}\partial_{\alpha}^{3}}v_{\tau n}(t)\Vert_{L^{2}})\leq C|m|$and$\frac{1}{2}|m|t^{-\xi(1-\frac{1}{p})}\leq||v_{n1}(t)||_{L^{p}}\leq 2|7n|t^{-g()}1-\perp p$

are true, where $p>4.$

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N. HAYASHI AND P. NAUMKIN

Theorem 4.3. Suppose that$\frac{1}{\sqrt{2\pi}}\int_{R}f_{\gamma n}(x)dx=\frac{1\wedge}{\sqrt{2\pi}}\int_{R}u_{O}(x)dx=m\neq 0.$

Let $u(t, x)$ andbe the solutions constructed in Theorem 4.1 and Theorem
4,2, respectively. Then there exists a $\gamma>0$ such that the asymptotics$v_{\gamma n}(t, x)$(4.1)$|u(t, x)-v_{7\gamma\iota}(t, x)|\leq C\epsilon t^{-\frac{1}{2}+\gamma}$

for $x>0$ and(4.2)$|u(t, x)-v_{rn}(t, x)|\leq C\epsilon t^{-\frac{1}{2}+\gamma}\langle Xf^{-\frac{1}{3}}\rangle^{-\frac{3}{4}}$for $x\leq 0$ are true for large$t\geq 1$. Also the sharp time decay estimate of solutions

is $valid_{J}$ namely there exist positive constants

such that$C_{1},$ $C_{2}$$C_{1}\epsilon t^{-\frac{1}{3}(1-\frac{1}{g})}\leq||u(t)||_{L^{q}}\leq C_{2}\epsilon t^{-\frac{1}{3}(1-\frac{1}{q}\rangle}$for $4<q<$ oo.
5. STRATEGY OF PROOFS IN 8
$[$

$]$

Local existence and uniqueness of solutions to the Cauchy problem (1.1) was
shown when $u_{O}\in H^{s},$and the estimate of solutions such that
$C$ for some time $T$
was also shown by C.E. Kenig-G. Ponce-L. Vega [9], [10]. By
using the local existence result, we have$f_{0}^{T}||\partial_{x}u(t)||_{L\infty}^{4}dt\leq$$s> \frac{3}{4}$Theorem 5.1. Assume that the initial data
$u_{\zeta)} \in H^{s}\cap H^{O,1}, s>\frac{3}{4}.$Then there unsts a unique tocal solutio $nu$ of the Cauchy problem (1.1) such that$\mathcal{U}(-t)u\in C([0, T];H^{S}\cap H^{O,1})$.

We can take $T>1$ if the data are small in $H^{s}\cap H^{0,1}$ and we may assume that
(5.1)$||\mathcal{F}\mathcal{U}(-t)u(1)||_{L\infty}+||Ju(1)||_{L^{2}}+||u(1)||_{Lp}\leq\epsilon,$

where $p>4$ . To get the result, in [8] we showed a priori estimates of solutions
under the following norm$||u||_{X_{T}}= \sup_{t\in[1,T\}}(||\mathcal{F}\mathcal{U}(-t)u(t\rangle||_{t\infty}+t^{-\not\in}||Ju(t)||_{L^{2}}+t^{\frac{1}{3}(n-\frac{1}{p})}||u(t)||_{Lz\}})$,

where $J=x-t\partial_{x}^{2}=\mathcal{U}(t)xt4(-t)$ . $Ir1$ particular, we use the factorization method to
get a priori estimates ofin Theorem 4.1 andin 4.2. In order to prove these estimates we introduce the free evolution group$||\mathcal{F}\mathcal{U}(-t)u(t)||_{L\infty}$$\Vert \mathcal{F}e^{-\yen}v_{m}\partial_{x}^{3}(t)\Vert_{L\infty}$$\mathcal{U}(t)=\mathcal{F}^{-1}e^{-\yen\xi^{3}}\mathcal{F},$dilation operator$\mathcal{D}_{t}\phi=|t|^{-\frac{1}{2}}\phi(xt^{-1})$,

scaling operator$(\mathcal{B}\phi)(x)=\phi(x|x|^{-\frac{1}{2}})$.

174
MODIFIED KORTEWEG DE VRIES EQUATION

Define the cut oﬀ function $\chi(\xi)\in C^{2}(R)$ such that$\chi(\xi)=0$

for$\xi\leq-\frac{1}{3},$$\chi(\xi)=1$

for$\xi\geq\frac{1}{3}$and$\chi(\xi)+\chi(-\xi)\equiv 1.$

Then we write$\mathcal{U}(t)\mathcal{F}^{-1}\phi$$= \mathcal{D}_{t}\mathcal{B}\frac{|t|^{\doteqdot}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{it(x^{2}\xi-*\xi^{3})}\phi(\xi)\chi(\xi x^{-1})d\xi$$+ \mathcal{D}_{t}\mathcal{B}\frac{|t|^{*}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-it\langle x^{2}\xi-\S\xi^{8}\rangle}\phi(-\xi)\chi(\xi x^{-1})d\xi$

for $x>0$ . Also we have$\mathcal{U}(t)\mathcal{F}^{-1}\phi=\mathcal{D}_{t}\mathcal{B}\frac{|t|^{*}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-it(x^{2}\xi+\frac{1}{3}\xi^{3})}\phi(\xi)d\xi$

for $x\leq$ O. Sincehence$u=\mathcal{U}(t)\mathcal{F}^{-1}\phi$is a real-valued function, we have$\phi(-\xi\rangle=\overline{\phi(\xi)},$$\mathcal{U}(t)\theta(x)\mathcal{F}^{-1}\phi$$= D_{t} \mathcal{B}\frac{|t|^{\neq}\theta(x)}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{it(x^{2}\xi-\S\xi^{3})}\phi(\xi)\chi(\xi x^{-1}\rangle d\xi$$+ \mathcal{D}_{t}\mathcal{B}\frac{|t|^{b}\theta(x)}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-it(x^{2}\xi-\not\in\xi^{3})}\overline{\phi(\xi)}\chi(\xi x^{-1})d\xi$$= \mathcal{D}_{t}\mathcal{B}(M\mathcal{V}\phi+\neg M\mathcal{V}\phi$with$\theta\langle x$) $=0$ for $x\leq 0$ , and 9 $(x)=1$ for $x>0$ , where the multiplication factor$M(t, x)=e^{2}\neq x^{s}$the phase function$S(x, \xi)=\frac{2}{3}x^{3}-x^{2}\xi+\frac{1}{3}\xi^{3},$and the operator$\mathcal{V}\phi=\frac{|t|^{*}\theta(x)}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-itS(x,\xi)}\phi(\xi)\chi(\xi x^{-1})d\xi.$Also we have$\mathcal{U}(t)\mathcal{F}^{-1}\phi=\mathcal{D}_{t}\mathcal{B}\mathcal{W}\phi$

for $x\leq 0$ , where the operator$\mathcal{W}\phi=\frac{|t|^{\neq}(1-\theta(x))}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-itS_{O}(x,\xi)}\phi(\xi)d\xi,$and the phase function $S_{O}(x, \xi)=x^{2}\xi+\frac{1}{3}\xi^{3}$ . If we define the new dependent variable, then we obtain the representation$\hat{\varphi}=\mathcal{F}\mathcal{U}(-t)u(t)$(5.2)$u(t)=\mathcal{U}(t)\mathcal{F}^{-1}\hat{\varphi}=D_{t}\mathcal{B}(M\mathcal{V}\hat{\varphi}+\overline{M\mathcal{V}\hat{\varphi}})+\mathcal{D}_{t}\mathcal{B}\mathcal{W}\hat{\varphi}$

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N. HAYASHI AND P. NAUMKIN

The first term of the right-hand side of (5.2) is the main term comparing with thesecond one.
We also need the representation for the inverse evolution group$\mathcal{F}\mathcal{U}(-t)$(5.3)$\mathcal{F}\mathcal{U}(-t)\phi=Q\overline{M}\mathcal{B}^{-\iota}D_{t}^{-1}\phi+’\mathcal{R}\mathcal{B}^{-1}\mathcal{D}_{t}^{-1}\phi,$where$\mathcal{D}_{t}^{-1}\phi=|t|^{\frac{1}{2}}\phi(xt) , (\mathcal{B}^{-1}\phi)(x)=\phi(x|x|)$and the operators$Q \phi=\frac{2t|t|^{-\frac{1}{2}}}{\sqrt{2\pi}}\int_{0}^{\infty}e^{itS(x,\xi)}\phi(x)xdx,$

and$\mathcal{R}\phi=-\frac{2t|t|^{-\not\equiv}}{\sqrt{2\tau\tau}}\int_{-\infty}^{0}e^{itS_{O}\langle x,\xi)}\phi(x)xdx.$

Since, withto equation $(1,1)$ we get with
$F\mathcal{U}(-t)\mathcal{L}=\partial_{t}\mathcal{F}\mathcal{U}(-t)$$\mathcal{L}=\partial_{t}-\frac{1}{3}\partial_{x}^{i*}$, applying the operator$\mathcal{F}\mathcal{U}(-t)$$\hat{\varphi}=\mathcal{F}\mathcal{U}(-t)u$$\partial_{t\hat{\varphi}}=\partial_{t}\mathcal{F}\mathcal{U}(-t)u=\mathcal{F}\mathcal{U}(-t)\mathcal{L}u=\mathcal{F}\mathcal{U}(-t)\partial_{x}(u^{3})=3\mathcal{F}u(-t)(u^{2}u_{x})$.

Then by (5.2) we find the following factorization property$\partial_{t}\hat{\varphi}=3\mathcal{F}\mathcal{U}(-t)(u^{2}u_{x}\rangle$$= 3t^{-1}Q\overline{M}(M\mathcal{V}\hat{\varphi}+\overline{M\mathcal{V}\hat{\varphi}})^{2}(M\mathcal{V}i\xi\hat{\varphi}+\overline{M\mathcal{V}i\xi\hat{\varphi}})+R.$

Note that$\overline{M}(M\mathcal{V}\hat{\varphi}+\overline{M\mathcal{V}\hat{\varphi}})^{3}=M^{2}(\mathcal{V}\hat{\varphi})^{3}+3(\mathcal{V}\hat{\varphi})^{2}(\overline{\mathcal{V}\hat{\varphi}})$$+3\overline{M}^{2}(\mathcal{V}\hat{\varphi})(\overline{\mathcal{V}\hat{\varphi}})^{2}+\overline{M}^{\sim}(\overline{\mathcal{V}\hat{\varphi}})^{3}$and for $\alpha\neq-1$$Q(t)M^{\alpha}\phi=E^{-\frac{\alpha\langle 2+\alpha)}{(1+\infty)^{2}}}D_{1+\alpha}Q(t(1+\alpha))\phi, E=e^{-{\}\xi^{3}}$Thus we obtain the equation for the new dependent variable$\hat{\varphi}(t, \xi)=\mathcal{F}\mathcal{U}(-t)u(t)$$\partial_{t}\hat{\varphi}(t, \xi\rangle$$= 3t^{-1}E^{-\ovalbox{\tt\small REJECT}}D_{3}Q(3t)(\mathcal{V}\hat{\varphi})^{2}(\mathcal{V}i\xi\hat{\varphi})$$+3t^{-1}Q(t)(2(\mathcal{V}\hat{\varphi})(\overline{\mathcal{V}\hat{\varphi}})(\mathcal{V}i\xi\hat{\varphi})+(\mathcal{V}\hat{\varphi})^{2}(\overline{\mathcal{V}i\xi\hat{\varphi}}))$$+3t^{-1}\mathcal{D}_{-1}Q(-t)((\overline{\mathcal{V}\hat{\varphi}})^{2}(\mathcal{V}i\xi\hat{\varphi})+2(\mathcal{V}\hat{\varphi})(\overline{\mathcal{V}\hat{\varphi}})(\overline{\mathcal{V}i\xi\hat{\varphi}}))$(S.4)$+3t^{-1}E^{-\S}9\mathcal{D}_{-3}Q(-3t)(\overline{\mathcal{V}\hat{\varphi}})^{2}(\overline{\mathcal{V}i\xi\hat{\varphi}})+R.$

Now we explain how to use equation (5.4) for estimating. For the real-valued solution , we have
suﬃcient to consider the case $\xi>0$ only.
respect to

$\xi$

$u$

$|\hat{\varphi}(t, \xi)|$$\hat{\varphi}(t, \xi)=\hat{\varphi}(t,$uniformly with, hence it is$-\xi\rangle$

176
MODIFIED KORTEWEG-DE VRIES EQUATIONThe second term of the right hand side. of (5.4) is a main term. We have
$3t^{-1}\mathcal{Q}(t)(2(\mathcal{V}\hat{\varphi})(\overline{\mathcal{V}\hat{\varphi}})(\mathcal{V}i\xi\hat{\varphi})+(\mathcal{V}\hat{\varphi})^{2}(\overline{\mathcal{V}i\xi\hat{\varphi}}))$$\simeq \frac{3}{2}t^{-1}\xi^{g}(t*\langle\xi t^{\S}\rangle^{2})t^{-\epsilon}\langle\xi t^{\xi}\rangle^{\neq}|\hat{\varphi}|^{2}\hat{\varphi}(\xi)$$\simeq \frac{3}{2}t^{-1}t^{\xi}\xi\langle\xi t^{\frac{1}{3}}\rangle^{-1}|\hat{\varphi}|^{2}\hat{\varphi}(\xi)$

,

where
$\mathcal{Q}(t)f(x)\simeq\xi^{\frac{1}{2}}f(\xi) , \mathcal{V}\hat{\varphi}\simeq t^{\delta}1\langle\xi t^{\xi}\rangle^{-\frac{1}{2}}, \mathcal{V}i\xi\hat{\varphi}\simeq t^{-\frac{1}{6}}\langle\xi t^{5}1\rangle^{\neq}$

The main term for the second summand of the right-hand side of (5.4) will be$\frac{3}{2}it^{-1}\xi t^{\xi}\langle\xi t^{g}\rangle^{-1}|\hat{\varphi}|^{2}\hat{\varphi}(\xi)$andTo justify the above procedure, we need the estimates of the derivatives, for details, see [S]. We have the desired a priori estimate of. In the similar way we have the result for the self-similar solution. Therefore Theorem 4.1 and Theorem 4.2 follow. To obtain Theorem 4.3 we consider thewith the same mass. Define
estimates for the diﬀerence of two solutions$\partial_{E}\mathcal{W}$$||\mathcal{F}\mathcal{U}(-t)u(t)||_{L\infty}=$$\partial_{\xi}\mathcal{V}$$||\hat{\varphi}(t)||_{L^{\infty}}$$u_{j}$$\Vert u_{1}-u_{2}||_{Y\tau}=\sup_{t\in[1,T]}(t\neq-\gamma||u_{1}-u_{2}||_{L\infty}+t^{-\gamma}\Vert \mathcal{J}(u_{1}-u_{2})\Vert_{L^{2}})$

with a small
O. Then we have Theorem 4.3 by the following lemma if we put$u_{1}=u$ and $u_{2}=v_{rn}=t^{-\S}f_{rn}(xt^{-\xi})$ .
$\gamma>$Lemma 5.2. Suppose that$j=1$ , 2, where is suﬃciently small.
$t\geq1$, where. Let
Let
for $j=1$ , 2,
$u_{2}=t^{-\frac{1}{3}}f(xt^{-\S})$ be-simdar solution. Then the estimate$||u_{j}||_{X_{T}}\leq C\epsilon,$$\hat{\varphi_{1}}(t, 0)=\hat{\varphi_{2}}(t, 0)$$\epsilon$$\hat{\varphi_{j}}(t, \xi)=\mathcal{F}\mathcal{U}(-t)u_{j}(t)$$a\mathcal{S}elf$$||u_{1}-u_{2}||_{Y_{T}}<C\epsilon$

is true for all $T>1.$

6. ASYMPTOTICS OF THE SELF SIMILAR SOLUTION
Let us consider the asymptotics of the self similar solutions. We assume thatsatisfies$\hat{\varphi}=\mathcal{F}e^{9^{\partial_{x}^{3}}}u$$\partial_{t}\hat{\varphi} = \frac{3}{2}it^{-1}\xi t\doteqdot\langle\xi t^{\frac{1}{3}}\rangle^{-1}|\hat{\varphi}(t, \xi)|^{2}\hat{\varphi}(t, \xi)+R$$= \frac{3}{2}it^{-1}\xi t^{\xi}\langle\xi t^{\xi}\rangle^{-1}|\hat{\varphi}(t,0)|^{2}\hat{\varphi}(t, \xi)+R$$= \frac{3}{2}i|m|^{2}\xi t^{-z}2\langle\xi t*\rangle^{-1}\hat{\varphi}(t, \xi)+R$

which suggests the self-similar solution is$\psi_{rn}(\xi t^{\perp}3)=me^{\frac{9}{2}i|rn|^{2}\log\langle\xi t\rangle}*.$177
N. HAYASHI AND P. NAUMKIN

Indeed$\partial_{t}\psi_{m}(\xi t^{\frac{1}{3}})=\frac{3}{2}i|m|^{2}m\langle\xi t^{\frac{1}{3}}\rangle\xi te^{9}2$$= git^{-1}\xi t^{\frac{1}{3}}\langle\xi t^{g}\rangle^{-1}|\psi_{m}(\xi f\xi)|^{2}\psi_{\gamma n}(\xi t^{\frac{1}{3}})$

It is possible to consider the diﬀerence between
suggests the self similar solution is$\psi_{m}(\xi f^{\xi})$

and$\hat{\varphi}(t, \xi)$. Therefore it$t^{-\frac{1}{3}}f_{m}(xt^{-\frac{1}{3}})$$= \frac{1}{\sqrt{2\pi}}t^{-\frac{1}{s}}\int e^{i(\eta-\S\eta)_{\psi_{rn}}}xt^{-*3}(\eta)d\eta$$=$$t^{-\frac{1}{\theta}}$$me$ $g_{i|m|^{2}\log\langle xt^{-4\rangle_{\sqrt{\pi}}}}\langle xt^{-g}\rangle^{-Z}1$$\cross\exp(\frac{2}{3}i|xt^{-\S}1|^{\frac{3}{2}}+i\frac{\tau r}{4})+O(t^{-\perp}3\langle xt^{-\doteqdot}\rangle^{-7/4})$

However it is not stated in [8] since the estimate ofto show the leading term ofof the above.

$t^{-\not\in}f_{\tau n}(xt^{-\frac{1}{3}})$$\psi_{rn}(\xi t^{\frac{1}{3}})-\hat{\varphi}(t, \xi)$.
is not enoughis the first term of the right hand side

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$s^{}$-$Vr\iota es$$V\gamma\eta es$$rI^{\backslash }$-178
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110133.
$86arrow 92.$

DEPARTMENT 0F MATHEMATICS, GRADUATE SCHOOL 0F SCIENCE,, OSAKA UNIVERSITY, Os, TOYONAKA 560-0043, JAPAN$E$-mailaddress: mhayashiQmath. sci. osaka-u. ac. jp$AKA,$$LIA$

CENTRO DE CIENCIAS MATEM\’ATICAS, UNAM CAMPUS MORELIA, AP 61-3 (XANGARI), MORECP 58089, MICHOAC\’AN, MEXICO-maSl address: $pavelni\Phi$atmor. unam. mx$E$