数理解析研究所講究録
1059 巻 1998 年 101-109

101

ASYMPTOTICS AND SCATTERING PROBLEM FOR
THE GENERALIZED KORTEWEG-DE VRIES EQUATION

1

NAKAO HAYASHI(林仲夫) $-1$ AND$\mathrm{p}\mathrm{A}\mathrm{v}\mathrm{E}\mathrm{L}$I. NAUMKIN 2

1 Department of Applied Mathematics,

Science University of Tokyo
1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, JAPAN-mail: nhayashi@rs.kagu.sut.ac.jp
and:2 Instituto de F\’isica
Matem\’aticas, Universidad
Michoacana, AP 2-82, CP 58040, Morelia, Michoacana, MEXICO-mail naumkin@ifml.ifm.umich.mx$\mathrm{e}$$\mathrm{y}$$\mathrm{e}$\S 1 Introduction. We consider the asymptotic behavior in time of solutions toequation
the Cauchy problem for the generalized Korteweg-de Vries$(\mathrm{g}\mathrm{K}\mathrm{d}\mathrm{v})$$u_{t}+(|u|^{\rho 1}-u)_{x}+ \frac{1}{3}u_{xxx}=0$$\{$$u(\mathrm{O}, x)=u_{0}(x)$,$x\in$

is a real valued function and,$t,$ $x\in \mathrm{R}$

,

(1.1)

R.

.

We denote the Sobolev space$H^{1,1}=\{\phi\in L^{2}; ||\phi||_{1,1}=||(1+x^{2})^{1/2}(1-\partial_{x}2)1/2\phi||L^{2}<\infty\}$ ,and the free Airyevolution groupHere$u_{0}$$\rho>3$$U(t)\phi=\mathcal{F}^{-1}e^{it}\hat{\phi}\xi \mathrm{s}/\mathrm{s}(\xi)$.

is the Fourier transform of the function
orHere and below. The inverse Fourier transformation
by.
given by the formula
Our purpose in this note is to explain the following result which waspaper [12].
$\hat{\phi}$$\mathcal{F}\phi$$\mathcal{F}\phi(\xi)=\frac{1}{\sqrt{2\pi}}\int e^{-ix\xi}\phi(x)dX$

definedis$\phi$$\mathcal{F}^{-1}$$\mathcal{F}^{-1}\phi(x)=\frac{1}{\sqrt{2\pi}}\int e^{ix\xi}\phi(\xi)d\xi$

AMS Classification$35\mathrm{Q}55$

oved in$\mathrm{p}\dot{\mathrm{r}}$102
Theorem 1.1. We assume that the initial data
are real, $u_{0}\in H^{1,1}$
is sufficiently small. Then there exists a unique global solution, of the Cauchy problem (1.1) with $\rho>3$ such that$u_{0}$$||u_{0}||_{1,1}=\epsilon$

and$u\in$$C(\mathrm{R};H^{1,1})$$||u(t)||_{L^{\beta}} \leq\frac{C\epsilon}{(1+t)^{\frac{1}{3}-\frac{1}{3\beta}}}$for alland for every
a unique final state
$t>0$

,

$||uu_{x}(t)||_{L} \infty\leq\frac{C\epsilon^{2}}{t^{\frac{2}{3}}(1+t)\frac{1}{3}}$$\beta\in(4, \infty]$,. Furthermore we show that there exists

such that$u_{+}\in L^{2}$$||u(t)-U(t)u_{+}||_{L^{2}}\leq C\epsilon t^{-\frac{\rho-3}{3}}$for$t\geq 1$

.

(1.2)

The Cauchy problem (1.1) was intensively studied by many authors and a large
amount of literature is devoted to investigate it. The existence and uniqueness ofsolutions to (1.1) in different Sobolev spaces were proved in [9, 10, 14, 15, 16, 19,
20, 23, 27]. The smoothing properties of solutions were studied in [3, 5, 6, 15,16] and the blow-up effect for the slowly decaying solutions of the Cauchy problem(1.1) was found in [2]. For the special cases of theequation itself and theequation ( $\rho=3$ in (1.1)) the Cauchy problem was solved by themodified
Inverse Scattering ]}(IST) method and the large time asymptotic behaviorof solutions was found (see [1, 7]). The IST method depends essentially on the
nonlinear character of the equation, although in the case ofequation $(\rho=$3) solutions decay with the same speed as in the corresponding linear case, i.e.
$\sup_{x\in \mathrm{R}}|u(t, X)|\leq C(1+t)^{-}1/3$. Now let us give a brief survey of theasprevious results on the large time asymptotic behavior of solutions to (1.1) which wereobtained by functional analytic methods. To state these results we introduce somefunction spaces. $L^{p}=\{\phi\in S’;||\phi||_{p}<\infty\}$ , where $|| \phi||_{p}=(\int|\phi(x)|^{p}d_{X)}1/p$ if$1\leq p<\infty$
and
if $p=\infty$ . For simplicity we$H^{m,s}$. Weighted Sobolev spaceletis defined by $H^{m,s}=\{\phi\in$$S’;||\phi||_{m,s}=||(1+X^{2})^{s/}2(1-\partial_{x}^{2})m/2|\phi|<\infty\}$ ,$m,$ $s\in$ R.
$\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}$$\mathrm{M}\mathrm{K}\mathrm{d}\mathrm{V}$$tarrow\infty$$||\phi||_{\infty}$$= \mathrm{e}\mathrm{s}\mathrm{s}.\sup_{x\in \mathrm{R}}|\phi(X)|$$||\phi||=||\phi||_{2}$In paper [25] Strauss proved
Proposition 1.1. Let $\rho>5$ , the initial data $u_{0}\in L^{1}\cap H^{1,0}$ and$||u_{0}||_{L}1+||u_{0}||H^{1,0}$be sufficiently small. Then the solution $u(t)$ of (1.1) satisfies
the time decay estimate
and there exists a final state$u_{+}\in L^{2}$such that $\lim_{tarrow\infty}||u(t)-U(t)u_{+}||=0$ .
$\epsilon=$$||u(t)||\infty\leq C\epsilon(1+t)^{-\frac{1}{3}}$In his method W. Strauss used the following large time decay estimateof thenorm of solutions to the Airy equation.

$\leq Ct^{-\frac{1}{3}}||u_{0}||_{1}$$||U(t)u0||_{\infty}$$L^{\infty}$

Later this result on the asymptotically hee evolution of solutions to (1.1) was
extended to the values $\rho>(5+\sqrt{21})/2\approx 4.79$ in [17, 18, 24, 26]. They obtained

103
Proposition 1.2. Assume that $\rho>(5+\sqrt{21})/2\approx 4.79$ , the initial dataandis sufficiently small. Thenthe solution $u(t)$ of (1.1) satisfies the time decay estimate $||u(t)||_{2}\rho\leq C\epsilon(1+$and there exists a final statesuch that$u_{0}\in$$L^{2\rho/(2-1)}\rho\cap H^{1,0}$$\epsilon=||u0||_{L}2\rho/(2\rho-1)+||u_{0}||_{H^{1,\mathrm{O}}}$$t)^{-\frac{1}{3}(1-)} \frac{1}{\rho}$$u_{+}\in L^{2}$$\lim_{tarrow\infty}||u(t)-U(t)u_{+}||=0$.

Their method is based on the following
decay estimate
for the solutions to the Airy equation.
$L^{p}$$||U(t)u0||_{2\rho}\leq$$ct^{-\frac{1}{3}(1-\frac{1}{\rho})}||u_{0}||_{2\rho}/(2\rho-1)$

In paper [22] Ponce and Vega improved the above result for the values of$(9+\sqrt{73})/4\approx 4.39$.

$\rho>$

Proposition 1.3. Let $\rho>(9+\sqrt{73})/4\approx 4.39$ , the initial data $u_{0}\in L^{1}\cap$
and $\epsilon=||u_{0}||L1+||u_{0}||_{H^{1,0}}$ be sufficiently small. Then the same result as
in Proposition 1.2 holds. Furthermore the solution $u(t)$ satisfies the time decay
estimate$H^{1,0}$$||(- \partial_{x}^{2})1/4)u(t||_{\infty}\leq c_{\epsilon}(1+.t)^{-}\frac{1}{2}$.

For the proof of Proposition 1.3 Ponce and Vega used the
decay estimatesof solutions to the Airy equation and the following
time decay estimateof the half derivative of solutions to the Airyequation.
$L^{p}$$L^{\infty}$$||(-\partial_{x}^{2})1/4(Ut)u_{0||_{\infty}}\leq Ct^{-\frac{1}{2}}||u_{0}||_{1}$Finally in [4] Christ and Weinstein extended the result of Ponce and Vega to thepowers $\rho>(23-\sqrt{57})/4\approx 3.86$ .

Proposition 1.4. Assume that $\rho>(23-\sqrt{57})/4\approx 3.86$ , the initial data,
and the norm $\epsilon=||u0||_{1}+||\partial xu0||1+||u0||_{2,0}$ is sufficiently
small. Then the same result as in Proposition 1.3 holds. Furthermore the solution$u(t)$
satisfies the time decay estimate
for $p>4$ .
$u_{0}\in$$L^{1}\cap H2,0$$u_{0}’\in L^{1}$$||u(t)||_{p} \leq C\epsilon(1+t)^{-\frac{1}{3}(1-)}\frac{1}{p}$$L^{p}$The proof of Proposition 1.4 is based on the previous methods. Also it uses the
decay estimates of solutions to the Airy equation$||U(t)u0||_{p} \leq Ct^{-\frac{1}{3}()}1-\frac{1}{p}||u_{0}||_{1}$

.

(1.3)

for all $p>4$ .
Thus we do not know a character of the large time asymptotic behavior of thesolutions to the Cauchy problem for the generalized Korteweg-de Vries equation(1.1) with $\rho<3.86$ . Our result in Theorem 1.1 fills a gap $3<\rho\leq 3.86$ . The

104
asymptotic expansion of the solutions to the Cauchy problem (1.1) was obtained in. The evaluation of the asymptotics in [21] is
21] for the integer values ofbased on the perturbation theory and essentially uses the explicit representation of
the Fourier transform of the nonlinearity and therefore does work only for the integer.
values of
The Airy free evolution group is defined by$[$$\rho\geq 4$$\rho$$U(t)\phi=\mathcal{F}^{-}1e\hat{\phi}it\xi 3/3(\xi)$$= \frac{1}{2\pi}\int dy\phi(y)\int d\xi e^{i}-yit/3=(x)+\xi \mathrm{s}\frac{1}{\sqrt[3]{t}}\xi{\rm Re}\int \mathrm{A}\mathrm{i}(\frac{x-y}{\sqrt[3]{t}})\phi(y)dy$,

is the Airy function (we take a slightly differwhereent definition of the Airy function, usually the real part of our function Ai is calledby the Airy function). The Airy function has the following asymptotics:
andas$\mathrm{A}\mathrm{i}(x)=\frac{1}{\pi}\int_{0}^{\infty_{e^{ii/3}}}xz+zdz3$$\mathrm{A}\mathrm{i}(\eta)=$$\frac{c}{\sqrt[4]{|\eta|}}\exp(-\frac{2}{3}i\sqrt{|\eta|^{3}}+i\frac{\pi}{4})+O(|\eta|^{-7/}4)$as$\mathrm{A}\mathrm{i}(\eta)=$$\eta=\tau_{\sqrt{t}^{-}}^{x}arrow-\infty$(see, e.g., [8]). In [12, Theorem 1.3] we showed that the solution of (1.1) has the same asymptotics as that of
decays as $xarrow\infty$ faster than any
the Airy function when the function
exponent.
$\frac{c}{\sqrt[4]{\eta}}e^{-\frac{2}{3}\sqrt{\eta^{3}}}+O(\eta^{-7/4\frac{2}{3}}e^{-\sqrt{\eta^{3}}})$$\eta=\sqrt[]{t}xarrow+\infty$$u_{0}$\S 2 Key linear estimates.
Our method uses the estimate (1.3) and the following time decay estimate ofsolutions to the Airy equation$||(U(t)u\mathrm{o})(U(t)u\mathrm{o})_{x}||_{\infty}\leq Ct^{-2/}(31+t)^{-1/3}|||u0|||_{X_{0}}$

,

(2.1)

where
$|||u_{0}|||\mathrm{x}_{\mathrm{O}}=||u_{0}||_{1,0}+||D^{\alpha}xu0||+||\partial_{x}xu_{0}||$and$\alpha=1/2-\gamma,$ $\gamma\in(0, \min(\frac{1}{2}, \triangle_{3}-\underline{3}))$

esti.mates.

,

The inequality (2.1) is obtained from the$|U(t)u \mathrm{o}(x)|\leq C(1+t)^{-1/3}(1+\frac{|x|}{\sqrt[3]{t}})^{-1/4}|||u_{0}|||\mathrm{x}_{0}$and$| \partial_{x}U(t)u_{0(}x)|\leq Ct^{-2/3}(1+\frac{|x|}{\sqrt[3]{t}})^{1/4}|||u_{0}|||\mathrm{x}_{\mathrm{o}}$.

For the proofs of the above estimates, see [12, Lemma 2.2]. Our method is close to thatof [11] in the point that here we also use the following operator
for obtaining estimates of the solution in the weighted Sobolev spaces. The operator$I=x+3t \int_{-\infty}^{x}\partial tdy$

105
almost commutes with the linear partof equation (1.1) andacts on the nonlinear term
as a first order differential operator. Note
operatorthat theis related with the operator $J=U(-t)xU(t)=(x-t\partial_{x}2)$
since we have $I-J=3t \int_{-\infty^{LdX}}^{x}$ .
In what follows we consider the positive time only. We define the function spaceas follows$I$$L= \partial_{t}+\frac{1}{3}\partial_{x}^{3}$$(|u|^{\rho 1}-u)_{x}$$I$$X_{T}$$X_{T}= \{\phi\in c([\mathrm{o}, T];L2) ; |||\phi|||\mathrm{x}_{T}=\sup_{t\in[0,\tau]}||\phi(t)||x<\infty\}$where$||\phi(t)||_{X}=||\phi(t)||1,0+||D^{\alpha}J\phi(t)||+||\partial J\phi(t)||$$u_{0}=U(-t)\phi(t)$.

,By virtue of (1.3) withand by the H\"older’s inequality we have for all$4<p\leq\infty$$||\phi(t)||_{p}\leq Ct^{-\frac{1}{3}(-\frac{1}{p}}1)||U(-t)\phi(t)||_{1}$$\leq ct^{-\frac{1}{3}(1-\frac{1}{p})}(||\phi(t)||+||XU(-t)\phi(t)||2/(1-2\alpha))$.

$\leq ct^{-\frac{1}{3}(1-\frac{1}{\mathrm{p}})}(||\phi(t)||+||D^{\alpha_{X}}U(-t)\phi(t)||)\leq Ct^{-\frac{1}{3}(-\frac{1}{\mathrm{p}}})|1||\phi|||X_{T}$(2.2)

Via (2.1) we also get the estimate$||\phi(t)\phi x(t)||\infty\leq Ct^{-2/3}(1+t)^{-1/2}\mathrm{s}|||\phi|||X_{T}$

.

(2.3)

Using estimates (2.2) and (2.3) we obtain in the next section the result of Theorem
1.1 by considering a-priori estimates of local solutions in the function space.
$X_{T}$\S 3 Proof of Theorem 1.1.
To clarify the idea of the proof of the Theorem 1.1 we only show a priori estimatesequation. For that purpose we use the following local
of local solutions toexistence theorem.
$\mathrm{g}\mathrm{K}\mathrm{d}\mathrm{V}$Theorem 3.1. We assume that $u_{0}\in H^{1,1}$ ,sufficiently mall. Then there exists a finite time interval
and a unique solutionof (1.1) with $\rho>3$ such that$||u_{0}||_{1,1}=\epsilon\leq\epsilon’$

$.s$

$u$

$[0, T]$

andiswith $T>1$.
$\epsilon’$$|||u|||\mathrm{x}_{\tau}\leq C\epsilon’$For the proof of Theorem 3.1, see, e.g., [9, 14, 15, 16, 20, 27].
Lemma 3.1.
Letbe the local solutions of the Cauchy problem (1.1) with$\rho>3$, where the constantstated in Theorem 3.1. Then we have$C$
does not depend on the time
existencesolutions.
of
of
$u$

$|||u|||_{X_{T}}\leq C\epsilon$$T$

Proof. We write the$\mathrm{g}\mathrm{K}\mathrm{d}\mathrm{V}$tiating it with respect to$x$equation in the form
we get$Lu_{x}=-(|u|^{\rho-}1u)_{xx}$$Lu=-(|u|^{\rho-}1u)x$.

.

Differen(3.1)

106Multiplying both sides of (3.1) by$u_{x}$and integrating by parts we obtain$\frac{d}{dt}||u_{x}||^{2}\leq C||u||^{\rho}\infty|-3|uu_{x}||_{\infty}||u_{x}||^{2}$

.

(3.2)

Using estimates (2.2), (2.3) in the right hand side of (3.2), we get$\frac{d}{dt}||u_{x}||^{2}\leq C|||u|||_{X^{+}}\rho 1t-\frac{2}{3}T(1+t)^{\frac{2}{3}-_{\mathrm{s}}}E$

.

(3.3)

Applying the operator
to the both sides of (3.1) and using the commutation
$[L,
I]=3
\int_{-\infty}^{x}LdX’$, [I, ] $=-1$ we haverelations$I$$\partial_{x}$$LIu_{x}=-\rho(|u|^{\rho-}1(Iux)_{x}+(\rho-1)|u|^{\rho-3}uuxIu_{x}+2|u|^{\rho 1}-u_{x})$Multiplying (3.4) by$Iu_{x}$(3.4)

and integrating by parts we obtain$\frac{d}{dt}||Iu_{x}||2\leq C|||u|||\rho \mathrm{x}_{T}^{-1}t-\frac{2}{3}(1+t)^{\frac{2}{3}-R}3(|||u|||2x_{T}+||Iu_{x}||2)$

Since

.

$(Iu)_{x}-(Ju)_{x}=3t(|u|^{\rho-}1u)x$

.

(3.5)

for the solution of equation (1.1) we have$||Ju_{x}||\leq C(||u||+||(Iu)_{x}||+3t||uux||_{\infty}||u||||u||^{\rho-3}\infty)$(3.6)$\leq C||u_{0}||+Ct^{1-R}3|||u|||\rho \mathrm{x}^{-1}\mathcal{T}^{\cdot}$

Hence (3.5) and (3.6) yield$\frac{d}{dt}||Iu_{x}||2\leq C|||u|||_{X}\rho+1t-\frac{2}{3}\tau(1+t)^{\frac{2}{3}-E}3$

.

(3.7)

Applying the operator
to (1.1), multiplying the result byusing inequalities (see, [12, Lemma 2.3])$D^{\alpha}Iu$$D^{\alpha}I$$||D^{\alpha}|u|^{\rho 1}-u||^{2}\leq C|||u|^{\rho-1}||^{2}(||uu_{x}||_{\infty}+||u||_{\infty}6\gamma||uu_{x}||_{\infty}1-\mathrm{s}_{\gamma})$and,$|(D^{\alpha}h,D^{\alpha}|u|^{\rho 1}-h_{x})|\leq C||D^{\alpha}h||(||D^{\alpha}h||+||\partial_{x}h||)(||u||^{\rho}\infty|-3|uux||_{\infty}$,$+||u||_{\infty}\rho-3-2\gamma||u||^{2}\gamma||uux||_{\infty}+||u||^{\rho-}\infty 3||u||2\gamma|\infty|uux||_{\infty}^{1-\gamma})$where$\alpha=1/2-\gamma$,$\gamma\in(0, \min(\frac{1}{2}, \frac{\rho-3}{3}))$,$h=Iu$

we get$\frac{d}{dt}||D^{\alpha}Iu||2=-2(D^{\alpha}Iu, \rho D\alpha|u|^{\rho}-1(Iu)_{x}+(3-\rho)D^{\alpha}(|u|^{\rho 1}-u))$$\leq C||D^{\alpha}Iu||((||D^{\alpha}Iu||+||\partial Iu||)(||u||^{\rho}\infty|-3|uux||_{\infty}$$+||u||^{\rho-}\infty-32\gamma||u||2\gamma||uux||_{\infty}+||u||_{\infty}^{\rho 3+}-2\gamma||uux||1-\gamma)\infty$$+|||u|^{\rho-1}||(||uu_{x}||_{\infty}^{1}/2+||u||_{\infty}^{3\gamma}||uu_{x}||^{\frac{1-3\gamma}{\infty^{2}}}))$

.

(3.8)

107
Since we take$\mu$

to be sufficiently small, we see that there exists a positive constant$\gamma$such that$\frac{d}{dt}||D^{\alpha}Iu||2\leq C|||u|||^{\rho 1}\mathrm{x}_{\tau}(+1+t)^{-}1-\mu$

where we have used the estimate(3.7), (3.8) and (3.9) yield$||D^{\alpha}Iu||\leq C|||u|||_{\mathrm{x}_{\tau}}$

,

(3.9)

. Thus inequalities (3.2),$||u(t)||_{1,0}+||D^{\alpha}Iu(t)||+|| \partial_{x}Iu(t)||\leq C||u_{0}||1,1+C|||u|||_{X}^{\rho+1}\tau\int_{0}^{t}(1+s)^{-}1-\mu d_{S}$.

The last inequality with estimate $||u(t)||\mathrm{x}\leq C(||u(t)||+||D^{\alpha}Iu(t)||+||\partial_{x}Iu(t)||)$
imply that if the initial data are sufficiently small thenfor
any
. This completes the proof of Lemma 3.1.
$|||u|||\mathrm{x}_{\tau}\leq C||u0||_{1},1$$T$$\square$. We take
Proof of Theorem 1.1 Via Lemma 3.1 we havesatisping. Then a standard continuation argument yields the a-priori$|||u|||_{X_{T}}\leq C\epsilon$$\epsilon$$C\epsilon\leq\epsilon’$for any
because the constantdoes not depend. Therefore it follows that there exists a unique global solution$u\in C(\mathrm{R};H^{1,1})$ ,of the Cauchy problem (1.1) with $\rho>3$ such that

estimateon the time$|||u|||\mathrm{x}_{T}\leq C\epsilon$

$T$

$C$

$T$

$||u(t)||_{\beta} \leq\frac{C\epsilon}{(1+t)\frac{1}{3}-\frac{1}{3\beta}}$,$||uu_{x}(t)||_{\infty} \leq\frac{C\epsilon^{2}}{t^{\frac{2}{3}}(1+t)\frac{1}{3}}$,for all $t>0$ and for every $\beta\in(4, \infty]$ . We next show the existence of thescattering state. Rewriting (1.1) in the form $(U(-t)u)_{t}=-U(-t)\partial_{x}(|u|^{\rho}-1)u$ ,we get$||U(-t)u(t)-U(-s)u(s)|| \leq C\int_{s}^{t}||u(\tau)||_{\infty}^{\rho 3}-||uu_{x}(\tau)||_{\infty}||u(\tau)||d_{\mathcal{T}}$$\leq C\epsilon\int_{s}^{t}\tau^{-\rho/3}d\tau\leq c_{\epsilon s^{-()}}\rho-3/3$(3.10)$1<s<t$ .

By virtue of (3.10) we find that there exists a unique function$tarrow\infty$$u_{+}\in L^{2}$insuch that $\lim_{tarrow\infty}||u(t)-U(t)u_{+}||=0$ .
We let(3.10) to get (1.2). This completes the proof of Theorem 1.1.

for$\square$Finally we note that in paper [13] we studied the asymptotic behavior for large
time of solutions to the Cauchy problem for the modified Korteweg-de Vries
equation:$(\mathrm{m}\mathrm{K}\mathrm{d}\mathrm{V})$$u_{t}+ \partial_{x}N(u)+\frac{1}{3}u_{xxx}=0$,$u(\mathrm{O}, x)=u_{0}(x)$,$(\mathrm{m}\mathrm{K}\mathrm{d})$$N(u)=a(t)u^{3}$ ,$a(t)\in$the nonlinear term is equal toare small enough beis real and bounded, and the initial data
and the integral $\int u_{0}(X)dX=0$ .
longing to the weighted Sobolev space

where$x,$ $t\in \mathrm{R}$,$C^{1}(0, \infty)$$u_{0}$$H^{1,1}$

108Then we show that the solution $u(t)$ satisfies the decay estimates$C(1+t)-1/3$ ,and $||u(t)ux(t)||L^{\infty}\leq Ct-2/3(1+t)-1/\mathrm{s}$ for all $t\geq 0$ . Moreoversuch that the
we proved that there exist unique functions $W$ andequation
following asymptotics for the solutions to the Cauchy problem for$||u(t)||_{L^{6}}\leq$$\Phi\in L^{\infty}$$\mathrm{m}\mathrm{K}\mathrm{d}\mathrm{V}$$u(t, x)= \frac{\sqrt{2\pi}}{\sqrt[3]{t}}\mathrm{R}\mathrm{e}\mathrm{A}\mathrm{i}(\frac{x}{\sqrt[3]{t}})W(\frac{x}{t})\exp(-3i\pi|W(\frac{x}{t})|^{2}\int_{1}^{t}a(\tau)\frac{d\tau}{\tau}$$-3i \pi\Phi(\frac{x}{t}))+O(\epsilon t^{-}/2-\lambda)1$, whereis valid for large time uniformly with respect toAlso in paper [13] we constructed the modified scattering states.
$x\in \mathrm{R}$$\lambda\in(0, \frac{1}{21})$.

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-$arrow\infty$$V_{7}\dot{\tau}es$-