Introduction

We study the following nonlinear Schrödinger equation in one space dimension:

$$\begin{split}iu_t(t,x) &= (-1/2) \partial^2 u(t,x) + f(u(t,x)), \\ u(0,x) &=u_0 (x). \end{split}$$ (1.1)

Here $(t,x) \in \Bbb{R}\times \Bbb{R}$, $\partial = \partial /\partial x$ and $f(u) = f_1(u)+f_2(u) = \lambda_1 \vert u\vert^{p_1-1} u +\lambda_2 \vert u\vert^{p_2-1} u$ with $\lambda_1, \lambda_2 \in \Bbb{R}$ and $1<p_1 < p_2$. The aim of this paper is to study the asymptotic behavior of the solution as $t\to \infty$, especially to obtain the second term of the asymptotic expansion of the solution in the case $p_1=3$.

There is a large literature on the equation (1.1); see [,3,4,,6,,8,9,,,14,16,,17,18,,22,23,,21] and references therein. The well-posedness of the Cauchy Problem (1.1) has been extensively studied, and the results already obtained are satisfactory for our study of the asymptotic behavior of the solution. To put it briefly, if $1\le p_1<p_2<5$, the equation (1.1) is (conditionally) well-posed in $L^2$, and moreover $U(-t) u(t) \in L^{2,s}$ if $u_0 \in L^{2,s}$ with $0<s<p_1$. Here $U(t) =\exp (it\partial^2 /2)$ is the free propagator and $L^{2,s}$ denotes the weighted $L^2$-space of order $s$; more precisely, see Proposition 2.4 below. In what follows, we simply call the solution obtained by Proposition 2.4 ``the solution to (1.1).''

The asymptotic behavior of the solution to (1.1) is usually explained in terms of the scattering theory. When $t\to \infty$, the solution is expected to decay by the dispersive effect of the equation. Hence we can expect that the nonlinearity in the equation decays rapidly enough and loses its effect as $t\to \infty$. Thus the expected profile of the solution to (1.1) is of the form $U(t)\phi$, which is a solution to the free Schrödinger equation; here $\phi$ is a suitable function called the scattering state of the solution. This observation is, however, correct only in case $3<p_1<p_2$, namely the short-range case (on the other hand, the nonlinear term with $p_1\le 3$ is called of long-range). Indeed, the followings are well known [,12,].

(I)

If $\lambda_1, \lambda_2 \ge 0$, $3<p_1<p_2<5$ and $u_0 \in L^{2,1}$, then there exists a function $\phi \in L^2$ satisfying

$$\lim_{t\to \infty} \Vert u(t) - U(t) \phi \Vert _2 =0,$$ (1.2)

where $u(t)$ is the solution to (1.1).

(II)

If $3<p_1<p_2<5$, $u_0 \in L^{2,1}$ and $\Vert u_0 \Vert _{L^{2,1}}$ is sufficiently small, then there exists a function $\phi \in L^{2,1}$ satisfying

$$\lim_{t\to \infty} \Vert U(-t) u(t) - \phi \Vert _{L^{2,1}} =0,$$ (1.3)

where $u(t)$ is the solution to (1.1).

On the other hand, we have

(III)

If $\lambda_1 \neq 0$, $p_1\le 3$ and $u_0 \in L^{2,1} \setminus \{ 0 \}$, then there does not exist a function $\phi \in L^2$ satisfying (1.2) for the solution $u(t)$ to (1.1).

From the results above, the critical exponent for the existence of the scattering state is $p_1=3$. In this case there does not exist usual scattering state, but if we introduce the modified free dynamics of the form $U(t)\exp \bigl( -iS(t,-i\partial)\bigr) \phi$, the situation is improved, where $S(t,\xi )=\lambda_1 \vert\Hat{\phi}(\xi )\vert^2 \log t$ is the modifier of the Dollard type. Indeed, the following is known [11].

(IV)

If $3=p_1<p_2<5$, $u_0 \in L^{2,s}$ for some $s>1/2$, and $\Vert u_0 \Vert _{L^{2,s}}$ is sufficiently small, then there exists a function $\phi \in L^2 \cap L^\infty$ satisfying

$$\lim_{t \to \infty} \Vert u(t) -U(t) \exp \bigl( -iS(t,-i\partial)\bigr) \phi \Vert _2 =0,$$ (1.4)

where $u(t)$ is the solution to (1.1) and $S(t,\xi)$ is defined as above.

These results give us the asymptotic profile of the solution to (1.1). Our concern is now to know the behavior of the difference of the nonlinear solution and the asymptotic profile. In the short-range case, the following has been proved [15]:

(V)

Let $3<p_1<p_2<5$, $u_0 \in L^{2,1}$ and $\Vert u_0 \Vert _{L^{2,1}}$ is sufficiently small. Then there exists a function $\phi \in L^{2,1}$ satisfying

$$\begin{split}u(t,x)-U(t)\phi(x)&= 2i^{1/2} (p_1-3)^{-1} \, t^... ...1(\Hat{\phi}) (x/t) \\ &{}\quad + o(t^{(2-p_1)/2}) \end{split}$$ (1.5)

uniformly in $\Bbb{R}$ as $t\to \infty$, where $u(t)$ is the solution to (1.1).

The nonlinear effect explicitly appears in the right-hand side. Proof of (1.5) is based on the method of stationary phase.

Remark 1.1   The scattering theory for (1.1), especially the existence and the completeness of wave operators, has been studied in various function spaces; for short-range case, see [,6,,9,16,,18,20,21] and long-range case [,19]. In the preceding, we have mentioned only the results directly related in this paper.

The purpose of this paper is to treat the case $p_1=3$. The equation (1.1) with $p_1=3$ appears various fields of mathematical physics, and hence it is considered to be important. Since this case is of long-range, mathematical treatment is more difficult than short-range case. Now we state our theorem.

Theorem 1   Let $3=p_1<p_2<5$. Let $u_0 \in L^{2,s}$ with $s>5/2$ and let $\Vert u_0 \Vert _{L^{2,s}}$ be sufficiently small. Let $u(t)$ be the solution to (1.1). Then there exists a function $\phi \in L^{2,2} \cap \mathcal{F}^{-1} H^{2,\infty}$ satisfying

$$\bigl\Vert U(-t) u(t) - \exp \bigl(-i\tilde{S} (t,-i\partial) \bi... ...hi +t^{-1} \tsum_{j=0}^1 (\log t)^j \phi_{1,j} \bigr) \bigr\Vert _2 =o(t^{-1}).$$ (1.6)

as $t\to \infty$. Here

$$\phi_{1,1} = -i{\lambda_1}^2 \mathcal{F}^{-1} \bigl(\partial^2 (\vert \Hat{\phi}\vert^2) \vert \Hat{\phi}\vert^2 \Hat{\phi}\bigr),$$ (1.7)

$$\phi_{1,0} = -\lambda_1 \mathcal{F}^{-1} \bigl( \Bar{\Hat{\phi}} (\partial \... ...Hat{\phi}\vert^2 + \Hat{\phi}^2 \partial^2 \Bar{\Hat{\phi}} \bigr) +\phi_{1,1},$$ (1.8)

and

$$\begin{split}\tilde{S}(t,\xi ) &= \lambda_1 \vert\Hat{\phi}(\... ...t^{-(p_2-3)/2} \vert\Hat{\phi}(\xi ) \vert^{p_2-1}. \end{split}$$ (1.9)

Furthermore,

$$\begin{split}u(t,x) &= (it)^{-1/2} \exp \bigl( i\vert x\vert^... ...j \psi_{1,j} (x/t) \bigr) \\ &{}\quad +o(t^{-3/2}) \end{split}$$ (1.10)

as $t\to \infty$ uniformly in $\Bbb{R}$. Here

$$\psi_{1,2} = (-i{\lambda_1}^2 /2) (\partial \vert\Hat{\phi}\vert^2)^2\, \Hat{\phi},$$ (1.11)

$$\psi_{1,1} = (-\lambda_1 /2) (\partial^2 \vert\Hat{\phi}\vert^2) \, \Hat{\ph... ...da_1 (\partial \vert\Hat{\phi}\vert^2) \, \partial \Hat{\phi}+\Hat{\phi}_{1,1},$$ (1.12)

$$\psi_{1,0} = (-i/2) \partial^2 \Hat{\phi}+ \Hat{\phi}_{1,0}.$$ (1.13)

Remark 1.2   The restriction $p_2<5$ is necessary only to guarantee the existence of $L^{2,s}$-solution. If we assume $u_0 \in H^1 \cap L^{2,s}$, we can remove this restriction.

This paper is organized as follows. In section 2, we give several basic estimates and introduce a result on the well-posedness of (1.1). In section 3, we prove key Proposition 3.2. The expansion formulae given in Proposition 3.2 (and Theorem) are so complicated that we sketch the formal derivation of them at the beginning of the proof. Our main theorem easily follows from this proposition.

In the previous paper [24], one of the authors showed an analogous result for the Hartree equation which includes the nonlinearity like $f(u) = (\vert x\vert^{-1} * \vert u\vert^2) u$. The treatment of (1.1) is more difficult than that of the Hartree equation by the following reason. While the nonlinear term in the Hartree equation can be differentiated as many times as we like, the short-range nonlinear perturbation $f_2(u)$ in (1.1) is differentiable at most $p_2$-times because of the singularity at $u = 0$. Thus, in case $p_2$ is close to $3$, we can differentiate the phase $\tilde{S}$ only $p_2-1 \approx 2$ times, which is not sufficient to prove the theorem as long as we rely on $H^s$-theory; indeed, we would need $s>5/2$ and hence $p_2>7/2$. To avoid this difficulty and prove the theorem for $p_2>3$, we state Proposition 3.2 in terms of the Besov space $B^\sigma _{r,2}$ with $\sigma \approx 2$, $r\gg 1$, which is continuously embedded in $H^{2,\infty}$.

We shall conclude this introduction by giving the notation used in this paper:

$\Hat{\psi} (\xi )=(\mathcal{F} \psi ) (\xi ) =(2\pi )^{-1/2} \int_{-\infty}^\infty e^{-ix \xi}\psi (x) dx$ is the Fourier transform of $\psi$, and $\mathcal{F}^{-1} \psi$ its inverse. For $1\le p \le \infty$, $\Vert \cdot \Vert _p$ means the usual $L^p$-norm; $L^{2,s}=\{ \phi \in \mathcal{S}' ; \Vert \phi \Vert _{L^{2,s}}= \Vert (1+\vert x\vert^2)^{s/2} \phi\Vert _2 <\infty \}$ denotes the weighted Lebesgue space; $H^{s,r}=\{ \phi \in \mathcal{S}' ; \Vert \phi \Vert _{H^{s,r}}=\Vert (1-\partial^2)^{s/2} \phi \Vert _r < \infty \}$ denotes the Sobolev space, $H^{s,2}$ is abbreviated to $H^s$; $B^s_{r,q}$ denotes the Besov space (see section 2). $U(t) =\exp (it\partial^2 /2)$, $M(t)=\exp (ix^2 /2t)$, $[D(t)\phi ](x) = (it)^{-1/2} \phi (x/t)$.