Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation

We study the asymptotic behavior in time of solutions to the Cauchy problem for the derivative nonlinear Schrodinger equation where a E R. We prove that if + is sufficiently small, then the solution of (DNLS) satisfies the time decay estimate where == s~; == + m, s E R. The above L° time decay estimate is very important for the proof of existence of the modified scattering states to (DNLS). In order to derive the desired estimate we introduce a certain phase function. @ Elsevier, Paris

, [19] to study the propagation of circular polarized Alfvén waves in plasma. The (DNLS) has an infinite family of conservation laws and can be solved exactly by the inverse scattering transform method [16]. The local existence of solutions to (DNLS) was proved in [23], [24] under the condition that uo E (s &#x3E; 3/2) and the global existence of solutions to (DNLS) was also proved in [23], [24] for the initial data Uo E ~2 ~ ° such that the norm is sufficiently small. These results were improved in [6], [7]. More precisely, the global existence of solutions to (DNLS) was shown in [6] if the initial data uo E are sufficiently small in the norm ~0~2 and in [7] the smallness condition on ~u0~L2 was given explicitly in the form I and also the smoothing effect of solutions was studied.
The final state problem for (DNLS) was studied in [9] and the existence of modified wave operators and the non existence of the scattering states in 161 ASYMPTOTICS IN TIME FOR DNLS EQUATION L2 were shown. For the cubic nonlinear Schrodinger equation, the modified wave operators were constructed and the non existence of scattering states in L2 were proved in paper [22] (see [4], [5] for the higher dimensional case). However the result in [9] does not say the asymptotics in large time of solutions to the Cauchy problem (DNLS). As far as we know there are no results concerning the time decay estimate of solutions to (DNLS). Our purpose in this paper is to prove L° decay of solutions of (DNLS) with the same decay rate as that of solutions to the linear Schrodinger equation and to give the large time asymptotic formula for the solutions which shows the existence of the modified scattering states. Our proof of the results is based on the choice of the function space which was done in the recent paper [20], [21], the gauge transformation method used in [6], [7], [17] and the systematic use of the operator J == ~ + The operator J was used previously to study (1) the scattering theory to nonlinear Schrodinger equations with power nonlinearities ( [3], [11], [25]), (2) the time decay of solutions ( [2], [10]) and (3) smoothing properties of solutions ( [12], [13]) to some nonlinear Schrodinger equations. The main results in [3], [ 1 1 ] , [25] are obtained through the following a priori estimate of solutions ~Ju~L2 ~ C.
This estimate along with the Sobolev type inequality (Lemma 2.2) in the one dimensional case yields the required L° time decay of solutions for the nonlinear Schrodinger equations with higher order power nonlinearity.
However it seems to be difficult to get the same estimate C in the case of cubic nonlinearity. To derive the desired a priori estimates of solutions in our function space taking L° time decay estimates of solutions into acount we have to introduce a certain phase function since the previous methods ( [3], [11], [25]) based solely on the a priori estimates of the value (x + without specifying any phase function does not work for (DNLS). The nonexistence of the usual L~ scattering states shows that our result is sharp. Some phase functions were used in [4], [22] to prove the existence of the modified wave operators to the nonlinear Schrodinger equations with the critical power nonlinearity. Their results were shown through an integral equation corresponding to the original Cauchy problem and therefore the derivative loss in the equation can not be canceled via integration by parts. The method presented here is general enough, since it is also applicable to a wide class of nonlinear Schrodinger equations with nonlinearities containing derivatives of unknown function and for the generalized and modified Benjamin-Ono equations (see [14]), where derivatives are treated via integration by parts. We finally note that the cubic nonlinear Schrodinger equation of the form N. HAYASHI AND P. 1. NAUMKIN was considered in [15] and the existence of scattering states in L~ were shown by making use of a priori estimate of Ju for the solutions of this equation. However it is clear that their method does not work for (DNLS) because the non existence of scattering states in L~ was shown in [9].
Let us explain the difference of our approach from the previous methods. The precise time decay estimate C(1+t,)-1~2 of solutions u of (DNLS) will be obtained in Section 3 from the asymptotics of solutions u of (DNLS) for large time (Lemma 2.5). The main term of this asymptotics is determined by the Fourier transform of U(-t)u(t) (where U(t) is the free evolution group associated with the linear Schrodinger equation) and we also need a certain phase function to obtain the desired estimates concerning the main term (see Lemma 3.2). The remainder term of the asymptotics is estimated by the norm (see Lemma  where 03B3, 03BB E R and p &#x3E; 3 if a = 0, p &#x3E; 4 0. The reason why we need the condition that p &#x3E; 4 when a # 0 comes from the fact that our method requires some regularity of solutions to obtain time decay estimates of solutions. On the other hand, if a = 0, by using the method of this paper we can obtain the asymptotic formula and the time decay estimate in the n space. We organize our paper as follows. In Section 2 we give some preliminary results. Sobolev type inequalities are stated in Lemmas 2.1-2.2 and Lemma 2.3 is used to treat the nonlinear terms. In Lemma 2.4 some estimates of functions are shown which are needed when the gauge transformation technique is used. Lemma 2.5 says that the time decay of functions can be represented by using the free evolution group of the Schrodinger operator. In Section 3 we first give the local existence theorem (Theorem 3.1 ) without proof and we prove the main results of this paper by showing a priori estimates of local solutions to (DNLS) in Lemma  provided that the right hand side is finite.
Proof. -By applying Lemma 2.1 with 03C8 = S(-t)f, we obtain where we have used the identity (C). We again apply (C) to the above inequality to get the desired result. Proof. -We only prove the last inequality in the lemma since the first one is proved in the same way. From  and a unique solution u of (DNLS) such that For the proof of Theorem 3.1, see [6]- [8]. In order to obtain the a priori estimates of solutions u to (DNLS) in XT we translate the original equation into another system of equations. In the same way as in the derivation of [6, (2.3)] we find that u(l) and U(2) defined by Lemma (3.4) where we have used the condition that the initial data is sufficiently small and Theorem 3.1. We again use Theorem 3.1 to get Applying the Gronwall inequality to (3.6), we obtain This implies In the same way as in the proof of (3.7) we have The lemma follows by applying (3.8), the condition that the initial data is sufficiently small and Theorem 3. Proof -By Sobolev's inequality (Lemma 2.1) and Lemma 3.1 we have We assume that t &#x3E; 1. From Lemma 2.5 and Theorem 3.1 it follows that We now consider the last term of the right hand side of (3.10). Multiplying both sides of (DNLS) by U(-t), we find that We put v(t) = U(-t)u(t). Then   Therefore we obtain where Substituting (3.12) into (3.11 ) and taking the Fourier transform, we obtain In order to eliminate the second term of the left hand side of the equation ( (3.13) where Integrating (3.15) with respect to t from 1 to t, we have for n = 0,1 By a simple calculation where we take j3 satisfying 0 j3 1/4. Since we have by (3.17), Holder's inequality and the identity J Vol. 68, n° 2-1998.