SMOOTHING EFFECTS FOR NONLINEAR DERIVATIVE SCHRÖDINGER EQUATIONS

PATRICK-NICOLAS PIPOLO

1. Introduction

This presentation deals with the study of smoothing and analyticity effects for some nonlinear derivative Schrödinger equations.

2. Smoothing effects for some derivative nonlinear Schrödinger equations

This a joint work with Pr. Nakao Hayashi and Pr. Pavel.I Naumkin. In this part we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations :

\begin{cases}ıu_{t} + u_{xx} = N (u, \overline{u}, u_{x}, \overline{u_{x}}), & x ∈ ℝ, t ∈ ℝ, \\ u(x, 0) = u_{0}(x) , & x ∈ ℝ, \end{cases} (1)

where the nonlinearity is

N (u, \overline{u}, u_{x}, \overline{u_{x}}) = K_{1}|u|^{2}u + K_{2}|u|^{2}u_{x} + K_{3}u^{2}\overline{u_{x}} + K_{4}|u_{x}|^{2}u + K_{5}\overline{u}u_{x}^{2} + K_{6}|u_{x}|^{2}u_{x}.

The function K_{j} = K_{j}(|u|^{2}) satisfy K_{j}(z) ∈ C^{l+3}([0, +∞); ℂ). If K_{5}(z) = \frac{1}{1+z} and K_{1} = K_{2} = K_{3} = K_{4} = K_{6} = 0 then the equation (1) is called classical pseudospin magnet model [5]. We introduce some functionnal spaces in order to state our results : L^{p}(ℝ) = {φ ∈ S′(ℝ) : ∥φ∥_{p} < ∞}, where ∥φ∥_{p} = (∫ |φ(x)|^{p}dx)^{1/p} if 1 ≤ p < ∞, if p = ∞ then ∥φ∥_{∞} = ess.sup{|φ(x)|; x ∈ ℝ}. In order to keep the notations clear, we write ∥φ∥ = ∥φ∥_{2}. The weighted Sobolev spaces are : H_{p}^{m,s} = {φ ∈ S′(ℝ) : ∥φ∥_{m,s,p} =

∥(1 + x^{2})^{s/2} (1 − ∂_{x}^{2})^{m/2} φ∥_{p} < ∞}, m, s ∈ ℝ, 1 ≤ p ≤ ∞. For simplicity, for p = 2 we write H^{m,s} = H_{2}^{m,s}

and ∥ϕ∥_{m,s} = ∥ϕ∥_{m,s,2}. We denote also H^{m,∞} = ∩_{s=1}^{∞}H^{m,s}. Let C(I; B) be the space of continuous functions from a time interval I to a Banach space B. Our main results of this section are the followings.

Theorem 1. We assume that the nonlinear term N does not depend on ū_{x} and the initial data u_{0} is such that u_{0} ∈ H^{3,l}with l ∈ ℕ. Then there exists T > 0 such that there exists a unique solution

u ∈ C ([−T, T]; H^{2,0}) ∩ L^{∞} (−T, T; H^{3,0}) ∩ C ([−T, T]\{0}; C^{l+2}(ℝ))

of the Cauchy problem (1) such that

sup_{t∈[−T,T]} |t|^{k} ∥(1 + x^{2})^{−k/2} ∂_{x}^{k}u(t)∥_{2,0} < ∞ for 0 ≤ k ≤ l.

For the case of nonlinearities depending on ū_{x}, we have to assume the smallness condition on the initial data. We have the following result.

Theorem 2. We assume that the nonlinear term N depends on ū_{x} and that the initial data u_{0} is such that u_{0} ∈ H^{3,l} with l ∈ ℕ and ∥u_{0}∥_{3,l} is small enough . Then there exists T > 0 such that there exists a unique solution which has the same results as in Theorem 1

Date: 25/01/2000.

In this work, we apply an operator which is defined by N. Hayashi, P.I. Naumkin and the author [3]. This operator is defined by S(φ) = cosh(φ) + i sinh(φ)H. φ is defined later as

φ(x, t) = \frac{1}{δ} ∑_{k=0}^{l} ∂_{x}^{−1}(|J^{k}ũ(x, t)|^{2} + |J^{k}ũ_{x}(x, t)|^{2} + ∑_{2≤j≤5} |J^{k}K_{j}ũ|^{2} + |J^{k}K_{6}ũ_{x}|^{2}),

where Hϕ = \frac{1}{π} Pv ∫ \frac{ϕ(x′)dx′}{x−x′}, ∂_{x}^{−1} = ∫_{−∞}^{x} … dx′ and ũ is related to solutions to (1). It enables us to avoid the use of the well known results of the pseudo-differential operators and so by virtue of simple explicit computations we treat the problem in the natural order Sobolev space H^{3,0}.

3. Smoothing effects for some derivative nonlinear Schrödinger equation without smallness condition

We consider the equation described in the previous part and we will remove the smallness condition on the data assumed in Theorem 1. To obtain the new result, we combine a diagonalization technique to the method used in part 1.

Theorem 3. We assume that the initial data u_{0} is such that u_{0} ∈ H^{3,l}with l ∈ ℕ. Then there exists T > 0 such that there exists a unique solution

u ∈ C ([−T, T]; H^{2,0}) ∩ L^{∞} (−T, T; H^{3,0}) ∩ C ([−T, T]\{0}; C^{l+2}(ℝ))

of the Cauchy problem (1) such that

sup_{t∈[−T,T]} |t|^{k} ∥(1 + x^{2})^{−k/2} ∂_{x}^{k}u(t)∥_{3,0} < ∞ for 0 ≤ k ≤ l.

Note that there is no more a any smallness condition on the initial data.

4. Analytical smoothing effects for some derivative nonlinear Schrödinger equations

This is a joint work with Pr. Nakao Hayashi and Pavel I. Naumkin. We study an analytical smoothing effect of solutions to the Cauchy problem,

\begin{cases}iu_{t} + u_{xx} = N (u, ū, u_{x}, ū_{x}), & x ∈ ℝ, t ∈ ℝ, \\ u(0, x) = u_{0}(x), & x ∈ ℝ, \end{cases} (2)

where the nonlinearity is

N (u, ū, u_{x}, ū_{x}) = K_{1}|u|^{2}u + K_{2}|u|^{2}u_{x} + K_{3}u^{2}ū_{x} + K_{4}|u_{x}|^{2}u + K_{5}ūu_{x}^{2} + K_{6}|u_{x}|^{2}u_{x},

The coefficients K_{j} = K_{j}(|u|^{2}) = K_{j}(f) are analytic and have analytic continuations z = f + ig in the circle |z| < ρ. Thus these functions can be written with their Taylor expansion

K_{j}(z) = ∑_{n=0}^{∞} a_{n,j}z^{n}, a_{n,j} = \frac{1}{n!}K_{j}^{(n)}(0) = \frac{1}{n!} \frac{d^{n}}{dz^{n}} K_{j}(0).

We also assume that ∑_{n=0}^{∞}|a_{n,j}||z|^{n} ≤ C(ρ) for |z| < ρ, where C(ρ) is a continuous function with ρ as a variable.

Our aim is to extend the results described in the first and second part and to the analytical case and to find a solution of (2) and study analytical problems. Precisely if the initial data u_{0} satisfy the condition (cosh βx)u_{0} ∈ H^{3,0} with the decay ∥(cosh βx)u_{0}∥_{3,0} < ρ in case of a nonlinearity N which does not depend on the term ū_{x}, then there exists a unique solution u of the Cauchy problem (2) and a time T > 0 which is analytic with respect to x and has an analytic continuation on the complex plane z = x+iy with |y| < 2|βt|, for all t ∈ [−T, T]\{0}. In case of a nonlinearity N depending on ū_{x}, we must assume again the additional smallness condition on the initial data u_{0}. So in this part we extend the estimations proved for the linear Schrödinger in the first section to the analytical case. Then the result stated above is proved by a contraction mapping method. We sum up these results in the following theorems.

Theorem 4. We assume that the nonlinear term N does not depend on ū_{x}, and the initial data u_{0} are such that u_{0} cosh βx ∈ H^{3,0}, where β ∈ ℝ and the norm ∥u_{0} cosh βx∥_{3,0} < ρ. Then for some time T > 0 there exists a unique solution u of the Cauchy problem (1) such that u ∈ C ([−T, T]; H^{2,0}) ∩ L^{∞} (−T, T; H^{3,0}) and the solution u has an analytic continuation u(t, z) to the strip {z = x + iy; −∞ < x < ∞, −2|tβ| < y < 2|tβ|, t ∈ [−T, T]\{0}} satisfying the estimate

sup_{−2|tβ|<y<2|tβ|} |u(t, x + iy)| ≤ C cosh βx∥u_{0} cosh βx∥_{3,0}
for all (t, x) ∈ [−T, T]\{0} × ℝ.

For the case of the nonlinearities depending on ū_{x} we have to assume the additional smallness condition on the initial data. We prove the following result.

Theorem 5. We assume that the nonlinear term N depends on ū_{x}, the initial data u_{0} are such that u_{0} cosh βx ∈ H^{3,0}, β ∈ ℝ and the norm ∥u_{0} cosh βx∥_{3,0} is sufficiently small. Then the same results as in Theorem 4 are true.

Through our recent research, we are able to prove an improvement of the previous theorem in this way :

Theorem 6. We assume that the initial data u_{0} are such that u_{0} cosh βx ∈ H^{3,0}, β ∈ ℝ. Then the same results as in Theorem 4 are true.

Therefore in this theorem, we do not assume anymore any smallness condition on the initial data. Therefore we improve theorem 5.

References

[1] H. Chihara : Gain of regularity for semilinear Schrödinger equations, preprint 1999.

[2] A.Friedman, ‘Partial Differential Equations’, (1983), Krieger.

[3] N.Hayashi, P. I. Naumkin, P-N Pipolo : Smoothing effects for some derivative nonlinear Schrödinger equations, in Discrete and Continuous Dynamical Systems, vol 5, n. 3, p. 685-695, 1999.

[4] K. Kato : New idea for Proof of analyticity of solutions to analytic nonlinear elliptic equations, in Science University of Tokyo journal of Mathematics, vol 32, n. 2, p. 157-161, 1996.

[5] V.G. Makhankov and O.K. Pashaev : Integrable pseudospin models in condensed matter,Sov. Sci. Rev. C. Math. Phys.,(1992), vol 9, p. 1-152.

Patrick-Nicolas Pipolo, Department of Applied Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162, Japan