The Cauchy problem for nonlinear quadratic interactions of the Schrödinger type in one dimensional space

Abstract

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations with quadratic nonlinearities, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for data in Sobolev spaces with low regularity. To obtain the local theory, I prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, I establish the global well-posedness in H s (R) × H s (R), for some negatives indexes s. The proof of the global result uses the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.

Well-posedness and lower bounds of the growth of weighted norms for the Schrödinger–Korteweg–de Vries interactions on the half-line

“…(a) We believe that the same approach used to prove Theorem 1.4 can provide similar results for other nonlinear dispersive systems on the half-line, for example, the quadratic Schrödinger system [3] or the coupled multicomponent NLS-gKortewegde Vries system [4]. We finally notice that there exists in the literature some results concerns the study of the dynamic for some nonlinear dispersive equations on the half-line.…”

The nonlinear Quadratic Interactions of the Schrödinger type on the half-line

“…In the mathematical context N. Hayashi, T. Ozawa and K. Tanaka in [13] obtained local well-posedness for the Cauchy problem (1.2) on the spaces L 2 (R n ) × L 2 (R n ) for n ≤ 4 and H 1 (R n ) × H 1 (R n ) for n ≤ 6. On the paper [2] the first author obtained local well posedness for the model posed on real line by assuming low regularity assumptions. In [20] the time decay estimates of small solutions to the systems under the mass resonance condition in 2dimensional space was revised.…”

On the low-regularity global well-posedness of a system of nonlinear Schrodinger Equation

“…If α = 1 2 , the Cauchy problem (1.2) is locally well-posed for s ≥ 0. If 1 2 < α < 1, the Cauchy problem (1.2) is locally well-posed for s ≥ − 1 2 . If 0 < α < 1 2 , the Cauchy problem (1.2) is locally well-posed for s > − 3 4 .…”

“…If 1 2 < α < 1, the Cauchy problem (1.2) is locally well-posed for s ≥ − 1 2 . If 0 < α < 1 2 , the Cauchy problem (1.2) is locally well-posed for s > − 3 4 . In this paper, we do not prove any ill-posed result to complement the cases when s is below the thresholds above, instead, we give some counterexamples in both ranges of α to illustrate that the desired bilinear estimate can not hold for s < 0.…”

“…If 1 2 < α < 1, the Cauchy problem (1.2) is locally well-posed for s > − 1 4 . If 0 < α < 1 2 , the Cauchy problem (1.2) is globally well-posed for s > − 5 8 . Notice that the global well-posedness in L 2 (s = 0) is the direct consequence of local well-posedness and conservation of mass, hence we will only proof the global existence result as well as the growth of Sobolev norms below L 2 .…”