Modified wave operators for the fourth-order nonlinear Schrödinger-type equation with cubic nonlinearity. (English) Zbl 1123.35072

The author considers the fourth-order nonlinear Schrödinger equation
\[ i\partial_tu-{1\over4} \partial_x^4u=\lambda| u| ^2u \] with cubic nonlinearity, a long range scattering problem where the solutions are not expected to be asymptotically free. Following the basic results in T. Ozawa [Commun. Math. Phys. 139, No. 3, 479–493 (1991; Zbl 0742.35043)], for final data \(v(t)\) that are small in a weighted Sobolev space with mean zero, a unique global solution \(u \in C(\mathbb R,L^2(\mathbb R)) \cap L^8_{loc} (\mathbb R,L^\infty(\mathbb R))\) is constructed such that \[ \| u(t)-v(t)\| _{L^2_x(\mathbb R)}=O(t^{-\alpha}) \] as \(t\to\infty\), for all \(\alpha\in (3/8,1)\). The proof consists of two steps: The first step is an application of the contraction mapping principle to find the solution \(u\) near \(\infty\), using Strichartz type estimates for the free evolution group \(W(t)\). In contrast to the second-order problem, here no “MDFM”-decomposition is known for \(W(t)\). This difficulty is overcome by using the method of stationary phase. In the second step global existence as \(t\to-\infty\) is shown.


35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs
81U05 \(2\)-body potential quantum scattering theory


Zbl 0742.35043
Full Text: DOI


[1] Fukumoto, Journal of Fluid Mechanics 417 pp 1– (2000)
[2] Segata, Differential and Integral Equations 16 pp 841– (2003)
[3] Segata, Proceedings of the American Mathematical Society 132 pp 3559– (2004)
[4] Lin, Journal of Functional Analysis 30 pp 245– (1978)
[5] Ginibre, Journal of Functional Analysis 32 pp 1– (1979)
[6] Barab, Journal of Mathematical Physics 25 pp 3270– (1984)
[7] Tsutsumi, Bulletin of the American Mathematical Society 11 pp 186– (1984)
[8] Ozawa, Communications in Mathematical Physics 139 pp 479– (1991)
[9] Carles, Communications in Mathematical Physics 220 pp 41– (2001)
[10] Hayashi, American Journal of Mathematics 120 pp 369– (1998)
[11] Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. American Mathematical Society: Providence, RI, 2003.
[12] Ginibre, Communications in Mathematical Physics 151 pp 619– (1993)
[13] Ginibre, Journal de Mathematiques Pures et Appliquees 64 pp 363– (1985)
[14] Hayashi, Electronic Journal of Differential Equations 2004 pp 16– (2004)
[15] Hayashi, Annales de l Institut Henri Poincaré Physique Théorique 48 pp 17– (1988)
[16] Ozawa, Sugaku 50 pp 337– (1998)
[17] Shimomura, Differential and Integral Equations 17 pp 127– (2004) · Zbl 1203.35183
[18] Tsutsumi, Annales de l Institut Henri Poincaré Physique Théorique 43 pp 321– (1985)
[19] Ben-Artzi, Comptes Rendus de l Academie des Sciences Paris Série I-Mathematique 330 pp 87– (2000)
[20] The Analysis of Linear Partial Differential Operators. I. Springer: New York, 1990.
[21] Kenig, Indiana University Mathematics Journal 40 pp 33– (1991)
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