Global existence of solutions for nonlinear Schroedinger equations in one space dimension

Journal Article · Sat Dec 31 04:00:00 EST 1994 · Communications in Partial Differential Equations

DOI:https://doi.org/10.1080/03605309408821079· OSTI ID:146814

Katayama, Soichiro ^[1]; Tsutsumi, Yoshio ^[2]

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1. Kyoto Univ. (Japan)
2. Univ. of Tokyo (Japan)

In this paper, we consider the initial value problems of nonlinear Schroedinger equations. Concerning the global existence of the solution, Klainerman-Ponce and Shatah showed that if F is a smooth function of degree 4 near the origin and satisfied (E), then there exists a unique global solution to (1.1) - (1.2) for any small initial data belonging to certain Sobolev spaces. One of the authors showed that if F is a homogeneous polynomial of degree satisfying the assumption and the {open_quotes}null gauge condition of order 3{close_quotes}, then there exits a global solution of (1.1) - (1.2) for the small initial data for the null gauge condition. In this paper we shall study the global existence for (1.1) - (1.2) when F satisfies the null gauge condition of order 3, without assuming (E). But, as the usage of the operator J = x+itD is essential in our proof, we need assume the gauge invariance of (1.1) to show the global existence of solution for (1.1)-F(1.2). The gauge invariance ensures us the chain rule for J.

• Sponsoring Organization: USDOE
• OSTI ID: 146814
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 11-12 Vol. 19; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
ANALYTICAL SOLUTION
DIFFERENTIAL CALCULUS
MATHEMATICAL LOGIC
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
SCHROEDINGER EQUATION