zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances. (English) Zbl 1075.35080
Summary: The nonlinear Schrödinger (NLS) equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. The purpose of this paper is to prove estimates, between the formal approximation, obtained via the NLS equation, and true solutions of the original system in case of nontrivial quadratic resonances. It turns out that the approximation property (APP) holds if the approximation is stable in the system for the three-wave interaction (TWI) associated to the resonance. We construct a counterexample showing that the NLS equation can fail to approximate the original system if instability occurs for the approximation in the TWI system. In the unstable case we give some arguments why the validity of the APP can be expected for spatially localized solutions and why it cannot be expected for non-localized solutions. Although, we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations, too.

35Q55NLS-like (nonlinear Schrödinger) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
78A60Lasers, masers, optical bistability, nonlinear optics
Full Text: DOI
[1] M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981. · Zbl 0472.35002
[2] Craig, W.; Sulem, C.; Sulem, P. L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5, 497-552 (1992) · Zbl 0742.76012
[3] Kalyakin, L. A.: Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium. Math. USSR sb. Surveys 60, No. 2, 457-483 (1988) · Zbl 0699.35135
[4] Kirrmann, P.; Schneider, G.; Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. roy. Soc. Edinburgh 122A, 85-91 (1992) · Zbl 0786.35122
[5] Schneider, G.: Validity and limitation of the Newell -- Whitehead equation. Math. nachr. 176, 249-263 (1995) · Zbl 0844.35120
[6] Schneider, G.: Approximation of the Korteweg -- de Vries equation by the nonlinear Schrödinger equation. J. differential equations 147, 333-354 (1998) · Zbl 0940.35179
[7] Schneider, G.: Justification of modulation equations for hyperbolic systems via normal forms. Nonlinear differential equations appl. (NODEA) 5, 69-82 (1998) · Zbl 0890.35082
[8] Schneider, G.; Uecker, H.: Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems. Asymptotic anal. 28, No. 2, 163-180 (2001) · Zbl 0988.35158
[9] Shatah, J.: Normal forms and quadratic nonlinear Klein -- Gordon equations. Comm. pure appl. Math. 38, 685-696 (1985) · Zbl 0597.35101
[10] Zakharov, V. E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. phys. J. appl. Mech. tech. Phys. 4, 190-194 (1968)