Functions operating on modulation spaces and nonlinear dispersive equations

2014/12/01 by Divyang G. Bhimani, Bhimani, Divyang G., P. K. Ratnakumar +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1412.0362

Abstract

The aim of this paper is two fold. We show that if a complex function F on \C operates in the modulation spaces M^p,1(\Rⁿ) by composition, then F is real analytic on \R² ≈ \C. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form |u|^αu. We also characterise the functions that operate in the modulation space M^1,1(\Rⁿ). The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonlinearities are also studied in some weighted modulation spaces M^p,q_s(\Rⁿ).

Citations

• Estimates for translation invariant operators in Lp spaces

Related

• Nonlinear operations on a class of modulation spaces
• Modulation Spaces and Nonlinear Evolution Equations
• The Hartree-Fock equations in modulation spaces
• Superposition in Modulation Spaces with Ultradifferentiable Weights
• Composition Operators on Wiener amalgam Spaces