Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity

On microlocal analyticity of solutions of first-order nonlinear PDE

— We study the microlocal analyticity of solutions u of the nonlinear equation ut = f(x, t, u, ux) where f(x, t, ζ0, ζ) is complex-valued, real analytic in all its arguments and holomorphic in (ζ0, ζ). We show that if the function u is a C2 solution, σ ∈ CharLu and 1 i σ([Lu, Lu]) < 0 or if u is a C3 solution, σ ∈ CharLu, σ([Lu, Lu]) = 0, and σ([Lu, [Lu, Lu]]) 6= 0, then σ / ∈WFau. Here WFau de...

EXISTENCE OF PERIODIC SOLUTIONS IN TOTALLY NONLINEAR NEUTRAL DIFFERANCE EQUATIONS WITH VARIABLE DELAY

Elliptic operators of totally characteristic type

This paper contains proofs of most of the results announced in [9], together with some further developments. The basic analytic question addressed here concerns the mapping properties, on the natural Sobolev spaces, of elliptically totally characteristic pseudodifferential operators, on a compact C∞ manifold with boundary. The ring, Ψb (X), of such operators was described in detail in [8] and i...

On Billiard Solutions of Nonlinear PDE ’ s ∗

This letter presents some special features of a class of integrable PDE’s admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discontinuities at peaks in their profiles. A connection is established between the peak locations and...

Positive Periodic Solutions of Nonlinear First-Order Functional Difference Equations

Ruyun Ma, Tianlan Chen, and Yanqiong Lu Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to Ruyun Ma, ruyun [email protected] Received 12 October 2010; Accepted 19 December 2010 Academic Editor: Marko Robnik Copyright q 2010 Ruyun Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which pe...