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On movable singularities of self-similar solutions of semilinear wave equations. (English) Zbl 1268.34184

Balser, Werner (ed.) et al., Formal and analytic solutions of differential and difference equations. Proceedings of the conference, Bȩdlewo, Poland, August 8–13, 2011. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-16-4/pbk). Banach Center Publications 97, 59-72 (2012).
The author continues (see [Appl. Math. Comput. 217, No. 22, 9451–9466 (2011; Zbl 1223.35237)]) to study the differential equation \[ (1-{{\rho }^{2}}){u}''+(\frac{n-1}{\rho }-\frac{2(p+1)}{(p-1)}\rho ){u}'-\frac{2(p+1)}{{{(p-1)}^{2}}}u+{{u}^{p}}=0\tag{1} \] for self-similar profiles \(u(\rho)\) of the semilinear wave equation \({{\Phi }_{tt}}-\Delta \Phi -{{\Phi }^{p}}=0\). He examines the radius of convergence for power series describing a local solution of (1) at \(\rho =0\) and \(\rho =1\). By means of the theory of the Lane-Emden equation, the author shows that the structure of the movable singularities for local analytic solutions at the origin of (1) is similar to those of the Lane-Emden equation, and he obtains a function describing approximately their position on the complex plane.
For the entire collection see [Zbl 1259.34003].

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
35L05 Wave equation

Citations:

Zbl 1223.35237
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