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Nikodem Szpak

Publications and source records attributed to Nikodem Szpak.

29 records · Page 2Linked to original sources
arXiv2011

Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2α} u =0$ on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution $S$ for intermediate times. The details of trapping are shown to depend on the power $α$, namely, we observe fast convergence to $S$ for $α>1$, slow convergence for $α=1$, and very slow (if any) convergence for $0<α<1$. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for $α>2$) by Krieger, Nakanishi, and Schlag \cite{kns}.

math-ph↗
arXiv2010

Late-time attractor for the cubic nonlinear wave equation

We apply our recently developed scaling technique for obtaining late-time asymptotics to the cubic nonlinear wave equation and explain appearance and approach to the two-parameter attractor found recently by Bizon and Zenginoglu.

math-ph↗
arXiv2010

Large data pointwise decay for defocusing semilinear wave equations

We generalize the pointwise decay estimates for large data solutions of the defocusing semilinear wave equations which we obtained earlier under restriction to spherical symmetry. Without the symmetry the conformal transformation we use provides only a weak decay. This can, however, in the next step be improved to the optimal decay estimate suggested by the radial case and small data results. This is the first result of that kind.

math.AP↗
arXiv2009

Comment on "Late-time tails of a self-gravitating massless scalar field revisited" by Bizon et al: The leading order asymptotics

In Class. Quantum Grav. 26 (2009) 175006 (arXiv:0812.4333v3) Bizon et al discuss the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in odd spatial dimensions. They derive explicit expressions for the leading order asymptotics for solutions with small initial data by using formal series expansions. Unfortunately, this approach misses an interesting observation that the actual decay rate is a product of asymptotic cancellations occurring due to a special structure of the nonlinear terms. Here, we show that one can calculate the leading asymptotics more directly by recognizing the special structure and cancellations already on the level of the wave equation.

gr-qc↗
arXiv2009

Global pointwise decay estimates for defocusing radial nonlinear wave equations

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.

math.AP↗
arXiv2009

Linear and nonlinear tails I: general results and perturbation theory

For nonlinear wave equations with a potential term we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method which reduces the wave equation to an algebraic one. We demonstrate that already first and second perturbation orders, satisfying linear equations, can provide precise information about the decay of the full solution to the nonlinear wave equation. In a forthcoming publication (part II) we address the issue of optimal decay estimates and precise asymptotics under spherical symmetry where the perturbation equations can be solved almost exactly.

math-ph↗
arXiv2007

Simple proof of a useful pointwise estimate for the wave equation

We give a simple proof of a pointwise decay estimate in 3+1 dimensions stated in two versions, making advantage of a particular simplicity of inverting the spherically symmetric part of the wave operator and of the comparison theorem. We briefly explain the role of this estimate in proving decay estimates for nonlinear wave equations or wave equations with potential terms.

math-ph↗
arXiv2007

Weighted-$L^\infty$ and pointwise space-time decay estimates for wave equations with potentials and initial data of low regularity

We prove weighted-$L^\infty$ and pointwise space-time decay estimates for weak solutions of a class of wave equations with time-independent potentials and subject to initial data, both of low regularity, satisfying given decay bounds at infinity. The rate of their decay depends on the asymptotic behaviour of the potential and of the data. The technique is robust enough to treat also more regular solutions and provides decay estimates for arbitrary derivatives, provided the potential and the data have sufficient regularity, but it is restricted to potentials of bounded strength (such that $-Δ-|V|$ has no negative eigenvalues).

math-ph↗
arXiv2006

Spontaneous particle creation in time-dependent overcritical fields

It is believed that in presence of some strong electromagnetic fields, called overcritical, the (Dirac) vacuum becomes unstable and decays leading to a spontaneous production of an electron-positron pair. Yet, the arguments are based mainly on the analysis of static fields and are insufficient to explain this phenomenon completely. Therefore, we consider time-dependent overcritical fields and show, within the external field formulation, how spontaneous particle creation can be defined and measured in physical processes. We prove that the effect exists always when a strongly overcritical field is switched on, but it becomes unstable and hence generically only approximate and non-unique when the field is switched on and off. In the latter case, it becomes unique and stable only in the adiabatic limit.

hep-th↗
arXiv2004

Quasinormal mode expansion and the exact solution of the Cauchy problem for wave equations

Solutions for a class of wave equations with effective potentials are obtained by a method of a Laplace-transform. Quasinormal modes appear naturally in the solutions only in a spatially truncated form; their coefficients are uniquely determined by the initial data and are constant only in some region of spacetime -- in contrast to normal modes. This solves the problem of divergence of the usual expansion into spatially unbounded quasinormal modes and a contradiction with the causal propagation of signals. It also partially answers the question about the region of validity of the expansion. Results of numerical simulations are presented. They fully support the theoretical predictions.

gr-qc↗