Extended lifespan of the fractional BBM equation

For 0 < α < 1, N ⩾ 2 and with initial data ‖ u 0 ‖ H N + α 2 = ε, sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂ t u + ∂ x u + u ∂ x u + | D | α ∂ t u = 0, can be extended from the hyperbolic existence time 1 ε to 1 ε 2 . For the proof we use a quasilinear modified energy method, based on a normal form transformation as in Hunter, Ifrim, Tataru, Wong (Proc. Am. Math. Soc., 143(8) (2015) 3407–3412).

On the Normal Form of the Kirchhoff Equation

Consider the Kirchhoff equation $$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$ ∂ tt u - Δ u ( 1 + ∫ T d | ∇ u | 2 ) = 0 on the d-dimensional torus $$\mathbb {T}^d$$ T d . In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

Simultaneous normal form transformation and model-order reduction for systems of coupled nonlinear oscillators

In this paper, we describe a direct normal form decomposition for systems of coupled nonlinear oscillators. We demonstrate how the order of the system can be reduced during this type of normal form transformation process. Two specific examples are considered to demonstrate particular challenges that can occur in this type of analysis. The first is a 2 d.f. system with both quadratic and cubic nonlinearities, where there is no internal resonance, but the nonlinear terms are not necessarily ε 1 -order small. To obtain an accurate solution, the direct normal form expansion is extended to ε 2 -order to capture the nonlinear dynamic behaviour, while simultaneously reducing the order of the system from 2 to 1 d.f. The second example is a thin plate with nonlinearities that are ε 1 -order small, but with an internal resonance in the set of ordinary differential equations used to model the low-frequency vibration response of the system. In this case, we show how a direct normal form transformation can be applied to further reduce the order of the system while simultaneously obtaining the normal form, which is used as a model for the internal resonance. The results are verified by comparison with numerically computed results using a continuation software.

CALCULATION OF NONLINEAR TUNE SHIFT USING BEAM POSITION MEASUREMENT RESULTS

The calculation of the nonlinear tune shift with amplitude based on the results of measurements and the linear lattice information is discussed. The tune shift is calculated based on a set of specific measurements and some extra information which is usually available, namely that about the size and particle distribution in the beam and the linear optics effect on the particles. The method to solve this problem uses the technique of normal form transformation. The proposed model for the nonlinear tune shift calculation is compared to both the numerical results for the nonlinear model of the Tevatron accelerator and the independent approximate formula for the tune shift by Meller et al. The proposed model shows a discrepancy of about 2%.

Streamline topology near non-simple degenerate critical points in two-dimensional flow with symmetry about an axis

The local flow patterns and their bifurcations associated with non-simple degenerate critical points appearing away from boundaries are investigated under the symmetric condition about a straight line in two-dimensional incompressible flow. These flow patterns are determined via a bifurcation analysis of polynomial expansions of the streamfunction in the proximity of the degenerate critical points. The normal form transformation is used in order to construct a simple streamfunction family, which classifies all possible local streamline topologies for given order of degeneracy (degeneracies of order three and four are considered). The relation between local and global flow patterns is exemplified by a cavity flow.

NINETY PLUS THIRTY YEARS OF NONLINEAR DYNAMICS: LESS IS MORE AND MORE IS DIFFERENT

I review the early (1885–1975) and more recent history of dynamical systems theory, identifying key principles and themes, including those of dimension reduction, normal form transformation and unfolding of degenerate cases. I end by briefly noting recent extensions and applications in nonlinear fluid and solid mechanics, with a nod toward mathematical biology. I argue throughout that this essentially mathematical theory was largely motivated by nonlinear scientific problems, and that after a long gestation it is propagating throughout the sciences and technology.