We examine the convergence in the Krylov–Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.

Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] derived two U (1) -invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schrdinger equation and admit peakons ($ non-smooth\solitons $). In the present paper, these two peakon PDEs are generalized to a family of U (1) -invariant peakon PDEs parametrized by the real projective line U (1) . All equations in this family are shown to posses $ conservative\peakon\solutions $(whose Sobolev U (1) norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon ODEs are shown to be Hamiltonian with respect to several Poisson structures. It is shown that the resulting Hamilonian peakon flows in the case of the two peakon equations derived in [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] form orthogonal families, while in general the Hamiltonian peakon flows for two different equations in the general family intersect at a fixed angle equal to the angle between two lines in U (1) parametrizing those two equations. Moreover, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the non-periodic time domain.

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We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.

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