We consider the focusing modified Zakharov–Kuznetsov (mZK) equation in two space dimensions. prove that solutions which blow up finite time \(H^1(\mathbb {R}^{2})\) norm have property they concentrate a non-trivial portion of their mass (more precisely, at least amount equal to ground state) blow-up time. For finite-time \(H^s(\mathbb {R}^2)\) for \(\frac{17}{18}< s < 1\), we slightly weaker re...

We numerically realize breather gas for the focusing nonlinear Schr\"odinger equation. This is done by building a random ensemble of N $\sim$ 50 breathers via Darboux transform recursive scheme in high precision arithmetics. Three types gases are synthesized according to three prototypical spectral configurations corresponding Akhmediev, Kuznetsov-Ma and Peregrine as elementary quasi-particles ...

Chinese members of the genus Aspidimerus Mulsant, 1850 are reviewed. Ten species are recognized, including two new species: A. zhenkangicus Huo & Ren, sp. n. and A. menglensis Huo & Ren, sp. n. A. kabakovi Hoàng is recorded from China for the first time. A. blandus (Mader, 1954) is recognized as synonymous with A. ruficrus Gorham, 1895 (syn. n.). Aspidimerus rectangulatus Kuznetsov & Pang, 1991...

We study a rather general class of optimal “ballistic” transport problems for matrix-valued measures. These naturally arise, in the spirit Brenier (Commun Math Phys 364(2):579–605, 2018), from certain dual formulation nonlinear evolutionary equations with particular quadratic structure reminiscent both incompressible Euler equation and Hamilton–Jacobi equation. The examples include ideal MHD, t...

This article is an introduction to the Petersson trace formula and Kuznetsov trace formula, both of which are now important, standard techniques in analytic number theory. To illustrate their applications to modular forms, we will explain their role in a proof of subconvexity bounds for Rankin-Selberg L-functions L(s, f ⊗ g) on the critical line σ = 1/2, where here and throughout, we write s = ...