On space-time properties of solutions for nonlinear evolutionary equations with random initial data

We consider space-time properties of periodic solutions of nonlinear wave equations, nonlinear Schrödinger equations and KdV-type equations with initial data from the support of the Gibbs’ measure. For the wave and Schrödinger equations we establish the best Hölder exponents. We also discuss KdV-type equations which are more difficult due to a presence of the derivative in the nonlinearity.

Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

Rogue waves in the KdV-type models

<p>In this study, we investigate the rogue-wave-type phenomena in the physical systems described by the Korteweg-de Vries (KdV)-like equation in the form $ u_t + [u^m \sgn{u}]_x + u_{xxx} = 0 $ with the arbitrary real coefficient $m>0$. The periodic waves (sinusoidal or cnoidal) described by this equation have been shown to suffer from the modulational instability if $m \ge 3$; the modulational growth results in the formation of rogue waves similar to the Peregrine, Kuznetsov-Ma or Akhmediev breathers known for the nonlinear Schrodinger equation. In this work we focus on the rogue wave occurrence in ensembles of soliton-type waves. First of all, the characteristics of the solitary waves are investigated depending on the power $m$. The existence of solitary waves with exponential tails, as well as algebraic solitons and compactons has been shown for different ranges of the parameter $m$ values. Their energetic stability is discussed. Two solitary wave/breathers interactions are studied as elementary acts of the soliton/breather turbulence. It is demonstrated that the property of attracting solitons/breathers is a necessity condition for the formation of rogue waves. Rigorous results are obtained for the integrable versions of the KdV-type equations. Series of numerical simulations of the rogue wave generation has been conducted for different values of $m$. The obtained results are applied to the problems of surface and internal waves in the ocean, and to elastic waves in the solid medium.</p><p>The research is supported by the RNF grant 19-12-00253.</p>