My research interests cover mathematical and physical aspects of nonlinear wave theory, including the following subjects:
1. General theory of nonlinear waves, Hamiltonian structures for conservative models of continuous media, canonical transformations and normal forms of these models. Envelope equations. Periodic stationary waves and their stability. Solitons and their stability. Wave collapses. Nonlinear stage of parametric instability in continuous media. Numerical simulation of nonlinear wave equations, including numerical study of wave collapse.
2. Mathematical theory of solitons – the Inverse Scattering Method (ISM). Integrable nonlinear wave equations, development of criteria for integrability. Asymptotic behavior of integrable systems. Reductions in integrable systems and their classification. The dressing method as a generator of new integrable equations and their exact solutions.
3. Applications of the Inverse Scattering method to Differential Geometry. Classification of N-orthogonal coordinate systems. Spaces of diagonal curvature and their classification. Hyperbolic systems of Hydrodynamic type and their connection to spaces of diagonal curvature. Application of the Inverse Scattering method to General Relativity. Construction of new exact solutions of the Einstein equations.
4. Wave turbulence. Statistical description of nonlinear wave fields. Derivation of kinetic equations for waves and corrections to them. Kolmogorov-type solutions of kinetic equations and their stability. Self-similar solutions of kinetic equations. Coexistence of weak wave turbulence and coherent structures: solitons, quasisolitons, collapses. Statistical description of integrable wave models. Solitonic turbulence. Intermittency in wave turbulence. Application of weak turbulence methods to the theory of nonlinear PDE’s. Numerical modeling of weak wave turbulence.
5. Waves in the ocean. Spectra of wind-driven sea waves on deep and shallow water. Analytical and numerical study of wave breaking. Interaction of ocean waves with currents. Formation of freak waves. Wave induced transport of passive impurities. Numerical methods for the solution of kinetic wave equation with application to development of the wave-predicting models.
6. Nonlinear waves in optical fibers. Optical solitons and optical turbulence. Dispersion-managed solitons.
7. Theory of plasma turbulence with application to astrophysics. Singular spectra of plasma turbulence. Application of methods of plasma turbulence to models of competition and evolution.
8. Vortex dynamics and formation of singularities in hydrodynamics and magnetohydrodynamics with application to the theory of hydrodynamic and MHD turbulence.
9. Weak-turbulent approach to the theory of Bose-condensation.
10. Weakly-nonlinear processes in bound systems – connections to number theory (in perspective).