Related papers: Consistent Riccati Expansion and Solvability
Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalisation of apparent horizons. I present an efficient method for locating CE…
An exact Craik-Criminale (CC) solution to the incompressible Navier-Stokes (NS) equations describes the instability of an elliptical columnar flow interacting with a single Kelvin wave. These CC solutions are extended to allow…
We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group.…
This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the…
To the spectral curves of smooth periodic solutions of the $n$-wave equation the points with infinite energy are added. The resulting spaces are considered as generalized Riemann surfcae. In general the genus is equal to infinity,…
Third-order ordinary differential equations with Lie symmetry algebras isomorphic to the nonsolvable algebra $\mathfrak{sl}(2,\mathbb{R})$ admit solvable structures. These solvable structures can be constructed by using the basis elements…
In this paper, we study the regularity problem of the 3D incompressible Navier\~nStokes equations. We prove that the strong solution exists globally for new regularity criteria. For negligible forces, we give an improvement of the known…
We are concerned with the asymptotic behaviour of classical solutions of systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is a…
We present a numerical scheme for the resolution of matrix Riccati equation used in control problems. The scheme is unconditionnally stable and the solution is definite positive at each time step of the resolution. We prove the convergence…
In this paper, we propose a new Robust Nonlinear Quadratic Gaussian (RNQG) controller based on State-Dependent Riccati Equation (SDRE) scheme for continuous-time nonlinear systems. Existing controllers do not account for combined noise and…
Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new…
An integrable system of two-component nonlinear Ablowitz-Ladik (AL) equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system.…
The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a…
This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution…
We consider existence and stability of an almost periodic solution of the quasilinear system of differential equations with piecewise constant argument of generalized type. The associated linear homogeneous system satisfies exponential…
We present a number of cases of optimal control of Volterra and Fredholm integral equations that are solvable in the sense that the problem can be reduced to a solvable integral equation. This is conceptually analogous to the role of the…
In the paper obtained equivalent system of Fredholm integral equations in the study of the Dirichlet problem for the generalized Manjeron equation with non-smooth coefficients in non-classical treatment (1), (4). When non-smooth conditions…
In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…
This paper proposes a framework to assess the stability of an ordinary differential equation which is coupled to a 1D-partial differential equation (PDE). The stability theorem is based on a new result on Integral Quadratic Constraints…
We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation…