Related papers: Consistent Riccati Expansion and Solvability
We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear…
A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In…
We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown…
We present quantum graphs with remarkably regular spectral characteristics. We call them {\it regular quantum graphs}. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly…
A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial…
We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one…
We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, whichn in turn is related to the stability algebra of Virasoro orbit. Using Darboux…
We investigate whether the recently proposed PT-symmetric extensions of generalized Korteweg-de Vries equations admit genuine soliton solutions besides compacton solitary waves. For models which admit stable compactons having a width which…
An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the…
A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…
Consider the positive, radial solutions of the nonlinear biharmonic equation $\Delta^2 u = u^p$. There is a critical power $p_c$ such that solutions are linearly stable if and only if $p\geq p_c$. We obtain their asymptotic expansion at…
It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the…
Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian…
Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art…
We prove a variant of Krein's resolvent formula expressing the resolvents of self-adjoint extensions through the associated boundary conditions. Applications to solvable quantum-mechanical problems are discussed.
The resonant-state expansion (RSE) provides a precise and computationally cheap tool to find resonant states in complex systems using the optical modes of a simpler system as a basis. We apply the RSE to a photonic crystal slab in order to…
We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear…
The discrete-time algebraic Riccati equation (DARE) have extensive applications in optimal control problems. We provide new theoretical supports to the stability properties of solutions to the DARE and reduce the convergence conditions…
The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and…
Third-order nonlinear dispersion equations (NDEs) are shown to admit both shock and rarefaction waves (as weak solutions), which are distinguished by a smooth deformation approach. Compacton-type travelling wave solutions are proved to be…