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A novel method for stability and instability study of autonomous dynamical systems using the flow and divergence of the vector field is proposed. A relation between the method of Lyapunov functions and the proposed method is established.…
In this paper, we describe a uniform and standardized approach for analytically verifying the stability of isotropic, incompressible hyperelastic material models. Here, we address {\sl stability} as fulfillment of the {\sc Hill} condition…
Historically, meteorological and climate studies have been prompted by the need for understanding precipitation to have better logistics in food production. Despite all efforts, nonlinearity in atmosphere dynamics is still a source of…
Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$ u_{tt} + u_{xxxx} + f(u)= g(x, t) $$ in bounded space-time…
Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these…
We prove robustness and uniqueness of equilibrium states for a class of partially hyperbolic diffeomorphisms with dominated splittings and H\"older continuous potentials with not very large oscillation.
The infinite-dimensional mechanics of fluids and plasmas can be formulated as "noncanonical" Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that…
In paper [1] unpredictable points were introduced based on Poisson stability, and this gives rise to the existence of chaos in the quasi-minimal set. This time, an unpredictable function is determined as an unpredictable point in the…
We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic…
Starting from a simple marginally stable model considered for Lyapunov based boundary control of flexible mechanical systems, we add a term driving an instability and prove that for an appropriate control condition the system can become…
We present a detailed non-perturbative analysis of the time-evolution of a well-known quantum-mechanical system - a particle between potential walls - describing the decay of unstable states. For sufficiently high barriers, corresponding to…
The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the N-level Friedrichs model, and is shown to depend on the initial population in unstable states. A…
We explore physics of unstable particles when mother particle mass is around the sum of its daughter particle masses. In this case, the conventional wave function renormalization factor is ill-defined. We propose a simple resolution of the…
We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$…
We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character - considering either one of…
Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the…
We study travelling fronts of equations of the form $u_{tt} + \phi(u) u_x = u_{xx} + f(u)$. A criterion for the transition from linear to nonlinear marginal stability is established for positive functions $\phi(u)$ and for any reaction term…
In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the…
We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem $\partial_{t}u - \rho_{\lambda}(x)u\Delta u = \rho_{\lambda}(x)g_{0}(x)u$ on the half line $(0,\infty)$. We find weak solutions from $L^p\ (p <…
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard…