Related papers: Hard loss of stability in Painlev\'e-2 equation
We investigate magnetic fluctuations of a mesoscopic critical region at the interface induced by smooth time-independent spacial changes of a control parameter across its critical value. Near the spatial critical point, the order parameter…
Axisymmetric steady solutions of Taylor-Couette flow at high Taylor numbers are studied numerically and theoretically. As the axial period of the solution shortens from about one gap length, the Nusselt number goes through two peaks before…
The bifurcation transition is studied for the onset of intermittency analogous to the Pomeau-Manneville mechanism of type-I, but generalized for the presence of a quasiperiodic external force. The analysis is concentrated on the…
By means of a linear scaling of the variables we convert a singular bifurcation equation in $\R^n$ into an equivalent equation to which the classical implicit function theorem can be directly applied. This allows to deduce the existence of…
We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u &=|x|^\alpha |u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on…
In this paper, we study a well known asymptotic limit in which the second Painlev\'e equation (P_II) becomes the first Painlev\'e equation (P_I). The limit preserves the Painlev\'e property (i.e. that all movable singularities of all…
The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the…
In this paper we describe physical properties arising in the vicinity of two coupled quantum phase transitions. We consider a phenomenological model based on two scalar order parameter fields locally coupled biquadratically and having a…
The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length {eta}, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of…
This paper presents a general framework to derive the weakly nonlinear stability near a Hopf bifurcation in a special class of multi-scale reaction-diffusion equations. The main focus is on how the linearity and nonlinearity of the fast…
We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense…
In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a $q$-difference Painlev\'e equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is…
The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e. in a connected union of thin rectangles with heights of order $\varepsilon<<1 $ and with bases of order 1 with smoothened boundary. An asymptotic…
We consider an anisotropic $(p,2)$-equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the…
The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the…
We investigate the solutions to the Lorentz-Dirac equation and show that its solution flow has a structure identical to the one of renormalization group flows in critical phenomena. The physical solutions of the Lorentz-Dirac equation lie…
The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of…
By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime…
For the fifth Painlev\'e equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of…
Plane Poiseuille flow shows turbulence at a Reynolds number that is lower than the critical one for the onset of Tollmien-Schlichting waves. The transition to turbulence follows the same route as the by-pass transition in boundary layers,…