Related papers: On Third-Order Evolution Systems Describing Pseudo…
The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived…
The symmetry approach to the classification of evolution integrable partial differential equations (see, for example \cite{MikShaSok91}) produces an infinite series of functions, defined in terms of the right hand side, that are conserved…
Zero curvature formulations, pseudo-potentials, modified versions, ``Miura transformations'', and nonlocal symmetries of the Korteweg--de Vries, Camassa-- Holm and Hunter--Saxton equations are investigated from an unified point of view:…
In this paper, we study the following semilinear Schr\"odinger system $$ -\triangle u+u=(1+K_\alpha(\epsilon x))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $\epsilon>0$, $\alpha>0$ are small parameters.…
We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the {\it converse problem}. Although we mainly study a method for…
The Arnowitt-Deser-Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of…
The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier--Stokes equations posed on a passively evolving smooth closed surface embedded in $\mathbb{R}^3$. The discrete…
We study existence and uniqueness of spherically symmetric solutions of S_k(D^2v)+beta xi\cdot\nabla v+\alpha v+\abs{v}^{q-1}v=0 in R^n, where \alpha,\beta are real parameters, n>2,\, q>k\geq 1 and S_k(D^2v) stands for the k-Hessian…
For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM)…
In the mean field limit, isolated gravitational systems often evolve towards a steady state through a violent relaxation phase. One question is to understand the nature of this relaxation phase, in particular the role of radial…
We establish a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathbf{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$,…
This paper considers solutions $u_\alpha$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_\alpha:=(-\alpha,\alpha)^3$ as the domain size $\alpha\to\infty$, and compares them to solutions of the same equations on…
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of integrable…
We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution…
We derive the general conditions for fully-nonlinear symmetry-integrable second-order evolution equations and their first-order recursion operators. We then apply the established Propositions to find links between a class of fully-nonlinear…
In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where…
We consider the $d$-dimensional nonlinear Schr\"odinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is…
We study noncentrosymmetric superconductors with the tetrahedral $T_d$, tetragonal $C_{4v}$, and cubic point group $O$. The order parameter is computed self-consistently in the bulk and near a surface for several different singlet to…
We develop efficient group-theoretical approach to the problem of classification of evolution equations that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable. We…
We lay out the foundations of the theory of second-order conformal superintegrable systems. Such systems are essentially Laplace equations on a manifold with an added potential: $(\Delta_n+V({\bf x}))\Psi=0$. Distinct families of…