Related papers: Singular Miura type initial profiles for the KdV e…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (\Delta H \equiv 0) hypersurface in \R^3…
We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_\kappa = \big ( \sum_{i < j} \frac{1}{\lambda_i+\lambda_j-2\kappa} \big )^{-1}$, where $\lambda_1 \leq \hdots \leq \lambda_n$ denote the…
This paper discusses the construction of a new $(3+1)$-dimensional Korteweg-de Vries (KdV) equation. By employing the KdV's recursion operator, we extract two equations, and with elemental computation steps, the obtained result is $…
We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern…
We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework…
We investigate the formation of singularities in a self-similar form of regular solutions of the Localized Induction Approximation (also referred as to the binormal flow). This equation appears as an approximation model for the self-induced…
For differential equations $P(y^{(k)},y)=0,$ where $P$ is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.
We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic…
Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on P^2_r respectively. We consider the…
In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,\mu}$ with initial $(t,\mu)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,\mu}$ with initial $(t,x)\in[0,T]\times…
We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global M\"{o}bius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the…
A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…
This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a…
We consider multi-breathers of (mKdV). Previously, a smooth multi-breather was constructed, and proved to be unique in two cases: first, if the class of super-polynomial convergence to the profile, and second, under the assumption that all…
The Miura map (introduced by Miura) is a nonlinear map between function spaces which transforms smooth solutions of the modified Korteweg - de Vries equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this paper we…
We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial…
We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field…
The dynamics of the highly nonlinear fifth order $KdV$-type equation is discussed in the framework of the Lagrangian and Hamiltonian formalisms. The symmetries of the Lagrangian produce three commuting conserved quantities that are found to…
We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and…