Related papers: Singular Miura type initial profiles for the KdV e…

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux…

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which…

We consider noncompact complete K\"ahler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms,…

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$…

For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by…

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove…

The KdV equation can be considered as a special case of the general equation u_{t} + f(u)_{x} - \delta g(u_{xx})_x = 0, \qquad \delta > 0, where f is non-linear and g is linear, namely $f(u)=u^2/2$ and g(v)=v. As the parameter $\delta$…

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak…

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -\Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing…

The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is…

The qualitative properties of the particle trajectories of the $N$-solitons solution of the KdV equation are recovered from the first order velocity field by the introduction of the stream function. Numerical simulations show an accurate…

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic…

In this paper, we study a class of non-homogeneous anisotropic fully nonlinear curvature flows in $\mathbb{R}^{n+1}$. More precisely, we consider a hypersurface $M$ in $\mathbb{R}^{n+1}$ deformed by a flow along its unit normal with its…

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $\tau$ function are presented. B\"acklund transformations of the…

We consider perturbations of the special pole-free joint solution $U(x,t)$ of the Korteweg--de Vries equation $u_t+uu_x+\frac{1}{12}u_{xxx}=0$ and $P_I^2$ equation $u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0$ under the action of the KdV…

Explicit hyperelliptic solutions of the modified Korteweg-de Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic $\al$-functions over non-degenerated hyperelliptic curve $y^2 = f(x)$ of…

In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H…

It is proved that for any positive number $\lambda$, $1<\lambda<2$; there exists a meromorphic function $f$ with logarithmic order $\lambda$= $\displaystyle\limsup_{r\to+\infty}\frac{\log T(r,f)}{\log\log r}$ such that $f$ has no Julia…

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-\Delta)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms…

The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary…