Related papers: The Korteweg-de Vries Equation on general star gra…

For a graph G, the k-total dominating graph D_{k}^{t}(G) is the graph whose vertices correspond to the total dominating sets of G that have cardinality at most k; two vertices of D_{k}^{t}(G) are adjacent if and only if the corresponding…

We carry out group analysis of a class of generalized fifth-order Korteweg-de Vries equations with time dependent coefficients. Admissible transformations, Lie symmetries and similarity reductions of equations from the class are classified…

In this work we investigate Cauchy problem and initial boundary value problem for time-fractional Airy equation on the graphs with infinite and finite bonds. We studied properties of potentials for this equation and using these properties…

We are concerned with the Cauchy problem for the KdV equation for nonsmooth locally integrable initial profiles q's which are, in a certain sense, essentially bounded from below and q(x)=O(e^{-cx^{{\epsilon}}}),x\rightarrow+\infty, with…

We prove local in time well-posedness for a class of quasilinear Hamiltonian KdV-type equations with periodic boundary conditions, more precisely we show existence, uniqueness and continuity of the solution map. We improve the previous…

A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: for what values of $k>d/2$ does the random $d$-regular graph have a $k$-star…

In this work we present a new method for solving of the Korteweg-de Vries (KdV) equation q'_t = - \dfrac{3}{2} q q'_x + \dfrac{1}{4} q"'_{xxx}. The proposed method is a particular case of the theory of evolutionary vessels, developed in…

In this note, we propose a straightforward method to produce an straight-line embedding of a planar graph where one face of a graph is fixed in the plane as a star-shaped polygon. It is based on minimizing discrete Dirichlet energies,…

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable…

The complete group classification problem for the class of (1+1)-dimensional $r$th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of $r$ greater than or equal to two. We find the…

A $k$-star is a complete bipartite graph $K_{1,k}$. A partial $k$-star design of order $n$ is a pair $(V,\mathcal{A})$ where $V$ is a set of $n$ vertices and $\mathcal{A}$ is a set of edge-disjoint $k$-stars whose vertex sets are subsets of…

A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope {\tau} from the inertial…

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein--Uhlenbeck and the harmonic…

We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical…

We derive a counting formula for the eigenvalues of Schr\"odinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue…

This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on…

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 0<x<L, \ t>0 \qquad (1) \end{equation} subject…

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in…

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/…

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction.