We study the integrability problem of the nonuniform Hirota (vc-Hirota) equation with variable coefficients by using the Bell polynomial method and the Darboux transformation (DT). The nonuniform vc-Hirota equation describes the wave propagation of picosecond pulses more accurately than the nonlinear Schrödinger (NLS) equation. Firstly, by using the Bell polynomial method, we construct the Bäcklund transformation (BT) and Lax pair of the equation to prove that the nonuniform vc-Hirota equation is Lax integrable. Interestingly, the Lax pair we constructed has variable coefficients and is nonisospectral, as opposed to having constant coefficients. Secondly, by pairing the Lax pair with the zero-curvature equation, we obtain the DT of the equation. Finally, we construct the soliton-like solutions of the equation.
We reveal the most general regularities of the surface distribution of compressions and stretches in a surfactant film distributed on a liquid surface, which is in an unstable state with respect to the excess of the surface electric charge.
In this paper, we provide a partial solution to the Birman–Williams problem on describing the planetary link of a fibered knot K⊂𝕊^3 . Using Nielsen’s classification theory of periodic homeomorphisms of surfaces and its close connection with the theory of gradient-like diffeomorphisms, we prove that the planetary link of the trefoil knot (the unique periodic fibered knot of genus 1 ) is the link consisting of the trefoil and the Hopf link. Furthermore, we classify all flows on the 3-sphere 𝕊^3 that realize the planetary link of the trefoil, i.e., structurally stable flows with a minimal number of periodic orbits transverse to the fibration of the trefoil.
We investigate a supercritical traveling wave as a solution to a hyperbolic semilinear partial differential equation and establish the conditions for its existence. We focus primarily on the stability problem, which leads us to analyze a spectral problem for a non-self-adjoint operator. Finally, we describe a wide class of initial perturbations with respect to which the wave exhibits asymptotic stability in the linear approximation.
We consider a periodic boundary value problem for the convective variant of the Cahn–Hilliard equation in the case where solutions depend on two spatial variables and time. For this boundary value problem, we study the existence and properties of traveling waves. These issues are reduced to the analysis of local bifurcations of an auxiliary boundary value problem. The possibility of the existence of traveling wave packets is demonstrated. We determine the stability of the invariant manifolds of the evolutionary boundary value problem under consideration, as well as the stability of the traveling waves forming them. Asymptotic formulas for these solutions are obtained.
