Self-similar planar curves related to modified Korteweg-de Vries equation
The planar curve \(z(s,t)= x(s,t)+ iy(s,t)\) with \(s\in\mathbb{R}\) the arclength and \(t\in\mathbb{R}\) the time evolution parameter is subjected to the flow equation \(z_t= -z_{sss}+ {3\over 2}\overline z_s z^2_{ss}\) and then the curvature \(k\) satisfies the modified Korteweg-de Vries equation \(k_t+ k_{sss}+{3\over 2} k^2 k_s= 0.\) The authors prove the existence of a solution satisfying the initial condition \(k(x, 0)= a\delta+\mu\text{ p.v. }{1\over x},\) where \(a\), \(\mu\) are constants close to zero and \(\delta\) is the Dirac function. The selfsimilar solutions \[ k(s, t)= {2\over (3t)^{1/3}}\,u\Biggl({s\over (3t)^{1/3}}\Biggr) \] are found. Then the problem is reduced to a thorough study of the ordinary differential equation \(u_{xx}- xu+ 2u^3=\mu\) in the main part of the article.
scientific article; zbMATH DE number 5144841