Loop-like kink breather and its transition phenomena for the Vakhnenko equation arising from high-frequency wave propagation in electromagnetic physics

We report a novel type of breather by studying the Vakhnenko equation (VE) describing high-frequency wave (HFW) propagation in electromagnetic physics. By extending the bilinear function into a mixed exponential and trigonometric cosine function in Hirota bilinear method, an analytical multiple-valued function solution is constructed, which is a verified loop-like kink breather. The propagation control and evolution based on the parameter are investigated for the breather. Several interesting transition phenomena are revealed, such as, the transitions from the soliton to breather, from the single loop to the double loops, and from the leaping background waves to the flat ones. The results are helpful to understand the compressed mechanism to pulses in ultrafast optics.