A. Mucalica and D.E. Pelinovsky
Solitons on the rarefaction wave background via the Darboux transformation
Proc. R. Soc. A 478 20220474 (17 pages)
Abstract:
Rarefaction waves and dispersive shock waves are generated from the step-like initial data in many nonlinear evolution equations including the classical
example of the Korteweg–de Vries (KdV) equation. When a solitary wave is injected on the step-like initial data, it is either transmitted over or trapped
inside the rarefaction wave background. We show that the transmitted soliton can be obtained by using the Darboux transformation for the KdV equation. On
the other hand, we show with the help of numerical simulations that the trapped soliton disappears in the long-time dynamics of the rarefaction wave.
Keywords:
Korteweg-de Vries equation, rarefaction waves, transmitted solitons, trapped solitons, Darboux transformation