M. Hoefer, A. Mucalica, and D.E. Pelinovsky
KdV breathers on a cnoidal wave background
J. Phys. A: Math. Theor. 56 (2023) 185701 (25pp)
Abstract:
Using the Darboux transformation for the Korteweg-de Vries equation,
we construct and analyze exact solutions describing the interaction of a solitary wave
and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these
wave patterns are referred to as breathers. Both elevation (bright) and depression
(dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate
that the bright (dark) breathers propagate faster (slower) than the background
cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the
cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated
by dark soliton solutions of the nonlinear Schrodinger equation. These
results provide insight into recent experiments on soliton-dispersive shock wave interactions
and soliton gases.
Keywords:
Korteweg-de Vries equation, cnoidal traveling waves, bright breathers, dark breathers, Darboux transformation.