Kavli Affiliate: Feng Yuan
| First 5 Authors: Feng Yuan, Jingsong He, Yi Cheng, ,
| Summary:
When the phase shift of X-shaped solutions before and after interaction is
finite but approaches infinity, the vertices of the two V-shaped structures
become separated due to the phase shift and are connected by a localized
structure. This special type of elastic collision is known as a quasi-resonant
collision, and the localized structure is referred to as the stem structure.
This study investigates quasi-resonant solutions and the associated localized
stem structures in the context of the KPII and KPI equations. For the KPII
equation, we classify quasi-resonant 2-solitons into weakly and strongly types,
depending on whether the parameter (a_{12} approx 0) or (+infty). We
analyze their asymptotic forms to detail the trajectories, amplitudes,
velocities, and lengths of their stem structures. These results of
quasi-resonant 2-solitons are used to to provide analytical descriptions of
interesting patterns of the water waves observed on Venice Beach. Similarly,
for the KPI equation, we construct quasi-resonant breather-soliton solutions
and classify them into weakly and strongly types, based on whether the
parameters (alpha_1^2 + beta_1^2 approx 0) or (+infty) (equivalent to
(a_{13} approx 0) or (+infty)). We compare the similarities and
differences between the stem structures in the quasi-resonant soliton and the
quasi-resonant breather-soliton. Additionally, we provide a comprehensive and
rigorous analysis of their asymptotic forms and stem structures. Our results
indicate that the resonant solution, i.e. resonant breather-soliton of the KPI
and soliton for the KPII, represents the limiting case of the quasi-resonant
solution as (epsilon to 0).
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