Curved asymptotic solitons of the Kadomtsev-Petviashvili equation

Higher-dimensional nonlinear integrable equations: Asymptotics of solutions and perturbations

“…Asymptotics of decreasing solutions of the DS equations and the KP-2 equation uniform in spatial variables were obtained in [16,17,39]. The asymptotic properties of decreasing solutions with functional arbitrariness for the KP-1 and KP-2 equations in the surgery region (solution front) were studied in [1,22].…”

“…As a result, we obtain the D-problem for a regular function on the whole complex plane. For the first column φ (1) , this problem reduces to the integral equation [28]:…”

Correctors for the Homogenization of Monotone Parabolic Operators

“…Asymptotic solitons (1.5) of the JE-I and the KP-I ( [9], [11]) coincide taking into account transformations (1.3) and (1.4).…”

“…Obviously this is not the fact for periodic initial data, which are not invariant with respect to this transformation, and investigation of the JE is an independent interesting problem in this case (for the KP see the corresponding theory, for example, in [35]- [38]). We are interested in the construction of a class of JE-I (α = i in (1.1)) non-decaying solutions, which are bounded for all (x, y, t) and vanish as x → +∞ for all fixed y and t. Such a kind of KP solutions was constructed and investigated firstly for KP-II in [7], [8], and then for KP-I in [9]- [13]. It turns out that all basic stages of the construction of the solutions of the KP and the JE, and the study of their asymptotic behaviour admit mutual recounting using the described mapping.…”

“…In [7]- [13] the method proposed by E.Ya. Khruslov ([17], [18]) was extended to the investigation of the asymptotic behaviour of non-decaying solutions of KP-type equations (KP, modified KP-I and 2D-Gardner equation) as t → ∞.…”