On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation

Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution

“…We study here the initial value problem for the Kadomtsev-Petviashvili (KP-I) equation (1) (u t + u xxx + uu x ) x − u yy = 0, where u = u(t, x, y), (x, y) ∈ R 2 , t ∈ R, with initial data (2) u(0, x, y) = φ(x, y) + ψ c (x, y),…”

“…Another possibility to see (3) as a solution of KP-I is to consider (1) posed on R × T. Global solutions of (1) for data on R × T, including data close to (3) were recently constructed in a work by Ionescu-Kenig [12]. In (2), the function ψ c may also be the profile of the Zaitsev [36] traveling waves (see also [30]) which is localized in x and periodic in y : (4) ψ c (x, y) = 12α 2 1 − β cosh(αx) cos(δy) (cosh(αx) − β cos(δy) Let us observe that the transform α → iα, δ → iδ, c → ic produces solutions of (1) which are periodic in x and localized in y. The profiles of these solutions are also admissible in (2), under the assumption |β| > 1.…”

“…This question is, as far as we know, still an open problem (see however [1] for a linear analysis of the instability and [10] for a linear instability analysis in the framework of the full Euler system). The instability scenario of the line soliton seems to be a symmetry breaking phenomenon : the line soliton should evolve towards the Zaitsev solitary wave (4).…”

Numerical comparisons of two long-wave limit models

“…In this section we consider a class of soliton for KP-I (24) periodic in one direction and exponentially decreasing in the direction of propagation (see [2]),…”

“…We recall that if σ > 1/3, this equation is KP-I, if σ < 1/3, it is KP-II. By setting a − b = σ − 1/3 = θµ with θ ∈ R independent of µ and doing exactly the same in (2) we obtain the fifth order KP-I equation (see [14])…”