On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

Abstract

We have recently solved the inverse scattering problem for oneparameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev -Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann -Hilbert problems on the real axis. In this paper we make use of the nonlinear Riemann -Hilbert problem of dKP, i) to construct the longtime behaviour of the solutions of its Cauchy problem; ii) to characterize a class of implicit solutions; iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.

Large Remanent Polarization in Sm-Substituted BiFeO3 Thin Film Formed by Chemical Solution Deposition

“…The Cauchy problem for the v = 0 reduction of (1), the dKP equation, was also presented in [24], while the Cauchy problem for the u = 0 reduction of (1), an integrable system introduced in [16], was given in [25]. This IST and its associated nonlinear Riemann-Hilbert (RH) dressing turn out to be efficient tools to study several properties of the solution space, such as (i) the characterization of a distinguished class of spectral data for which the associated nonlinear RH problem is linearized and solved, corresponding to a class of implicit solutions of the PDE (as it was done for the dKP equation in [26] and for the Dunajski generalization [27] of the second heavenly equation in [28]); (ii) the construction of the longtime behaviour of the solutions of the Cauchy problem [26]; (iii) the possibility to establish whether or not the lack of dispersive terms in the nonlinear PDE causes the breaking of localized initial profiles and, if yes, to investigate in a surprisingly explicit way the analytic aspects of such a wave breaking (as it was recently done for the (2+1)-dimensional dKP model in [26]). …”

“…In section 2 we present the dressing scheme for equations (2) and (12), given in terms of a vector nonlinear RH problem. As for the dressing of dKP presented in [26], since the normalization of the eigenfunctions turns out to depend on the unknown solution of 2ddT, a closure condition is necessary, allowing one to construct the solution of 2ddT through an implicit system of algebraic equations, whose inversion is responsible for the wave breaking of an initial localized profile. In section 3 we present the IST for the 2ddT equation (12) and use it to obtain the formal solution of the Cauchy problem for such equation.…”

An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

“…In the linear limit v ≪ 1 the scattering datum σ(τ, λ) reduces to the Radon transform of v x (x, y) [6].…”

Interpolating differential reductions of multidimensional integrable hierarchies

“…where the functions Ψ 0 in (λ, t) and Ψ k in (λ, t) are analytic inside the unit circle and the functions Ψ 0 out (λ, t) and Ψ k out (p, t) are analytic outside the unit circle and have an expansion of form (3), (4). We assume that the functions F 0 and F k (at least locally) define a diffeomorphism in C N +1 , F ∈ Diff(N + 1), and we call them the dressing data.…”

“…introduced in [2] (also see [3], [4]). It was shown in [5] that a simple differential reduction αu = v x , where α = const, of Manakov-Santini system (1) corresponds to the interpolating system introduced in [5] as "the most general symmetry reduction of the second heavenly equation by a conformal Killing vector with a null self-dual derivative."…”