We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.
It was observed long ago [1] that the commutation of multidimensional vector fields can generate integrable nonlinear partial differential equations (PDEs) in arbitrary dimensions. Some of these equations are dispersionless (or quasiclassical) limits of integrable PDEs, having the dispersionless Kadomtsev -Petviashvili (dKP) equation [2], [3] as universal prototype example; they arise in various problems of Mathematical Physics and are intensively studied in the recent literature (see, f.i., [4] - [22]). In particular, an elegant integration scheme applicable, in general, to nonlinear PDEs associated with Hamiltonian vector fields, was presented in [8] and a nonlinear ∂ -dressing was developed in [14]. Special classes of nontrivial solutions were also derived (see, f.i., [13], [16]).
Distinguished examples of PDEs arising as the commutation conditions [ L1 (λ), L2 (λ)] = 0 of pairs of one parameter families of vector fields, being λ ∈ C the spectral parameter, are the following. 1. The vector nonlinear PDE in N + 4 dimensions [23]:
where U(t 1 , t 2 , z 1 , z 2 , x) ∈ R N , x = (x 1 , . . . , x N ) ∈ R N and ∇ x = (∂ x 1 , .., ∂ x N ), associated with the following pair of (N + 1) dimensional vector fields
- Its dimensional reduction, for N = 2 [23]:
obtained renaming the independent variables as follows: t 1 = z, t 2 = t, x 1 = x, x 2 = y, associated with the two-dimensional vector fields:
- The Hamiltonian reduction ∇ x • U = 0 of (3), the celebrated second heavenly equation of Plebanski [24]:
θ tx -θ zy + θ xx θ yy -θ 2 xy = 0, θ = θ(x, y, z, t) ∈ R, x, y, z, t ∈ R, (5) describing self-dual vacuum solutions of the Einstein equations, associated with the following pair of Hamiltonian two-dimensional vector fields
- The following system of two nonlinear PDEs in 2 + 1 dimensions [25]:
arising from the commutation of the two-dimensional vector fields
and describing a general integrable Einstein-Weyl metric [26]. 5. The v = 0 reduction of (7), the dKP equation
(the x-dispersionless limit of the celebrated Kadomtsev-Petviashvili equation [27]), associated with the following pair of Hamiltonian two-dimensional vector fields [7,8]:
describing the evolution of small amplitude, nearly one-dimensional waves in shallow water [28] near the shore (when the x-dispersion can be neglected), as well as unsteady motion in transonic flow [2] and nonlinear acoustics of confined beams [3]. 6. The u = 0 reduction of (7) [15,17]:
associated with the non-Hamiltonian one-dimensional vector fields [18] L1
- The two-dimensional dispersionless Toda (2ddT) equation [29,30]:
(or ϕ ζ 1 ζ 2 = (e ϕ ) tt , ϕ = φ t ), associated with the pair of Hamiltonian vector fields [6]:
describing integrable heavens [31,32] and Einstein -Weyl geometries [33], [34], [35]; whose string equations solutions [10] are relevant in the ideal Hele-Shaw problem [36]- [40].
The Inverse Spectral Transform (IST) for 1-parameter families of multidimensional vector fields, developed in [23], has allowed one to construct the formal solution of the Cauchy problem for the nonlinear PDEs (3) and ( 5) in [23], for equations ( 7) and ( 9) in [25], for equation (11) in [41] and for the wave form (e φt ) t = φ xx + φ yy of equation ( 13) in [42]. This IST, introducing interesting novelties with respect to the classical IST for soliton equations [43,28], turns out to be, together with its associated nonlinear Riemann -Hilbert (NRH) dressing, an efficient tool to study several properties of the solution space of the PDE under consideration: 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 (for the dKP and 2ddT equations respectively in [44] and in [42], and for the Dunajski generalization [46] of the heavenly equation in [45]) and for equations (11) and ( 5) in [47]; ii) the construction of the longtime behaviour of the solutions of the Cauchy problem (for the dKP, 2ddT and heavenly equations respectively in [47], [44] and [47]); iii) the possibility to establish whether or not the lack of dispersive terms in the nonlinear PDE causes the breaking of localized initial profiles (for the dKP, 2ddT and heavenly equations respectively in [44], in [42] and in [47]) and, if yes, to investigate in a surprisingly explicit way the analytic aspects of such a wave breaking, as it was done for the dKP equation in [44]. Recent results on integrable differential constraints on the hierarchy associated with the nonlinear system (7) and their connection to the associated NRH problems can be found in [48].
In this paper, generalizing the examples of solvable NRH problems illustrated in [42,44,47], we present, in §2, a general procedure to construct solvable NRH problems for integrable PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of
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