64 research outputs found
Multi-Hamiltonian structure of Plebanski's second heavenly equation
We show that Plebanski's second heavenly equation, when written as a
first-order nonlinear evolutionary system, admits multi-Hamiltonian structure.
Therefore by Magri's theorem it is a completely integrable system. Thus it is
an example of a completely integrable system in four dimensions
Fragmented in space: the oral history narrative of an Arab Christian from Antioch, Turkey
This study uses the case of Can Kılçıksız, an Arab Christian refugee youth from Antioch, Turkey, to argue that globalization may result in fragmented families and subjectivities and can also accelerate processes initiated by modernity and the construction of national identities. Can Kılçıksız and his siblings now live in Turkey, Germany, France and Finland. His life story suggests that males of Arab Christian origin from Antioch who had access to schooling are more likely to be involved in politics whereas females tend to be drawn to evangelical Christian organizations. The case also suggests that sibling ties might prove more durable in the course of transnational migration than conjugal ties. The case of Can Kılçıksız shows that the time/space linked to childhood through memory can play an important role in identity construction of subjects circulating in transnational space
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
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 . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex 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 plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
On the solutions of the second heavenly and Pavlov equations
We have recently solved the inverse scattering problem for one parameter
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 connected with the commutation of multidimensional
vector fields, like the heavenly equation of Plebanski, the dispersionless
Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless
Toda (2ddT) equation, as well as with the commutation of one dimensional vector
fields, like the Pavlov equation. We also showed that the associated
Riemann-Hilbert inverse problems are powerfull tools to establish if the
solutions of the Cauchy problem break at finite time,to construct their
longtime behaviour and characterize classes of implicit solutions. In this
paper, using the above theory, we concentrate on the heavenly and Pavlov
equations, i) establishing that their localized solutions evolve without
breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime
behaviour of the solutions of their Cauchy problems; iii) characterizing a
distinguished class of implicit solutions of the heavenly equation.Comment: 16 pages. Submitted to the: Special issue on nonlinearity and
geometry: connections with integrability of J. Phys. A: Math. and Theor., for
the conference: Second Workshop on Nonlinearity and Geometry. Darboux day
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