In this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations (PDEs) arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S V Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, (i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, (ii) on its first commuting flow, and (iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time

The inverse spectral transform for the Dunajski hierarchy and some of its reductions: I. Cauchy problem and longtime behavior of solutions / Yi, Ge; Santini, Paolo Maria. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 48:21(2015), p. 215203. [10.1088/1751-8113/48/21/215203]

The inverse spectral transform for the Dunajski hierarchy and some of its reductions: I. Cauchy problem and longtime behavior of solutions

YI, GE;SANTINI, Paolo Maria
2015

Abstract

In this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations (PDEs) arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S V Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, (i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, (ii) on its first commuting flow, and (iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time
2015
Dunajski hierarchy; inverse scattering transform; longtime behavior; Mathematical Physics; Physics and Astronomy (all); Statistical and Nonlinear Physics; Modeling and Simulation; Statistics and Probability
01 Pubblicazione su rivista::01a Articolo in rivista
The inverse spectral transform for the Dunajski hierarchy and some of its reductions: I. Cauchy problem and longtime behavior of solutions / Yi, Ge; Santini, Paolo Maria. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 48:21(2015), p. 215203. [10.1088/1751-8113/48/21/215203]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/845566
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