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

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2015-05, Vol.48 (21), p.215203-25
• Main Authors: Yi, G, Santini, P M
• Format: Article
• Language: English
• Subjects: Cauchy problem; Dunajski hierarchy; Hierarchies; Inverse; inverse scattering transform; longtime behavior; Mathematical analysis; Partial differential equations; Reduction; Spectra; Transforms
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/48/21/215203

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.