Hodograph transformations for a Camassa–Holm hierarchy in 2 + 1 dimensions

Abstract

A generalization of the negative Camassa-Holm hierarchy to 2 + 1 dimensions is presented under the name CHH(2+1). Several hodograph transformations are applied in order to transform the hierarchy into a system of coupled CBS (Calogero-Bogoyavlenskii-Schiff) equations in 2 + 1 dimensions that pass the Painlevé test. A non-isospectral Lax pair for CHH(2+1) is obtained through the above mentioned relationship with the CBS spectral problem.

Reciprocal transformations for a spectral problem in 2+1 dimensions

“…Nevertheless, we can sometimes identify nontrivial transformations that transform an equation into a form for which the Painlevé methods work. For instance, in [5], we applied a reciprocal transformation [6]- [9] to a (2+1)-dimensional Camassa-Holm hierarchy that allowed transforming it into a system of equations with which the singular manifold method can be successfully used [10].…”

Generalized Qiao hierarchy in 2+1 dimensions: Reciprocal transformations, spectral problem and non-isospectrality

“…• In [12], the authors compare (1) with the Harry-Dym case [15], which reads (6) is also known as the first member of the positive flow of the Camassa-Holm hierarchy [4,7,9]. In contrast the celebrated Camassa-Holm equation [1] is the first member of the negative flow.…”

“…• In [4], an integrable generalization to 2 + 1 dimensions of the Camassa-Holm hierarchy was presented. By using reciprocal transformations, the spectral problem for such a hierarchy was obtained.…”

Преобразования Двойственности Для Спектральной Задачи В Размерности $2+1$

“…Тем не менее иногда можно указать нетривиальные преобразования, которые приводят уравнение к виду, допускающему применение метода Пенлеве. Например, в работе [5] мы применили преобразование двойственности [6]- [9] к (2 + 1)-мерной иерархии Камассы-Холма, что позволило преобразовать ее к системе уравнений, для которых можно с успехом применить метод сингулярного многообразия [10].…”