Starting from nonlocal symmetries related to Bäcklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact form which comes from BT is presented and used to make researches in the following three subjects: two sets of negative pKdV hierarchies and their corresponding bilinear forms are constructed; the nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions; some other models both in finite dimensions and infinite dimensions are generated by comprising the original BT and evolution under new nonlocal symmetry. The finitedimensional models are completely integrable in Liouville sense, which are shown equivalent to the results given through the nonlinearization method for Lax pair.

In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlevé waves. Actually, even if for the most well-known prototypical models such as the Kortewet-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlevé II solutions.

Female patients with sepsis have better clinical outcomes than male patients in terms of mortality and length of hospitalization and ICU stay.This is an open access article distributed under the terms of the Creative Commons Attribution-Non Commercial-No Derivatives License 4.0 (CCBY-NC-ND), where it is permissible to download and share the work provided it is properly cited. The work cannot be changed in any way or used commercially without permission from the journal. http://creativecommons.org/licenses/by-nc-nd/4.0.

No abstract