TY - JOUR

T1 - Bäcklund transformations for new fourth Painlevé hierarchies

AU - Conde, J. M.

AU - Gordoa, P. R.

AU - Pickering, A.

N1 - Funding Information:
The work of PRG and AP was supported in part by the Ministry of Science and Innovation of Spain under contract MTM2009-12670 and the Universidad Rey Juan Carlos via the project M743, and currently by the Ministry of Economy and Competitiveness of Spain under contract MTM2012-37070 .

PY - 2014/3

Y1 - 2014/3

N2 - We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems. Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.

AB - We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems. Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.

KW - Boussinesq equation

KW - Bäcklund transformations

KW - Integrable systems

KW - Painlevaé hierarchies

UR - http://www.scopus.com/inward/record.url?scp=84885189435&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2013.07.006

DO - 10.1016/j.cnsns.2013.07.006

M3 - Artículo

AN - SCOPUS:84885189435

SN - 1007-5704

VL - 19

SP - 448

EP - 458

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

IS - 3

ER -