TY - JOUR

T1 - Exact solutions of a novel integrable partial differential equation

AU - Conde, Juan M.

AU - Gordoa, Pilar R.

AU - Pickering, Andrew

N1 - Funding Information:
The work of PRG and AP is supported in part by the Ministry of Science and Innovation of Spain under contract MTM2009–12670, and that of JMC, PRG and AP by the Universidad Rey Juan Carlos via project M743.

PY - 2012/6

Y1 - 2012/6

N2 - We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P II with its shift in parameter. We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel's equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.

AB - We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P II with its shift in parameter. We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel's equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.

KW - Bäcklund transformations

KW - Exact solutions

KW - Nonlinear superposition

UR - http://www.scopus.com/inward/record.url?scp=84855236204&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2011.10.007

DO - 10.1016/j.cnsns.2011.10.007

M3 - Artículo

AN - SCOPUS:84855236204

SN - 1007-5704

VL - 17

SP - 2309

EP - 2318

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

IS - 6

ER -