Coupled quartic anharmonic oscillators, Painlevé analysis, and integrability

A detailed and systematic investigation of the Painlevé (P) properties of coupled quartic anharmonic oscillators is presented. Considering the two coupled oscillators we show that there exist four different parametric cases possessing P properties, two are identified with strong P property and the other two with weak P property. For each of these four cases explicit second integrals of motion can also be constructed directly. We then consider the three-coupled-oscillator system and identify three cases, one with strong P and the other two with weak P nature. We have explicitly derived the second and third integrals of motion for a special case of the strong P case, but for the remaining cases they have not yet been found. Finally, we extend the procedure to the N coupled oscillators and succeed to show that there exist three cases possessing P properties, which are the natural generalizations of the three-coupled-oscillator system.