Some PDE's and their rogue waves +

Overview & Synopsis

We give some of our results over the past few years about rogue waves. For the (NLS) and (KP) equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order 2N . These solutions are called solutions of order N . In the case of the (NLS) equation, the solutions explicitly constructed, appear as deformations of the Peregrine breathers PN as the last one can be obtained when all parameters are equal to zero. At order N , these solutions are the product of a ratio of two polynomials of degree N (N +1) in x and t by an exponential depending on time t and depending on 2N -2 real parameters: they are called quasi rational solutions. For the (KPI) equation, we obtain explicitly solutions at order N depending on 2N -2 real parameters. We present different examples of rogue waves for the (LPD) and Hirota equations.