Multisoliton interactions approximating the dynamics of breather solutions

Nowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N-soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super-regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.