Multisoliton interactions approximating the dynamics of breather solutions

Breather solutions are considered to be generally accepted models of rogue waves. However, breathers are not localized, while wavefields in nature can generally be considered as localized due to the limited spatial dimensions. 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 multi-soliton 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 the constructed with our method multi-soliton 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 multi-breather solutions, breathers on a nontrivial background (e.g., cnoidal waves) and other integrable systems. The constructed multi-soliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronizations conditions represents a challenging problem for future studies.