The first part of this talk deals with the numerical approximation to nonlinear dispersive equations, such as the prototypical nonlinear Schrödinger or Korteweg-de Vries equations. We introduce integration techniques allowing for the construction of schemes which perform well both in smooth and non-smooth settings. Higher order extensions will be presented, following techniques based on decorated trees series.

In the second part, we introduce a new approach for solving some nonlinear and nonlocal integrable PDEs, based upon recent theoretical breakthroughs on explicit formulas for these equations. It opens the way to numerical approximations which are far more accurate and efficient for simulating these equations, allowing for the study of the asymptotic behavior of the solutions, with applications to the soliton resolution conjecture and in wave turbulence theory.