We discover an abstract structure behind several nonlinear dispersive equations (including the NLS, NLKG and GKdV equations with generic defocusing power-law nonlinearities)
that is reminiscent of hyperbolic conservation laws. The underlying abstract problem admits an "entropy" that is formally conserved. The entropy is determined by a strictly convex function that naturally generates an anisotropic Orlicz space. For such problems, we introduce the dual matrix-valued variational formulation in the spirit of Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605 and prove several results related to existence, uniqueness and consistency on large time intervals. As an application, we show that no subsolution of the problems that fit into our framework is able to dissipate the total entropy earlier or faster than the strong solution on the interval of existence of the latter. This result (we call it Dafermos' principle) is new even for "isotropic" problems such as the incompressible Euler system.