  In this paper we study two multidimensional nonlinear dispersive systems: the
Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows,
and Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing
small compressible gas bubbles. These models are Euler-Lagrange equations for a
given Lagrangian and share common mathematical structure, namely the dependence
of the pressure on material derivatives of macroscopic variables. We develop a
generic dispersive model such that SGN and IKW systems become its special cases
if only one specifies the appropriate Lagrangian, and then use the extended
Lagragian approach proposed in Favrie and Gavrilyuk (2017) to build its
hyperbolic approximation. The new approximate model is unconditionally
hyperbolic for both SGN and IKW cases, and accurately describes dispersive
phenomena, which allows to impose discontinuous initial data and study
dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as
an example for numerical simulations and apply a second order implicit-explicit
scheme in order to numerically integrate the system. The obtained 1-D and 2-D
results are in close agreement with available exact solutions and numerical
tests.
  \textbf{Keywords: } dispersive shallow water equations, bubbly fluids,
Euler-Lagrange equations, hyperbolic conservation laws, multidimensional waves,
implicit-explicit numerical methods
