  A higher-order dispersive equation is introduced as a candidate for the
governing equation of a field theory. A new class of solutions of the
three-dimensional field equation are considered, which are not localized
functions in the sense of the integrability of the square of the profile over
an infinite domain. For this new class of solutions, the gradient and/or the
Hessian/Laplacian are square integrable. In the linear limiting case, an
analytical expression for the pseudolocalized solution is found and the method
of variational approximation is applied to find the dynamics of the centers of
the quasi-particles (QPs) corresponding to these solutions. A discrete
Lagrangian can be derived due to the localization of the gradient and the
Laplacian of the profile. The equations of motion of the QPs are derived from
the discrete Lagrangian. The pseudomass ("wave mass") of a QP is defined as
well as the potential of interaction. The most important trait of the new QPs
is that at large distances, the force of attraction is proportional to the
inverse square of the distance between the QPs. This can be considered
analogous to the gravitational force in classical mechanics.
