Bright soliton of the focusing nonlinear Schrödinger equation

For negative frequency $E$, the one-dimensional focusing nonlinear Schrödinger equation has the localized stationary profileGalilean boosts generate moving profiles with a translational collective coordinate and constant velocity.

A collective-coordinate approximation substitutes a finite-parameter soliton ansatz into the field action and integrates over space. Its Euler-Lagrange equations approximate the slow motion of the soliton parameters.