# A Breather Solution in the Causal Interpretation of Quantum Mechanics

A Breather Solution in the Causal Interpretation of Quantum Mechanics

A breather solution appears often in nonlinear wave mechanics, in which a nonlinear wave has energy concentrated in a localized oscillatory manner. This Demonstration studies a breather solution with a hyperbolic secant envelope of the focusing nonlinear Schrödinger (NLS) equation , with = and so on, also known as the Gross–Pitaevskii equation in the causal interpretation, developed by Louis de Broglie and David Bohm.

iψ-ψ-β(ψ)ψ=0

∂

t

1

2

∂

x,x

*

ψ

∂

t

∂

∂ t

The NLS equation could be interpreted as the Schrödinger equation (SE) with the nonlinear potential term with , although for most situations it has no relationship with the quantum Schrödinger equation other than in name. In the studied breather, there are large density amplitudes at certain times, which could be interpreted as rogue waves.

V

V=-β(ψ)

*

ψ

The graphic on the left shows the density (blue), the quantum potential (red), and the velocity (green). On the middle and on the right, you can see the density and the trajectories in space and the quantum potential and the trajectories in space. The velocity and the quantum potential on the left side are scaled to fit.

(x,t)

(x,t)