Acceleration of Particles in a Wave Obeying the Korteweg-de Vries Equation

One of the advantages of the causal interpretation of quantum mechanics developed by David Bohm is that the concept of single particle trajectories in which the particles are guided by a so-called “quantum potential” can be applied to nonlinear partial differential equations obeying the continuity equation as a conservation law. In this Demonstration, the particles follow the wave density , which is a solution of the famous Korteweg–de Vries (KdV) equation, and in which a quantum-like potential, here called wave potential or fluid pressure, is responsible for the acceleration or deceleration of the particles. For two very special cases, the dynamics (like trajectories, velocities, acceleration, and density) are given analytically. The waves could be described as a rational solution and a solitary wave of the KdV equation [1]. It is notable that a fluid pressure, which likely has the same form as the quantum potential in the causal interpretation, appears in the acceleration term of the KdV equation, which of course introduces highly non-Newtonian behavior. The "wave potential" differs slightly in the definition. Here, it is a nonlinear function of the density itself. But as in the causal interpretation, the effect of the quantum-like potential varies with the form of the wave function and not its intensity. The potential acts instantaneously and does not diminish as the distance between two particles increases. The time evolution of the position could be interpreted as streamlines of the wave or as idealized test particles that do not interact on each other and do not perturb the wave. The graphic on the left shows the density (black), the quantum potential (red), and the acceleration (blue). On the right, you can see the contours and the trajectories in -space.


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The KdV equation can be written , where ​, and so on. The motion of the particles is governed by the current flow , where is the velocity, which is derived from the continuity equation . The guiding equation is based only on the velocity term, , with . We get the starting points of possible trajectories inside the wave that are linearly distributed according to the wave density and lead to single trajectories: If the velocity could be expressed by , then the total derivative is simplified to and it follows that the acceleration is for the single particles in the wave. In this case, the acceleration is based only on the velocity term , where the wave potential has a similar form of the quantum potential of the causal interpretation of quantum mechanics. Only for the solitary wave is unequal to zero, which cancels out the term proportional to , which further leads to a zero acceleration. Keep in mind that, for the rational solution, the conservation of total mass fails. Further investigation will be necessary to answer many open questions.
[1] A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, London: Chapman & Hall/CRC Press, 2003.
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