I have a question. I read a paper where the authors refer to the well-known dark soliton solution (Tanh(x)) as a probability density. How can it be, if it is not normalizable, meaning the probabilities over the space domain do not add up to 1?
No, the dark soliton solution in its standard form, which is expressed using the hyperbolic tangent function (Tanh), is not a probability density. In the context of nonlinear wave equations, such as the nonlinear Schrödinger equation, the dark soliton is a particular type of solitary wave that can arise in certain physical systems, particularly in the study of Bose-Einstein condensates in ultracold atomic gases.
The dark soliton solution describes a localized density variation within the medium, characterized by a notch or a dark region surrounded by a higher-density background. It arises due to a balance between dispersion and nonlinear effects. While it can be interpreted as a wave-like solution, it is not a probability density because it does not satisfy the conditions for a valid probability density function. A probability density function must be non-negative everywhere and integrate to unity over the entire space.
In the case of the dark soliton, the density variation becomes zero at the position of the soliton (the notch) and is non-zero in other regions. Therefore, the non-negativity condition is violated, making it unsuitable as a probability density function.
However, in certain situations, the dark soliton solution can be used to describe the density profile of the condensate in the context of Bose-Einstein condensates, even though it is not a valid probability density. Instead, it provides valuable insights into the behavior of the system, such as the presence of solitonic structures and their dynamics
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