Solitons

I'm spending April this year recovering from March, so this is not so much diving into a new topic per se, but one that I keep running into, perhaps even in my master's thesis. Physicists absolutely love wave equations. They're so convenient that we start seeing periodic solutions even where there are none. Most waves however, aren't very stable. Simple obstacles will create a these interference patterns that are commonly used to tech children about quantum mechanics. Sometimes we a mores table wave would be convenient, not only for intuition, but also for some experimental setup (not for mathematics, we will get into that later).

Solitons are wave packets that are strongly stable and self-reinforcing. When two solitons collide, both will retain their structure. To achieve that, the nonlinear and dispersive effects acting on the wave packet need to be cancelled out. This also means that, unfortunately, the wave equation will also need to be nonlinear, and thus a bunch more complicated than those beautiful sine/cosine waves. They are always partial differential equations, which of course also means that they might not always have a convenient solution.

The first wave equation for solitons was a mathematical description of the waves on shallow water surfaces. This is because a lot of the dispersive effects of waves, happen when "sheets" of (even the same) media interact in complicated ways with one another. Hence waves that utilize treats the entirety of the medium for their excitation are almost automatically solitonic. The mathematical description for this kind of wave is the Korteweg-De Vries equation, which one might have encountered as an example for integrable partial differential equations.

We assume that this waves propagates in the x-direction only. The constant 6 is arbitrary and holds no real significance. The first term is a normal time-dependency that every wave equation has, the second describes the dispersion, which only really shows up at third (or higher) order terms. The last term describes the transport of the wave. The proper term for this is "advection".

As an integrable partial differential equation, it can have any number of solutions, and this equation can model any set of separate single solitons. For a single soliton, we of course make a few assumptions about these waves. For one, we know that it has to maintain its shape, due to stability, and it travels at a constant phase speed c. By virtue of these properties, a soliton can never be a standing wave. It's description will then relate to the traveling wave equation. Visually, solitons tend to consist of a single, often straight, wave-front.

The function f is meant to satisfy Newton's equation of motion for a particle in a cubic potential, which fixes the integration constant A that will appear upon integration of the above equation.

For a vanishing integration constant, there is a local maximum at V(f=0), meaning that a solution exists where f(X) begins at 0, infinitely far back in time, and sliding down to a local minimum of the potential, oscillating around it. This is the characteristic shape of the solitary wave solution

Generalized, the N-soliton solution describes a wave packet separating into N separate single solitons.

Treating u(x, t) as a density with flux Φ and creation rate g in a region with smooth boundary, for which Gauss Law holds.

We can take g(x, t) to be zero, because the soliton isn't being "created" in most observations. With that, if the flux vanishes at infinity, then the integral over u is actually independent of time. This yields a conservation law about the components of u. For shockwave solutions u, given a shock path s, the Rankine-Hugoniot Jump Condition applies.

Depending on the breadth of the soliton pulse, it might become subject to other wave scattering phenomenon. For short solitons the optical spectrum for example becomes so broad that the tail carries a longer wavelength, which can then experience Raman scattering, draining power from the component of the tail that has a shorter wavelength. This causes an overall frequency shift, referred to as a "soliton self-frequency shift". The shorter a soliton is, the higher the peak power is, which translates into a broader optical spectrum. Since Raman scattering also smears out the soliton, this effect grows weaker with time.

Since most solitons propagate chiefly in one direction, there is no issue in flipping the time and space axes in the equations. This suggests that not only are there temporal solitons, there are also spatial solitons. The nonlinearity of the medium then cancels the diffraction. In this way, a stable beam can be formed in inhomogenous mediums. In this case, photorefractive effects are centered to achieve the soliton.