Tao why are solitons stable

All these models are motivated by physics e. Hi Terence, The equation directly above 2. Terence Tao. Does anyone have any guesses as to what happens when two solitons of similar speed and width collide? Does there exist some kind of spectrum of soliton solutions? If they both have a positive topological charge, they scatter at a 90 degree angle: They first merge into an excited bound state and then the bound state splits into two.

Though it depends at what speed they hit each other. If it is very slow, then they might stay in the bound state if it is energetically favorable. The same is true for 3D. If they have exactly opposite topological charge, they merge and annihilate each other and create a lot of waves with zero topological charge. The interesting thing is that 90 degree scattering and annihilation processes are typically quantum phenomena that these topological solitons exhibit, too.

Some speculate that quantum mechanics is really just the fact that elementary particles are really solitons and neither waves nor particles. Though interesting, no-one has really written down a convincing and consistent theory. I should also mention that string theorists and quantum field theorists are also dealing with solitons. For solitons in completely integrable models e. KdV, 1D cubic NLS, sine-Gordon, … what happens is that the solitons pass through each other, but they get shifted in location by the collision though the velocity remains unchanged.

For non-integrable models, the situation seems to be rather complicated; numerics suggest that one gets a wide range of behaviours ranging from integrable-type behaviour possibly shedding a little bit of mass and energy as radiation to total disruption into radiation.

Solitons can often be expressed as critical points of certain Lyapunov functionals which typically are combinations of such conserved quantities as the mass and energy, the former of which can also be interpreted as a Lagrange multiplier.

The ground state soliton tends to be the minimiser of such a functional, which tends to make it stable. But there are also excited states which correspond to non-minimising critical points.

Typically there are a discrete set of these modulo symmetries , and near each such soliton there are a finite number of unstable directions and a cofinite number of stable directions in which the dynamics can evolve. If one inserts a parameter into the nonlinearity and varies that parameter continuously, it is possible for some states to coalesce or branch; there are some interesting mathematical results analysing these sorts of things but I am not really an expert in these topics.

On the other hand, there are certainly discrete analogues of the KdV or NLS equations in which space and sometimes time is replaced with a discrete space such as , and then by restricting attention to spatially periodic solutions one can then study such evolutions on for various p.

But in all such situations, the solutions remain complex-valued or real-valued, so it is not a true finite characteristic ps ituation. I would imagine that for a nonlinear Schrodinger or wave equation with a potential that was not of power type thus breaking scale invariance , one would obtain a one-parameter family of soliton solutions after quotienting out by the remaining symmetries translation, Galilean, rotation etc.

Do you know where to find it on the web? Dear Weidig, Are these stable: Gravitational solitons and the squashed 7-sphere, Classical Quantum Gravity 24 , no. Hi Weidig, 4th order KdV has a one-parameter family of solitons with radiating tails. How to choose the stable one?

Hi Thomas, I wonder why physicists name some patterns as soliton-stripes. Probably because interfaces are joined by smooth layers. If you are interested in patterns, some of them are complicated, e. Thus it is a minimizer of an unknown order variational thermodynamics functional, an extension of the Ginzburg-Landau one. I heard it will be presented in Canberra. Everyday entropy: soliton — bonefactory. You are commenting using your WordPress.

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