The heat equation is:

the Lagrangian is

then if u(t,x) and x does not depend on t

if there are equation with u, then the action has ever a extremal along the trajectory, because of the lagrangian is ever null along the trajectory, and it is not null for different trajectories: if there exist a term $L_u$, then the Euler Lagrange equation has a solution that is the same differental equation:

where F=0 is a solution. I am sure that this Lagrangian is right, and I am not perfectly sure about the Euler-Lagrange equation, because of it is possible to have Lagrangian of higher order $L(u,u_t,u_x,u_{xx})$ with these generalized variables, with Euler-Lagrange of higher order.

Hmm, this approach to getting a Lagrangian is new to me: it seems you’re taking the differential equation, writing it in the form

and then using as the Lagrangian. But when you work out the Euler–Lagrange equation from this Lagrangian, it seems you don’t get back.

What happens if you do this with the heat equation?

]]>Well when a stripe splits, typically one branch just persists for a few temporal grid steps (or horizontal pixels if you prefer). The lifetime of a split is the time between the split itself and the first death in a resulting branch. Note that we can’t really guarantee the death “of” a resulting branch versus “in”, because you could have a series of evanescent splits. I’m not sure offhand if my code actually does “in” or “of”–I’ll have to circle back to that, but it’s tricky enough as is that I wouldn’t try to adapt it further.

Rather than try to tweak the definition of a stripe, I think it’s more sensible to demonstrate that any phenomenological unpleasantness is fleeting, whether it’s for deaths (birth was typically just a moment ago) or splits (death is typically just a moment later).

I’m sure you’ve noted the “adjunction/dagger” between a (binary) stripe diagram and the negation of its horizontal flip–this restricts to births/deaths/merges/splits. My code tries to exploit this as much as possible, but a split lifetime seems to require a different treatment than a birth lifetime for topological reasons.

]]>Interesting! What do you mean by the lifetime of a split?

]]>An extremal Lagrangian for the equation is

because of only along the trajectory the Lagrangian has a global minimum.

The Euler-Lagrange equation is:

that is the derivative of the differential Kurasamoto-Sivashinsky equation, so that it contain the solution of the equation.

The Hamiltonian of the Kurasamoto-Sivashinsky equation is simply (I don’t use the Legendre trasformation of the Lagrangian but the equations of motion):

where f(h) is an arbitrary function, the Hamilton’s equation are:

so that it is possible to write the Schrodinger equation

where I have a problem with the arbitrariness of the f(h).

The Hamiltonian dynamics is arbitray along the p axis, in the classical dynamics, so that all the integral invariant are preserved in the motion.

I try to obtain Lagrangian and Hamiltonian to obtain the conserved quantity, but the problem is that in the Noether‘s theorem the continuous simmetry is on x, and the variable here is h. ]]>

My major sticking point at present, and the cause of two late nights in a row, has been to find a computationally efficient way to determine the lifetimes of splits (versus stripes per se). I haven’t been able to exploit the “adjunction” births merges / deaths splits in this particular regard, though it works fine in identifying the loci of events.

Anyway if one ignores (or perhaps it might be better to say properly accounts for) the evanescent stripes I claim there is preliminary support for John’s quantitative conjectures.

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