Dynamic-similarity solver for PDEs

The KdV equation $u_t + 6u\,u_x + u_{xxx} = 0$ admits a self-similar solution where the relevant combination of variables is $\eta = x\,t^{-1/3}$ and the amplitude decays as $t^{-2/3}$. Those exponents follow from requiring each term in the PDE to scale identically, a constraint that reduces to a linear system; working it out by hand means matching term by term, and the algebra is error-prone enough that valid reductions frequently go unnoticed.

The package automates this via the dilation method: it assigns a formal scaling exponent to each variable and term, writes the invariance constraint as a linear system in those exponents, and reads the reduced ODE directly from the null space of that linear system. No candidate powers need to be guessed or scanned; if a power-law similarity exists, the method finds it exactly. A string-based wrapper (find_similarity_v2) is also available for users who prefer not to work with Symbolics.jl expressions directly.

using SimilaritySolver, Symbolics

@variables x t u(..)
Dt = Differential(t); Dx = Differential(x)
kdv = Dt(u(x,t)) + 6*u(x,t)*Dx(u(x,t)) + Dx(Dx(Dx(u(x,t))))

results = find_ode_dilation(kdv; indep_vars=[x,t], dep_vars=[u(x,t)])
# results[1]["similarity_variable"]  =>  x * t^(-1//3)
# results[1]["gamma"]                =>  -2//1   (u scales as t^(-2/3))

The method returns all valid similarity reductions, not just one. It currently handles two independent variables with power-law scalings; extending to three or more variables and to non-power-law symmetries (logarithmic, spiral) is ongoing.