Preview. Results were computed and written by AI models; statements labelled 'agent-proof' have been read by a second AI model but not checked by a person.
Six equations, each with the kind of answer it gets. A link opens the result on the page Solve; where the catalogue has no report for the exact text, the engine computes it, in the browser or on the server.
u_t + u*u_x + u_xx + u_xxxx = 0
A published theorem gives the class of all meromorphic solutions of the travelling-wave equation (rational, simply periodic or elliptic, with one pole in a period), and the list within that class is complete: the constants and two simply periodic families.
Open the result typed as u_t + u*u_x + u_xx + u_xxxx = 0
w'' + 5/sqrt(6) w' + w - w^2 = 0
At this speed the travelling-wave equation has, besides the constants and a simply periodic family, a family of solutions with d = 2: an exponential times a Weierstrass function of an exponential.
Open the result typed as w'' + 5/sqrt(6) w' + w - w^2 = 0
i q_t + q_xx + g |q|^2 q = 0
Typed as an evolution equation for a complex field: the envelope reduction q = w(z)·exp(iθ) leads to a real equation of second order with a cubic term, and all its meromorphic solutions are listed.
Open the result typed as i q_t + q_xx + g |q|^2 q = 0
w'' = a w + b w^3 + c w^5
For c ≠ 0 the only meromorphic solutions are constants; the square u = w² satisfies an equation that has nonconstant meromorphic solutions, and the result lists them.
Open the result typed as w'' = a w + b w^3 + c w^5
w'''' + 2 w'' + (1 - r) w + w^3 = 0
Every meromorphic solution is rational, simply periodic or elliptic with at most two poles in a period; the engine in the browser lists those with one pole and reports those with two as not computed.
Open the result typed as w'''' + 2 w'' + (1 - r) w + w^3 = 0
w''' = 12 w w' + 1
Every solution with a pole is a first Painlevé transcendent, so d = 3: the result describes these solutions through an earlier record and gives no closed form.
Open the result typed as w''' = 12 w w' + 1