Solitons on Twisted Closures: How a Möbius Twist Creates, Selects and Sustains Nonlinear Waves

A twisted closure joins a field to its own negative after one circuit, as a Möbius ring or a resonator with a phase defect of π does. This article finds that the twist selects, creates and sustains nonlinear waves, and marks the ones that circulate. On the double cover of the ring the twisted fields are the odd modes, the twisted sector of the MKT canon. An odd nonlinearity, as in the NLS, mKdV, φ⁴ and sine-Gordon equations, keeps that sector closed. An even one converts it in a homogeneous medium, where KdV makes a second harmonic; on a twisted closure the quadratic products stay twisted, and no second harmonic forms. What the twist creates depends on how it acts on the ground states. When it exchanges two of them, as in φ⁴, it forces a kink, which needs a ring longer than π. When it turns a continuous phase by half a turn, as in a repulsive condensate, it forces a current of half a quantum with a spontaneous direction. When it maps a ground state onto itself, as in sine-Gordon, it forces nothing. A forward and a backward soliton are paired by the mirror unless a term of the equation is odd under it, and likewise by the time reversal; their product survives self-steepening, which is odd under both. On a twisted closure the pairing is unchanged, and the sign of a twin is fixed by the mirror axis. A soliton circulating on a twisted ring changes sign at every circuit, so the lines of its pulse train shift by half a spacing. Most of these phenomena are known in separate fields; the contribution is the route through the twisted sector and its design rules. The results are measured on model equations, not on a fabricated device.