Solitons and orbital stability

Fix the whole-line focusing cubic convention i u_t plus u_xx plus |u|²u equals zero. Substitute u(t,x)=exp(i omega t)Q(x), with omega positive and Q real and decaying. The profile equation is minus Q double prime plus omega Q minus Q cubed equals zero. The positive localized solution is square root of twice omega times sech of square root of omega times x. Phase multiplication and spatial translation give further profiles, and Galilean symmetry generates traveling waves with an additional spatial phase factor.

Direct differentiation verifies the profile without implying anything about perturbations. Its height is square root of twice omega and its width scale is one over square root of omega. Integrating sech squared gives mass four times square root of omega. Thus changing omega changes the mass and width; it is not merely a translation or phase change within the fixed-frequency orbit. This distinction matters because the variational argument minimizes energy at a prescribed mass rather than comparing arbitrary frequencies as if the constraint stayed fixed.

For fixed omega, the orbit consists of exp(i theta)Q_omega(x−y), with arbitrary real phase and translation. Orbital H¹ stability means that for every positive error tolerance there is an initial-distance tolerance such that any H¹ datum within that distance of the orbit generates a global solution remaining within the error tolerance of the orbit for all times. The optimizing phase and center may depend on time. This is the standard stable ground-state setting for one-dimensional focusing cubic NLS, with perturbations on the same whole line.

The conclusion is weaker than asymptotic stability, which would require a specified long-time convergence or radiation decomposition. It also does not assert that a fixed absolute phase or fixed center is stable without adjustment. Nearby solutions can carry small velocity and drift. Higher-dimensional perturbations of a one-dimensional wave introduce a different transverse-stability problem, and other powers may lie in mass-critical or supercritical regimes with instability. The theorem stated here should not be transferred to those settings merely because they also have a sech-like or localized profile.

The variational route fixes positive mass m and minimizes the focusing cubic energy over H¹ profiles of that mass. A sharp Gagliardo–Nirenberg inequality bounds the negative quartic term by a fixed mass factor times the gradient norm, so energy is bounded below on the constraint. The minimizers are the translated, phase-rotated Q family with matching mass. Their energy is minus m cubed over ninety-six under the current normalization. This is a constrained minimum; the unconstrained energy is not being claimed globally positive.

To turn minimization into orbital stability, a near-minimizing sequence must be compact after translations and phases. On the unbounded line, ordinary compactness fails because a profile can drift to infinity. Concentration-compactness rules out the incompatible vanishing and splitting scenarios in this subcritical ground-state problem. Conservation of mass and energy then prevents a nearby evolving solution from leaving the minimizing orbit by a fixed amount. This paragraph describes the proof mechanism, not a full compactness proof. A critical point of the energy under a constraint can also be a saddle, so the profile equation alone cannot replace minimization and compactness.