Nonlinear Schrödinger evolution and conservation

Use i u_t plus the Laplacian of u equals kappa |u| to the p minus one times u, with real kappa. Positive kappa is the defocusing convention in this lesson; negative kappa is focusing. For a smooth decaying solution on whole space, or a smooth periodic solution with matching boundary terms, multiply by the complex conjugate and take the appropriate real or imaginary part. The Laplacian term integrates to a real gradient integral and the nonlinear term is real pointwise, so the derivative of the squared L² norm vanishes.

The energy is half the gradient norm squared plus kappa over p plus one times the integral of |u| to the p plus one. Differentiation pairs the variational derivative with u_t. The equation makes that pairing purely imaginary, so its real part is zero. These identities need sufficient integrability and regularity to justify differentiation and boundary cancellation. For energy-space solutions obtained by approximation, conservation is established by passing from smooth solutions with the necessary continuity, not by performing unjustified pointwise operations on distributions.

For the defocusing cubic equation on the real line, let the initial datum belong to H¹. The one-dimensional local H¹ theory supplies a lifespan depending on its controlling norm and a blow-up alternative. Conservation extends from smooth approximations. Because both terms of the energy are nonnegative, the gradient norm squared is at most twice the initial energy. The mass controls the remaining L² part of H¹. These uniform bounds prevent the local control norm from blowing up, so the solution extends globally in both time directions.

The argument's architecture is more general than its current example, but the assumptions cannot simply be removed. In an energy-critical setting, bounded energy does not automatically provide every spacetime norm needed for continuation. Extra global analysis may be required even with a positive conserved energy. On other domains or for other powers, nonlinear differentiability and local theory also change. State exactly which continuation norm your local theorem uses before claiming that a conserved quantity rules out its breakdown.

With a negative nonlinear coefficient, energy is a difference and no longer directly bounds the gradient. That does not mean every focusing solution blows up. The one-dimensional focusing cubic equation is mass-subcritical; a Gagliardo–Nirenberg estimate bounds its quartic term using mass and a lower power of the gradient, allowing absorption and global H¹ control. It also supports localized standing waves, which will appear next. Other dimensions and powers can permit finite-time concentration under additional hypotheses, so the sign alone is not a complete classification.

Nor does conserved mass imply that the solution becomes small pointwise or approaches a linear wave. A soliton can retain a localized profile forever, and periodic domains can support recurrent nondecaying motion. Scattering is a separate asymptotic statement comparing the nonlinear solution with a free evolution in a specified norm. Weak formulations and nonconservative approximations need particular care: conservation may become an inequality, fail to pass to the limit, or require a stronger solution class. Every invariant should be paired with its derivation and its actual control consequence.