KdV evolution and dispersive balancing

Use u_t+6u u_x+u_xxx=0 for a real-valued field on the whole line. Smooth rapidly decaying solutions permit the integrations by parts used below. For the linearized equation, insert exp(i(kx−omega t)); the dispersion relation is omega=−k cubed. Thus phase velocity is minus k squared and group velocity is minus three k squared. These signs belong to this particular convention. Reversing the spatial coordinate or the sign of the third derivative changes the interpretation.

The linear Fourier multiplier is exp(i k cubed t), which has modulus one. It changes phase rather than damping each Fourier mode. The nonlinear term is the derivative of three u squared and moves different amplitudes at different local speeds. Dimensional balance compares amplitude squared over width with amplitude over width cubed, suggesting amplitude proportional to inverse width squared. That scaling predicts the shape relation, while the traveling-wave equation determines the exact coefficient and profile.

Set u(t,x)=Q(x−ct−x₀), with positive speed c and a profile decaying together with its derivatives. Substitution gives minus c Q prime plus six Q Q prime plus Q triple prime equal to zero. Integrating once and using decay sets the integration constant to zero, leaving Q double prime minus cQ plus three Q squared equal to zero. The localized solution is c over two times sech squared of square root of c times the traveling coordinate divided by two.

Consequently, larger positive pulses are taller, narrower, and faster. Their positive rightward travel does not contradict the leftward group velocity of the linearized zero-background equation; nonlinear transport changes the balance. Integrating the profile gives two times square root of c, and its squared L² integral is two thirds c to the three halves. These are whole-line quantities. Periodic KdV has different traveling families and integration constants, so imposing zero at infinity would not be justified for every periodic profile.

For sufficiently regular decaying KdV solutions, integrating the equation preserves the spatial integral of u. Multiplying by u and integrating preserves the squared L² norm because both the transport and dispersive contributions become boundary terms. A Hamiltonian is the integral of one half u_x squared minus u cubed under this normalization. Deriving these invariants requires the same control of boundary terms as the traveling-wave integration. The identities do not mean the profile remains a soliton for arbitrary initial data.

KdV scales as lambda squared times u at time lambda cubed t and position lambda x. In one dimension the homogeneous H to the s norm gains lambda to the s plus three halves, giving formal critical order minus three halves. This dimensional value is not a sharp low-regularity well-posedness theorem; derivative nonlinearities and resonant interactions create additional difficulties. Likewise a linear superposition argument fails because squaring a sum creates cross-interactions. Exact multisoliton constructions use more structure than adding pulses, and a plot of separated pulses is not proof that those interaction terms vanish.