  We study one-dimensional quantum systems near the classical limit described
by the Korteweg-de Vries (KdV) equation. The excitations near this limit are
the well-known solitons and phonons. The classical description breaks down at
long wavelengths, where quantum effects become dominant. Focusing on the
spectra of the elementary excitations, we describe analytically the entire
classical-to-quantum crossover. We show that the ultimate quantum fate of the
classical KdV excitations is to become fermionic particles and holes. We
discuss in detail two exactly solvable models exhibiting such crossover, the
Lieb-Liniger model of bosons with weak contact repulsion and the quantum Toda
model. We argue that the results obtained for these models are universally
applicable to all quantum one-dimensional systems with a well-defined classical
limit described by the KdV equation.
