Asymptotic integrability of nonlinear wave equations

Abstract: The notion of asymptotic integrability is based on the asymptotic theory of propagation of high-frequency wave packets along large-scale and time-dependent backgrounds. We assume that the evolution of the background obeys the dispersionless (hydrodynamic) limit of the nonlinear wave equation under consideration and demand that the Hamilton equations for the packet's propagation have an additional integral of motion independently of the initial conditions for the background dynamics. This condition is studied for systems described by one or two wave variables, and it is shown that it imposes strong restrictions on the dispersion relation for linear harmonic waves in the case of two wave variables. Existence of the integral of Hamilton’s equations leads to important consequences: (1) it allows one to calculate the number of solitons produced from an intensive initial pulse; (2) this formula can be generalized in a natural way to the Bohr-Sommerfeld quantization rule for parameters of solitons produced from such a pulse; (3) if the condition of asymptotic integrability is only fulfilled approximately, then the Bohr-Sommerfeld rule provides the solitons’ parameters with good accuracy even for not completely integrable equations; (4) if it is fulfilled exactly, then the appearing in the theory integral can be identified with the quasiclassical limit of one of the equations of the Lax pair for the corresponding completely integrable equation with the same dispersion relation and equations of the dispersionless limit, moreover, the second equation of the Lax pair is related to the phase velocity of linear waves; (5) “quantization” of the quasiclassical limit allows one to restore the full expressions for the Lax pair equations; (6) analytical continuation of the integral into the complex plane of wave numbers yields the expression for the soliton’s inverse half-width as a function of the background wave variables; (7) existence of such an integral for soliton motion leads to formulation of Hamiltonian dynamics of solitons moving along not-uniform and time-dependent background. The theory is illustrated by examples, and it is confirmed by comparison with numerical simulations.

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods