The Zabolotskaya-Khokhlov Equation and the Inverse Scattering Problem of Classical Mechanics

  • John Gibbons
Part of the Springer Series in Synergetics book series (SSSYN, volume 30)


The Zabolotskaya-Khokhlov equation,
$${U_{TX}} + {\left( {{U^2}/2} \right)_{XX}} = {U_{YY}}$$
which models beams of nonlinear, diffracting sound waves in two dimensions, may be written as the compatibility condition between two classical Hamiltonian flows, one evolving in Y with Hamiltonian H2= P2/2 + U(X,Y,T), and the other evolving in T, with Hamiltonian H3= P3/3 + P U(X,Y,T) + V(X,Y,T). Some solutions of the system may consequently be derived from solutions of the Benney hierarchy, a class of moment equations which arose originally as a description of long waves on a shallow perfect fluid, while the initial value problem is equivalent to the open problem of finding the potential U(X,Y,T) in the Hamiltonian H2, given the asymptotics of its orbits.


Poisson Bracket Compatibility Condition Moment Equation Vlasov Equation Semiclassical Limit 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • John Gibbons
    • 1
  1. 1.Department of MathematicsImperial CollegeLondonGreat Britain

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