Spontaneously generated waves in perturbed evolution equations

The Burgers and KdV equations provide approximations to more complex dynamical systems. Both equations can be viewed as conservation laws and are integrable. When the small terms, left out in the derivation of the two equations from the original systems, are re-instated, the resulting perturbed equations may lose the conservation-law structure. In addition, integrability may be lost even in the asymptotic sense (i.e. in the order-by-order perturbation analysis). These losses are independent occurrences. However, they can both be traced to a specific contribution, which may exist in the original perturbation and/or may surface in the higher-order analysis. Identification of that contribution is based on the observation that, in the case of a single-wave (front, soliton) zero-order approximation to the solution, the conservation-law structure of the perturbed equation is preserved, and obstacles to asymptotic integrability do not emerge. We first identify the component in the perturbation which is responsible for the fact that the perturbed equation does not have a conservation-law structure. A driving term of an identical structure may emerge in the equations which determine higher-order contributions to the solution and lead to the loss of asymptotic integrability. That both properties are not lost in the case of a single-wave solution is a consequence of the fact that this component in the perturbation vanishes identically. This component can be viewed as a coupling between the solution one is seeking and a new wave that is generated spontaneously by the perturbation. In the case of the Burgers equation, the spontaneously generated wave is a solution of the equation but different from the solution one is seeking. In the case of the perturbed KdV equation, the new wave is constructed from non-KdV solitons.