Control of Integrable Nonlinear Waves
Abstract
In this poster we study the generic properties of the control of integrable nonlinear waves. The control term contains both conservative and dissipative terms and targets an exact solution of the undriven system. We take as our model system a driven nonlinear Schrodinger equation (NLS), iq_{t} + q_xx + 2q^2q = iɛ (q_0q), where q_{0} is an exact solution to the undriven NLS (ɛ = 0). The undriven NLS is the generic equation for the slow space and time dynamics of weakly nonlinear nearly monochromatic wave trains in a disperive medium. The phase of the complex control gain, ɛ, has an effect on the controllability to the target by dictating the strength of the conservative term with respect to the dissipative term. We show when a controller is applied to integrable nonlinear wave systems a highly degenerate bifurcation can occur which implies extreme noise sensitivity [1]. We motivate our results using one and two mode truncations of the driven NLS previously mentioned, which in the undriven case are 1 and 2dimensional integrable Hamiltonian systems, respectively. [1] C. W. Kulp & E. R. Tracy, Control of Integrable Hamiltonian Systems and Degenerate Bifurcations, available online at http://arxiv.org/abs/nlin.PS/0306018.
 Publication:

APS Division of Plasma Physics Meeting Abstracts
 Pub Date:
 October 2003
 Bibcode:
 2003APS..DPPCP1037K