Studies of the nonlinear Schrodinger equation and its application to water waves

• Main Author: Martin, David U.
• Format: Others
• Language: en
• Published: 1981
• Online Access: https://thesis.library.caltech.edu/3928/1/Martin_du_1981.pdf; Martin, David U. (1981) Studies of the nonlinear Schrodinger equation and its application to water waves. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/by3r-zk43. https://resolver.caltech.edu/CaltechETD:etd-10052006-082151 <https://resolver.caltech.edu/CaltechETD:etd-10052006-082151>

Various properties of the nonlinear Schrodinger equation and special solutions thereof are investigated, with emphasis on applications to water waves. The spreading of modal energy for initial conditions corresponding to unstable perturbations of a uniform wave is investigated numerically and analytically. For a one-dimensional surface, the upper limit of the Benjamin-Feir instability interval appears to provide a good estimate of the maximum spread of the modal energy. The analytical estimate of Thyagaraja is shown to be insufficiently sharp to account for this effective maximum. In the case of a two-dimensional surface, the instability region obtained with the nonlinear Schrodinger equation is infinite in extent, and the numerical results suggest that energy may leak to arbitrarily high unstable harmonics in a quasi-recurring fashion. Stability results for plane-wave envelopes on a two-dimensional surface are calculated and verified numerically with the nonlinear Schrodinger equation. For standing-wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains, but that associated with the even modes disappears. This is consistent with previous results on the stability of solitons. In addition, traveling-wave instabilities are identified for even mode perturbations which are absent in the longwave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities may also be present for the envelope solitons. Thus the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, traveling-wave perturbations. Bifurcation techniques are used to obtain a new class of small-amplitude water waves of permanent form. Stokes waves are used as a starting point, and the critical value of steepness at which bifurcation can occur is computed for various choices of modulation wavelength and angular orientation. It is found that, for a two-dimensional surface, bifurcation can occur at small values of wave steepness. Second order corrections to the wave amplitude, modulation, frequency, and speed, which apply when one moves off the bifurcation point onto a new branch of solutions, are also computed. Two types of new solutions are found, one symmetric with respect to the carrier wave propagation direction, and one asymmetric. Finally, the nonlinear Schrodinger equation is used to study the interaction of deep-water gravity waves with currents. The case of a uniform wave encountering a steady current is treated numerically and analytically, and the results are shown to correspond to known results in the linear limit. As an example of modulated waves encountering a current, the development of an instability of Benjamin-Feir type is calculated in both the presence and absence of current. Finally, the case of an envelope soliton encountering a current is treated numerically, and the results are compared to those obtained by applying a perturbation scheme.