Dynamics of wave propagation in nonlinear optics and hydrodynamics

Abstract

Several significant wave propagation problems in the fields of nonlinear optics and hydrodynamics are studied in this thesis. In optics, the physical model considered is the two-core optical fiber (TCF), which is an essential component of lightwave technology. In hydrodynamics, the motion of a wave packet on the free surface of water of finite depth allowing modulations from two mutually perpendicular and horizontal directions, governed by the famous Davey-Stewartson (DS) equations, is taken into account. The main contributions of this thesis are: In optics, the effects of the intermodal dispersion (IMD) and the birefringence induced effects, both of which always exist in the TCFs, have been ignored in the previous studies of the modulation instability (MI) of continuous waves (CWs) in the TCFs. In this thesis, a detailed analysis of these effects on the MI spectra has been done. It is found that IMD does not seriously affect the MI spectra of the symmetric/antisymmetric CW states, but can significantly modify the MI spectra of the asymmetric CW states. In exploring the birefringence induced effects, a particular class of asymmetric CW states, which admits analytical solutions and has no counterpart in the single-core fibers, is focused on. It is found that the MI spectra of a birefringent TCF in the normal dispersion regime can be distinctively different from those of a zero-birefringence TCF especially for the circular-birefringence TCF. All the findings of MI analysis can be well verified by the wave propagation dynamics. Another contribution of this thesis is that we find the dramatic pulse distortion and even pulse splitting phenomenon due to IMD in TCFs, which is unexpected in many situations, can be effectively suppressed and even avoided by Kerr nonlinearity, which has never been reported in the literatures in the studies of TCFs. In hydrodynamics, DS equations describe the evolution of weakly nonlinear, weakly dispersive wavepackets with slow spanwise dependences on a fluid of finite depth. Generally, DS equations are divided into two types e, i.e. DSI and DSII equations, depending on the specific fluid configurations (fluid depth, wavelength of the water wave, surface tension etc). Due to the importance of DS equations, many exact solutions have been derived by different nonlinear wave methods over the years in the literature. In this thesis, two new exact doubly periodic wave patterns of DS equations are derived by the use of properties of the theta functions, or equivalently, the Jacobi elliptic functions, and the corresponding solitary waves are also deduced in the long wave limits. The new feature of the two wave patterns found is that they can be applied to both DSI and DSII systems at the same time.