Abstract
The nonlinear Schrödinger equation (NLS) is a canonical evolution equation, which describes the dynamics of weakly nonlinear wave packets in time and space in a wide range of physical media, such as nonlinear optics, cold gases, plasmas and hydrodynamics. Due to its integrability, the NLS provides families of exact solutions describing the dynamics of localised structures which can be observed experimentally in applicable nonlinear and dispersive media of interest. Depending on the co-ordinate of wave propagation, it is known that the NLS can be either expressed as a space- or time-evolution equation. Here, we discuss and examine in detail the limitation of the first-order asymptotic equivalence between these forms of the water wave NLS. In particular, we show that the the equivalence fails for specific periodic solutions. We will also emphasise the impact of the studies on application in geophysics and ocean engineering. We expect the results to stimulate similar studies for higher-order weakly nonlinear evolution equations and motivate numerical as well as experimental studies in nonlinear dispersive media.
Highlights
The theory of weakly nonlinear water waves has been found to be very useful for the modelling of ocean waves [1,2,3]
The nonlinear Schrödinger equation (NLS) provides exact analytical solutions that describe the evolution of localised structures on the water surface in time and space, allowing subsequently the study and understanding of the dynamics of fundamental localised structures
The family of Akhmediev breathers (ABs) [18] and Peregrine breathers [18,19] are strongly connected to the modulation instability (MI), known as Benjamin–Feir instability [20], of Stokes waves [21]
Summary
The theory of weakly nonlinear water waves has been found to be very useful for the modelling of ocean waves [1,2,3]. The validity of the NLS has been experimentally confirmed even in the modelling of extreme localisations, beyond its well-known asymptotic limitations [4,5,6,7,8], and due to its interdisciplinary character analogies being able to be built into other nonlinear dispersive media, such as in optics [9], a research field in which several NLS applications have found strong interest [9,10,11,12,13,14]. The experimental investigation of exact solutions of the NLS, either numerically or in water wave facilities, has increased the degree of understanding of nonlinear and unstable water waves, as well as allowing the characterisation of the limitations of weakly nonlinear hydrodynamic models [8]. A detailed analysis of this feature, and its consequences and potential applications, will be discussed
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
