The hierarchy of rogue wave solutions in nonlinear systems

Oceanic freak waves, optical spikes and extreme events in numerous contexts can arguably be modelled by modulationally unstable solutions within nonlinear systems. In particular, the fundamental nonlinear Schroedinger equation (NLSE) hosts a high-amplitude spatiotemporally localised solution on a plane-wave background, called the Peregrine breather, which is generally considered to be the base-case prototype of a rogue wave. Nonetheless, until very recently, little was known about what to expect when observing or engineering entire clusters of extreme events. Accordingly, this thesis aims to elucidate this matter by investigating complicated structures formed from collections of Peregrine breathers. Many novel NLSE solutions are discovered, all systematically classifiable by their geometry. The methodology employed here is based on the well-established concept of Darboux transformations, by which individual component solutions of an integrable system are nonlinearly superimposed to form a compound wavefunction. It is primarily implemented in a numerical manner within this study, operating on periodically modulating NLSE solutions called breathers. Rogue wave structures can only be extracted at the end of this process, when a limit of zero modulation frequency is applied to all components. Consequently, a requirement for breather asymmetry ensures that a multi-rogue wavefunction must be formed from a triangular number of individual Peregrine breathers (e.g. 1, 3, 6, 10, ...), whether fused or separated. Furthermore, the arrangements of these are restricted by a maximum phase-shift allowable along an evolution trajectory through the relevant wave field. Ultimately, all fundamental high-order rogue wave solutions can be constructed via polynomial relations between origin-translating component shifts and squared modulation frequency ratios. They are simultaneously categorisable by both these mathematical existence conditions and the corresponding visual symmetries, appearing spatiotemporally as triangular cascades, pentagrams, heptagrams, and so on. These parametric relations do not conflict with each other, meaning that any arbitrary NLSE rogue wave solution can be considered a hybridisation of this elementary set. Moreover, this hierarchy of structures is significantly general, with complicated arrangements persisting even on a cnoidal background

Numerical Simulations of Water Waves\u27 Modulational Instability Under the Action of Wind and Dissipation

Since the work of Benjamin & Feir (1967), water waves propagating in infinite depth are known to be unstable to modulational instability. The evolution of such wave trains is well described through fully nonlinear simulations, but also by means of simplified models, such as the nonlinear Schrödinger equation. Segur et al. (2005) and Wu et al. (2006) studied theoretically and numerically the evolution of this instability, and both concluded that a long term restabilization occurs in these conditions. More recently, Kharif et al. (2010) considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation, and discussed the range of parameters for which this behavior is still valid. This work aims to demonstrate how numerical simulations are useful to analyze their theoretical predictions. Since we are dealing with long term stability, results are especially complicated to obtain experimentally. Thus, numerical simulations of the fully nonlinear equations turn out to be a very useful tool to provide a validation for the model. Here, the evolution of the modulational instability is investigated within the framework of the two-dimensional fully non linear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles theory. The introduction of dissipation in the equations is briefly discussed. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, the long term evolution of the wave trains can be obtained through the numerical simulations, and it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon

Sasa--Satsuma hierarchy of integrable evolution equations

We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth- order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly

Modulation instability and convergence of the random-phase approximation for stochastic sea states

The nonlinear Schrödinger equation is widely used as an approximate model for the evolution in time of the water wave envelope. In the context of simulating ocean waves, initial conditions are typically generated from a measured power spectrum using the random-phase approximation, and periodized on an interval of length L. It is known that most realistic ocean waves power spectra do not exhibit modulation instability, but the most severe ones do; it is thus a natural question to ask whether the periodized random-phase approximation has the correct stability properties. In this work, we specify a random-phase approximation scaling, so that, in the limit of L→∞ ,the stability properties of the periodized problem are identical to those of the continuous power spectrum on the infinite line. Moreover, it is seen through concrete examples that using a too short computational domain can completely suppress the modulation instability.Peer reviewe

Modulation instability and convergence of the random phase approximation for stochastic sea states

The nonlinear Schrödinger equation is widely used as an approximate model for the evolution in time of the water wave envelope. In the context of simulating ocean waves, initial conditions are typically generated from a measured power spectrum using the random-phase approximation, and periodized on an interval of length L. It is known that most realistic ocean waves power spectra do not exhibit modulation instability, but the most severe ones do; it is thus a natural question to ask whether the periodized random-phase approximation has the correct stability properties. In this work, we specify a random-phase approximation scaling, so that, in the limit of L → ∞ , the stability properties of the periodized problem are identical to those of the continuous power spectrum on the infinite line. Moreover, it is seen through concrete examples that using a too short computational domain can completely suppress the modulation instability.<br/

A hybrid model for simulating rogue waves in random seas on a large temporal and spatial scale

A hybrid model for simulating rogue waves in random seas on a large temporal and spatial scale is proposed in this paper. It is formed by combining the derived fifth order Enhanced Nonlinear Schrödinger Equation based on Fourier transform, the Enhanced Spectral Boundary Integral (ESBI) method and its simplified version. The numerical techniques and algorithm for coupling three models on time scale are suggested. Using the algorithm, the switch between the three models during the computation is triggered automatically according to wave nonlinearities. Numerical tests are carried out and the results indicate that this hybrid model could simulate rogue waves both accurately and efficiently. In some cases discussed, the hybrid model is more than 10 times faster than just using the ESBI method, and it is also much faster than other methods reported in the literature

Generalized Sasa--Satsuma equation: Densities approach to new infinite hierarchy of integrable evolution equations

We derive the new infinite Sasa-Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 x 3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa-Satsuma hierarchy functionals. All these operators are parts of a generalized Sasa-Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multi-parameter dynamical system

A hybrid model for large scale simulation of unsteady nonlinear waves

A hybrid model for simulating rogue waves in random seas on a large time and space scale is proposed in this thesis. It is formed by combining the derived fifth order Enhanced Nonlinear Schrödinger Equation based on Fourier transform (ENLSE-5F), the fully nonlinear Enhanced Spectral Boundary Integral (ESBI) method and its simplified version. The numerical techniques and algorithm for coupling three models on time scale are provided. Using them, and the switch between the three models during the computation is triggered automatically according to wave nonlinearities. Numerical tests are carried out and the results indicate that this hybrid model could simulate rogue waves both accurately and efficiently. In some cases showed, the hybrid model is more than 10 times faster than just using the ESBI method

Nonlinear wave patterns in the complex KdV and nonlinear Schrodinger equations

This thesis is on the theory of nonlinear waves in physics. To begin with, we develop from first principles the theory of the complex Korteweg-de Vries (KdV) equation as an equation for the complex velocity of a weakly nonlinear wave in a shallow, ideal fluid. We show that this is completely consistent with the well-known theory of the real KdV equation as a special case, but has the advantage of directly giving complete information about the motion of all particles within the fluid. We show that the complex KdV equation also has conserved quantities which are completely consistent with the physical interpretation of the real KdV equation. When a periodic wave solution to the real KdV equation is expanded in the quasi-monochromatic approximation, it is known that the amplitude of the wave envelope is described by the nonlinear Schrodinger (NLS) equation. However, in the complex KdV equation, we show that the fundamental modes of the velocity are described by the split NLS equations, themselves a special case of the Ablowitz-Kaup-Newell-Segur system. This is a directly physical interpretation of the split NLS equations, which were primarily introduced as only a mathematical construct emerging from the Zakharov-Shabat equations. We also discuss an empirically obtained symmetry of the rational solutions to the KdV equations, which seems to have been unnoticed until now. Solutions which can be written in terms of Wronskian determinants are well-known; however, we show that these are actually part of a more general family of rational solutions. We show that a linear combination of the Wronskians of orders $n$ and $n+2$ generates a new, multi-peak rational solution to the KdV equation. We next move on to the integrable extensions of the NLS equation. These incorporate higher order nonlinear and dispersive terms in such a way that the system keeps the same conserved quantities, and is thus completely integrable. We obtain the general solution of the doubly-periodic solutions of the class I extension of the NLS equation, and discuss several special cases. These are the most general one-parameter first order solutions of the (class I) extended NLS equation. Building on this, we also discuss second order solutions to the extended NLS equation. We obtain the general 2-breather solutions, and discuss several special cases; among them, semirational breathers, the degenerate breather solution, the second-order rogue wave, and the rogue wave triplet solution. We also discuss the breather to soliton conversion, which is a solution which does not exist in the basic NLS equation where only the lowest order dispersive and nonlinear terms are present. Finally, we discuss a few possibilities for future research based on the work done in this thesis

Solitons, Breathers and Rogue Waves in Nonlinear Media

In this thesis, the solutions of the Nonlinear Schrödinger equation (NLSE) and its hierarchy are studied extensively. In nonlinear optics, as the duration of optical pulses get shorter, in highly nonlinear media, their dynamics become more complex, and, as a modelling equation, the basic NLSE fails to explain their behaviour. Using the NLSE and its hierarchy, this thesis explains the ultra-short pulse dynamics in highly nonlinear media. To pursue this purpose, the next higher-order equations beyond the basic NLSE are considered; namely, they are the third order Hirota equation and the fifth order quintic NLSE. Solitons, breathers and rogue wave solutions of these two equations have been derived explicitly. It is revealed that higher order terms offer additional features in the solutions, namely, ‘Soliton Superposition’, ‘Breather Superposition’ and ‘Breather-to-Soliton’ conversion. How robust are the rogue wave solutions against perturbations? To answer this question, two types of perturbative cases have been considered; one is odd-asymmetric and the other type is even-symmetric. For the odd-asymmetric perturbative case, combined Hirota and Sasa-Satsuma equations are considered, and for the latter case, fourth order dispersion and a quintic nonlinear term combined with the NLSE are considered. Indeed, this thesis shows that rogue waves survive these perturbations for specific ranges of parameter values. The integrable Ablowitz-Ladik (AL) equation is the discrete counterpart of the NLSE. If the lattice spacing parameter goes to zero, the discrete AL becomes the continuous NLSE. Similar rules apply to their solutions. A list of corresponding solutions of the discrete Ablowitz-Ladik and the NLSE has been derived. Using associate Legendre polynomial functions, sets of solutions have been derived for the coupled Manakov equations, for both focusing and defocusing cases. They mainly explain partially coherent soliton (PCS) dynamics in Kerr-like media. Additionally, corresponding approximate solutions for two coupled NLSE and AL equations have been derived. For the shallow water case, closed form breathers, rational and degenerate solutions of the modified Kortweg-de Vries equation are also presented