Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

This Special Issue focuses on recent progress in a new area of mathematical physics and applied analysis, namely, on nonlinear partial differential equations on metric graphs and branched networks. Graphs represent a system of edges connected at one or more branching points (vertices). The connection rule determines the graph topology. When the edges can be assigned a length and the wave functions on the edges are defined in metric spaces, the graph is called a metric graph. Evolution equations on metric graphs have attracted much attention as effective tools for the modeling of particle and wave dynamics in branched structures and networks. Since branched structures and networks appear in different areas of contemporary physics with many applications in electronics, biology, material science, and nanotechnology, the development of effective modeling tools is important for the many practical problems arising in these areas. The list of important problems includes searches for standing waves, exploring of their properties (e.g., stability and asymptotic behavior), and scattering dynamics. This Special Issue is a representative sample of the works devoted to the solutions of these and other problems

Soliton generation and control in engineered materials

Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities. New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis. We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two components allows the stabilization of the whole soliton state. Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case. Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed. We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes. Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption

Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity

Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the self-modulation of the continuous “envelope waves” with respect to small periodic perturbations that precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales

Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches

Impact of PVT Properties of the Fluid on the LBM Scheme Within the Scale Integration for Shale Reservoirs

Modelling of the performance of shale gas reservoirs is known for the presence of multiple scales. The latter includes pore-scale, fracture scale and field scale. The nature of flow-mechanisms at various scales is different. Therefore, separate treatment of the physical processes is required. On the other hand, an integrated approach is highly beneficial for practical implementation. One of the candidates for seamless integration concerned is the Lattice-Boltzmann Method. The latter fact together with the demands of the industry provides the major motivation for the present work. In this study the novel Lattice-Boltzmann Model for pore-scale simulations has been introduced. The major advantage of the approach concerned is that the mathematical formulation of the model has a high degree of self-consistency. The latter means that it does not have an artificially introduced terms like pseudo-potentials, which are common for conventional Lattice-Boltzmann schemes. Despite the advantages of the approach in terms of mathematical formulation, there exist certain limitations because of the issues with numerical stability. One of the most important results of the present work is that the issues concerned can not be resolved by the reasonable increase of the number of lattice vectors in the model. The limitations involved make the scheme impractical for fieldscale simulations. Therefore, an alternative formulation of Lattice-Boltzmann method for reservoir modelling is required. In the present work, a novel pseudo-potential model for field-scale simulations has been introduced. The model concerned demonstrates a reasonable agreement with the analytical techniques in the case of steady-state flow. However, further investigation shows significant deviations because of the numerical diffusion. Moreover, it has been shown that significant numerical diffusion is a feature of the majority of the existent pseudo-potential models. The numerical effect concerned is critically important in the case of the multiphase flow, because it can lead to non-physical solutions. In order to resolve the problem concerned a novel Lattice-Boltzmann Scheme has been introduced. The scheme demonstrates reasonable agreement with analytical methods and with simulations performed with trusted programs for reservoir modelling. Finally, the major contribution of the present work includes the development of selfconsistence approach for simulations at pore-scale, the proof of fundamental limitations of the model introduced, observation of numerical diffusion in pseudo-potential Lattice- Boltzmann Methods, and the solution of the latter issue through the development of the novel Lattice-Boltzmann scheme for field-scale simulations

Polaron Dynamics in the Alpha Helix: Models of Electron Transport in Hydrogen-Bonded Polypeptides

In this thesis, I present two mathematical models which are capable of explaining the phenomenon of directed electron transport in α-helical regions of protein macromolecules. The models are built upon the framework of polaron theory, which originated in condensed matter physics, and which I argue is applicable to biophysical systems such as an extra electron interacting electromagnetically with peptide units in an α-helix. The two models concern the electron’s coupling to, respectively, picosecond-scale intrapeptide oscillators and nanosecond-scale hydrogen bond phonons in the α-helix. I show that the models permit the auto-localisation of the electron in stationary polaron states, and that certain electromagnetic fields cause the polaron to propagate along the polypeptide, transporting the electron in a solitonic manner. Taking effects of the cell environment into account, I demonstrate that stochastic forces arising from thermal fluctuations can enhance the electron transport, and that the stability of the polaron dynamics exhibit contrasting degrees of tolerance to temperature in the two models. When interpreting my results, I describe their biological implications, as well as the physical realisability of the models’ forcing parameters. In particular, I establish that some electromagnetic fields which can facilitate directed electron transport are intrinsic physical features of the cell