Two-soliton interaction as an elementary act of soliton turbulence in integrable systems

Two-soliton interactions play a definitive role in the formation of the structure of soliton turbulence in integrable systems. To quantify the contribution of these interactions to the dynamical and statistical characteristics of the nonlinear wave field of soliton turbulence we study properties of the spatial moments of the two-soliton solution of the Korteweg-de Vries (KdV) equation. While the first two moments are integrals of the KdV evolution, the 3rd and 4th moments undergo significant variations in the dominant interaction region, which could have strong effect on the values of the skewness and kurtosis in soliton turbulence

Criteria for the Transition from a Breaking Bore to an Undular Bore

Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchive

Phase mixing of Alfven waves in axisymmetric non-reflective magnetic plasma configurations

We study damping of phase-mixed Alfvén waves propagating in non-reflective axisymmetric magnetic plasma configurations. We derive the general equation describing the attenuation of the Alfvén wave amplitude. Then we applied the general theory to a particular case with the exponentially divergent magnetic field lines. The condition that the configuration is non-reflective determines the variation of the plasma density along the magnetic field lines. The density profiles exponentially decreasing with the height are not among non-reflective density profiles. However, we managed to find non-reflective profiles that fairly well approximate exponentially decreasing density. We calculate the variation of the total wave energy flux with the height for various values of shear viscosity. We found that to have a substantial amount of wave energy dissipated at the lower corona, one needs to increase shear viscosity by seven orders of magnitude in comparison with the value given by the classical plasma theory. An important result that we obtained is that the efficiency of the wave damping strongly depends on the density variation with the height. The stronger the density decrease, the weaker the wave damping is. On the basis of this result, we suggested a physical explanation of the phenomenon of the enhanced wave damping in equilibrium configurations with exponentially diverging magnetic field lines

Critical density of a soliton gas

We quantify the notion of a dense soliton gas by establishing an upper bound for the integrated density of states of the quantum-mechanical Schr\"odinger operator associated with the KdV soliton gas dynamics. As a by-product of our derivation we find the speed of sound in the soliton gas with Gaussian spectral distribution function.Comment: 7 page

Two-soliton interaction as an elementary act of soliton turbulence in integrable systems

Two-soliton interactions play a definitive role in the formation of the structure of soliton turbulence in integrable systems. To quantify the contribution of these interactions to the dynamical and statistical characteristics of the nonlinear wave field of soliton turbulence we study properties of the spatial moments of the two-soliton solution of the Korteweg-de Vries (KdV) equation. While the first two moments are integrals of the KdV evolution, the 3rd and 4th moments undergo significant variations in the dominant interaction region, which could have strong effect on the values of the skewness and kurtosis in soliton turbulence

Dynamics of irregular wave ensembles in the coastal zone

Les vagues et les ondes internes de gravité ont un impact important surl’hydrodynamique et l’hydrologie de la zone côtière. Les vagues extrêmes sontparticulièrement intéressantes à étudier, car elles sont une menace sérieuse pour letransport maritime, les plates-formes pétrolières, les installations portuaires et leszones touristiques de la côte. Ces ondes entravent aussi les activités humainesdéveloppées à la côte. Les ondes internes non linéaires affectent la biosphèreaquatique, notamment le transport de sédiments et créent des affouillements à labase des plates-formes et des pipelines. Elles affectent également la propagationdes signaux acoustiques. Les vagues scélérates provoquent d’importants dégâtsmatériels et de nombreuses pertes en vies humaines. Par conséquent, l’étude de laformation des ondes scélérates dans la zone côtière est d’une importance capitale.L'objectif principal de la thèse est l'étude de la formation d’ondes océaniquesanormales dans la zone côtières pour différentes profondeurs d’eau et différentschamps d'ondes. Il est montré que le mécanisme de focalisation dispersive àl’origine de la formation d’ondes scélérates est pertinent quand les ondesinteragissent avec une paroi verticale. Il est démontré que juste avant la formationde l’onde maximale, celle-ci change rapidement de forme, d'une haute crête vers uncreux profond. La durée de vie de l’onde scélérate augmente avec le nombred’ondes individuelles contenues dans le paquet d'ondes anormales et lorsque laprofondeur de l'eau diminue.Il est démontré que l'interaction de paires de solitons unipolaires conduit à unediminution des facteurs de dissymétrie et d’aplatissement du champ d'ondes. Il estprouvé que dans le cas d'interactions hétéropolaires de solitons, le facteurd’aplatissement augmente.La dynamique non linéaire de champs de solitons unipolaires aléatoires estétudiée dans le cadre de l’équation de Korteweg - de Vries (KdV) et de l’équationde Korteweg - de Vries modifiée (mKdV). Il est montré que les coefficients dedissymétrie et d'aplatissement du gaz de solitons sont réduits à la suite de collisionsde solitons. Les fonctions de distribution des amplitudes des ondes sont obtenues.Le comportement des champs solitoniques dans le cadre de ces modèles estqualitativement similaire. Il est démontré que l'amplitude des ondes extrêmesdiminue en moyenne en raison des interactions entre multi-solitons.Dans le cadre de l'équation de Korteweg-de Vries modifiée, les interactionsnon linéaires entre le soliton de plus petite amplitude et les autres solitons du gazont pour effet de réduire sa célérité qui devient négative et de modifier ainsi sadirection de propagation.A partir de l'équation de Korteweg-de Vries modifiée, il est prouvé que dans ungaz de solitons héteropolaires, des ondes scélérates peuvent se former. Laprobabilité d’occurrence et l’amplitude des ondes scélérates dans de tels systèmesaugmente avec la densité du gaz de solitons.Surface and internal gravity waves have an important impact on the hydrological regime ofthe coastal zone. Intensive surface waves are particularly interesting to study because they canbe a serious threat to ships, oil platforms, port facilities and tourist areas on the coast; suchwaves hampered the implementation of human activities on the shelf. Nonlinear internal wavesaffect the underwater biosphere and cause sediment transport, they create washouts soil at thebase of platforms and pipelines, affect the propagation of acoustic signals. Freak waves have aparticularly strong impact, and they are studied in this thesis. Therefore, the study of freak waveformation in the coastal zone is relevant and practically significant.The main goal of the thesis is the study of particularities of abnormal wave formation incoastal zones under different assumptions on the water depth and wave field form. In particular,it is demonstrated that the mechanism of dispersion focusing of freak wave formation "works"for waves interacting with a vertical barrier. It is shown that just before the maximum waveformation a freak wave quickly changes its shape from a high ridge to a deep depression.Lifetime of freak wave increases with the growth of number of individual waves in anomalouswave packet, and lifetime of freak wave increases with water depth decreasing.It is demonstrated that pair interaction of unipolar solitons leads to decrease of the thirdand fourth moments of the wave field. It is shown that in the case of heteropolar solitoninteraction the fourth moment increases.The nonlinear dynamics of ensembles of random unipolar solitons in the framework of theKorteweg - de Vries equation and the modified Korteweg - de Vries equation is studied. It isshown that the coefficients of skewness and kurtosis of the soliton gas are reduced as a resultof soliton collision, the distribution function of wave amplitudes are defined. The behavior ofsoliton fields in the framework of these models is qualitatively similar. It is shown that in thesefields the amplitude of the big waves is decreased in average due to multi-soliton interactions.A new braking effect of soliton with a small amplitude and even changing of its direction inmulti-soliton gas as a result of nonlinear interaction with other solitons is found in the frameworkof the modified Korteweg-de Vries equation.It is shown that in heteropolar soliton gas abnormally big waves (freak waves) appear inthe frameworks of the modified Korteweg - de Vries equation. With increasing of soliton gasdensity the probability and intensity of freak waves in such systems increases

Dynamique de champs de vagues irréguliers en zone côtière

Surface and internal gravity waves have an important impact on the hydrological regime ofthe coastal zone. Intensive surface waves are particularly interesting to study because they canbe a serious threat to ships, oil platforms, port facilities and tourist areas on the coast; suchwaves hampered the implementation of human activities on the shelf. Nonlinear internal wavesaffect the underwater biosphere and cause sediment transport, they create washouts soil at thebase of platforms and pipelines, affect the propagation of acoustic signals. Freak waves have aparticularly strong impact, and they are studied in this thesis. Therefore, the study of freak waveformation in the coastal zone is relevant and practically significant.The main goal of the thesis is the study of particularities of abnormal wave formation incoastal zones under different assumptions on the water depth and wave field form. In particular,it is demonstrated that the mechanism of dispersion focusing of freak wave formation "works"for waves interacting with a vertical barrier. It is shown that just before the maximum waveformation a freak wave quickly changes its shape from a high ridge to a deep depression.Lifetime of freak wave increases with the growth of number of individual waves in anomalouswave packet, and lifetime of freak wave increases with water depth decreasing.It is demonstrated that pair interaction of unipolar solitons leads to decrease of the thirdand fourth moments of the wave field. It is shown that in the case of heteropolar solitoninteraction the fourth moment increases.The nonlinear dynamics of ensembles of random unipolar solitons in the framework of theKorteweg - de Vries equation and the modified Korteweg - de Vries equation is studied. It isshown that the coefficients of skewness and kurtosis of the soliton gas are reduced as a resultof soliton collision, the distribution function of wave amplitudes are defined. The behavior ofsoliton fields in the framework of these models is qualitatively similar. It is shown that in thesefields the amplitude of the big waves is decreased in average due to multi-soliton interactions.A new braking effect of soliton with a small amplitude and even changing of its direction inmulti-soliton gas as a result of nonlinear interaction with other solitons is found in the frameworkof the modified Korteweg-de Vries equation.It is shown that in heteropolar soliton gas abnormally big waves (freak waves) appear inthe frameworks of the modified Korteweg - de Vries equation. With increasing of soliton gasdensity the probability and intensity of freak waves in such systems increases.Les vagues et les ondes internes de gravité ont un impact important surl’hydrodynamique et l’hydrologie de la zone côtière. Les vagues extrêmes sontparticulièrement intéressantes à étudier, car elles sont une menace sérieuse pour letransport maritime, les plates-formes pétrolières, les installations portuaires et leszones touristiques de la côte. Ces ondes entravent aussi les activités humainesdéveloppées à la côte. Les ondes internes non linéaires affectent la biosphèreaquatique, notamment le transport de sédiments et créent des affouillements à labase des plates-formes et des pipelines. Elles affectent également la propagationdes signaux acoustiques. Les vagues scélérates provoquent d’importants dégâtsmatériels et de nombreuses pertes en vies humaines. Par conséquent, l’étude de laformation des ondes scélérates dans la zone côtière est d’une importance capitale.L'objectif principal de la thèse est l'étude de la formation d’ondes océaniquesanormales dans la zone côtières pour différentes profondeurs d’eau et différentschamps d'ondes. Il est montré que le mécanisme de focalisation dispersive àl’origine de la formation d’ondes scélérates est pertinent quand les ondesinteragissent avec une paroi verticale. Il est démontré que juste avant la formationde l’onde maximale, celle-ci change rapidement de forme, d'une haute crête vers uncreux profond. La durée de vie de l’onde scélérate augmente avec le nombred’ondes individuelles contenues dans le paquet d'ondes anormales et lorsque laprofondeur de l'eau diminue.Il est démontré que l'interaction de paires de solitons unipolaires conduit à unediminution des facteurs de dissymétrie et d’aplatissement du champ d'ondes. Il estprouvé que dans le cas d'interactions hétéropolaires de solitons, le facteurd’aplatissement augmente.La dynamique non linéaire de champs de solitons unipolaires aléatoires estétudiée dans le cadre de l’équation de Korteweg - de Vries (KdV) et de l’équationde Korteweg - de Vries modifiée (mKdV). Il est montré que les coefficients dedissymétrie et d'aplatissement du gaz de solitons sont réduits à la suite de collisionsde solitons. Les fonctions de distribution des amplitudes des ondes sont obtenues.Le comportement des champs solitoniques dans le cadre de ces modèles estqualitativement similaire. Il est démontré que l'amplitude des ondes extrêmesdiminue en moyenne en raison des interactions entre multi-solitons.Dans le cadre de l'équation de Korteweg-de Vries modifiée, les interactionsnon linéaires entre le soliton de plus petite amplitude et les autres solitons du gazont pour effet de réduire sa célérité qui devient négative et de modifier ainsi sadirection de propagation.A partir de l'équation de Korteweg-de Vries modifiée, il est prouvé que dans ungaz de solitons héteropolaires, des ondes scélérates peuvent se former. Laprobabilité d’occurrence et l’amplitude des ondes scélérates dans de tels systèmesaugmente avec la densité du gaz de solitons

Wave Dynamics in the Channels of Variable Cross-Section

Dynamics of long sea waves in the channels of variable depth and variable rectangular cross-section is discussed within various approximations – from the shallow water equations to those of nonlinear dispersion theory. General approach permitting to find traveling (non-reflective) waves in inhomogeneous channels is demonstrated within the framework of the shallow water linear theory. The appropriate conditions are determined by solving a system of ordinary differential equations. The so-called self-consistent channel in which the width is connected with its depth in a specific way is studied in detail. Within the linear theory of shallow water, a wave does not reflect from the bottom irregularities. The wave shape remains unchanged on the records of the wave gauges (mareographs) fixed along the channel axis, but it varies in space. Nonlinearity and dispersion lead to the wave transformation in such a channel. Within the framework of the shallow water weakly nonlinear theory, the wave shape is described by the Riemann solution, and the wave breaks (gradient catastrophe) quicker in the zones of decreasing depth. The modified Korteweg – de Vries equation describing evolution of a solitary wave of weak but finite amplitude in a self-consistent channel, the depth of which can vary arbitrary, is derived. Some examples of a solitary wave transformation in such a channel are analyzed (particularly, a soliton adiabatic transformation in the channel with the slowly varying parameters, and a solitary wave fission into the group of solitons after it has passed the zone where the depth changes abruptly. The obtained solutions extend the class of those represented earlier by S.F. Dotsenko and his colleagues