Surface Of Finite-depth Fluid Research Articles

A Fifth-Order Nonlinear Schrödinger Equation for Waves on the Surface of Finite-Depth Fluid

We derive a high-order nonlinear Schr¨odinger equation with fifth-order nonlinearity for the envelope of waves on the surface of a finite-depth irrotational, inviscid, and incompressible fluid over the flat bottom. This equation includes the fourth-order dispersion, cubic-quintic nonlinearity, and cubic nonlinear dispersion effects. The coefficients of this equation are given as functions of one dimensionless parameter kℎ, where k is the carrier wave number, and ℎ is the undisturbed fluid depth. These coefficients stay bounded in the infinite-depth limit.

Bright and dark solitons on the surface of finite-depth fluid below the modulation instability threshold

Bright and dark solitons are usually considered from the perspective of modulation instability of the carrier wave train: bright solitons are observed when the carrier wave is unstable with respect to long-wave modulations, while dark solitons are observed when the carrier wave is modulationally stable. For the slowly modulated wave trains with narrow bandwidth on the surface of ideal finite-depth fluid, the corresponding modulation instability threshold is known to lie at kh≈1.363, k being the carrier wave number and h the undisturbed fluid depth. In this case, bright solitons are known to exist above the modulation instability threshold (kh≳1.363), while dark solitons are known to exist below this threshold (kh<1.363). We demonstrate for the first time that besides these classical results bright solitons can also be observed along with dark solitons in a certain water depth range around kh≈1.2 that lies below the modulation instability threshold. We provide initial conditions at which these new bright solitons can be generated in water wave tank experiments.

High-Order Nonlinear Schrödinger Equation for the Envelope of Slowly Modulated Gravity Waves on the Surface of Finite-Depth Fluid and Its Quasi-Soliton Solutions

We consider the high-order nonlinear Schr\"odinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter $kh$, where $k$ is the carrier wavenumber and $h$ is the undisturbed fluid depth. We show that one-soliton solutions of the classical nonlinear Schr\"{o}dinger equation are transformed into quasi-soliton solutions with slowly varying amplitude when the high-order terms are taken into consideration. These quasi-soliton solutions represent the secondary modulations of gravity waves.

The modulational instability of Stokes waves on the surface of finite-depth fluid

We study the modulational instability of Stokes waves without the traditional assumption as to the movement of a mean flow with the group velocity of the first harmonic. We have shown that the well-known limit k h = 1.363 , where h is the fluid depth and k is the wave number, for the transition between the states of modulationally stable and unstable fluids in the case of weakly nonlinear waves can be obtained without this assumption. This limit is shown to be shifted to greater kh, as the nonlinearity is strengthened. It is also shown that the disappearance of the modulational instability (restabilization) for small wave numbers of a perturbation, which was predicted on the basis of the Zakharov equations and numerical calculations of the exact equations, does not follow also from a weakly nonlinear theory, provided that the above assumption is not used.

Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Bridges Thomas J. 1996Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface wavesPhil. Trans. R. Soc. A.354533–574http://doi.org/10.1098/rsta.1996.0019SectionRestricted accessArticlePeriodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Published:15 March 1996https://doi.org/10.1098/rsta.1996.0019AbstractSpace and time periodic waves at the two-dimensional surface of an irrotational inviscid fluid of finite depth are considered. The governing equations are shown to have a new formulation as a generalized Hamiltonian system on a multisymplectic structure where there is a distinct symplectic operator corresponding to each unbounded space direction and time. The wave-generated mean flow in this framework has an interesting characterization as drift along a group orbit. The theory has interesting connections with, and generalizations of, the concepts of action, action flux, pseudofrequency and pseudowavenumber of the Whitham theory. The multisymplectic structure and novel characterization of mean flow lead to a new constrained variational principle for all space and time periodic patterns on the surface of a finite-depth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for three-dimensional travelling wavesFootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Hu W, Xiao C and Deng Z (2023) Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System Geometric Mechanics and Its Applications, 10.1007/978-981-19-7435-9_3, (89-201), . Clamond D and Dutykh D (2012) Practical use of variational principles for modeling water waves, Physica D: Nonlinear Phenomena, 10.1016/j.physd.2011.09.015, 241:1, (25-36), Online publication date: 1-Jan-2012. Bridges T and Donaldson N (2010) VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME‐DEPENDENT MOVING MESH, Mathematika, 10.1112/S0025579310001233, 57:1, (147-173), Online publication date: 1-Jan-2011. Sedletsky Y (2006) A new type of modulation instability of Stokes waves in the framework of an extended NSE system with mean flow, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/39/31/L03, 39:31, (L529-L537), Online publication date: 4-Aug-2006. Sedletsky Y (2005) The modulational instability of Stokes waves on the surface of finite-depth fluid, Physics Letters A, 10.1016/j.physleta.2005.04.076, 343:4, (293-299), Online publication date: 1-Aug-2005. Laine-Pearson F and Bridges T (2004) Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves, SIAM Journal on Applied Mathematics, 10.1137/S0036139903423753, 64:6, (2096-2120), Online publication date: 1-Jan-2004. Bridges T (2003) Benjamin Memorial Lecture: Stability of Solitary Waves: Geometry, Symplecticity and Three-Dimensionality Wind over Waves II, 10.1533/9780857099532.178, (178-188), . Bridges T (1999) A new framework for studying the stability of genus-1 and genus-2 KP patterns, European Journal of Mechanics - B/Fluids, 10.1016/S0997-7546(99)80044-2, 18:3, (493-500), Online publication date: 1-May-1999. Dias F and Kharif C (1999) NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES, Annual Review of Fluid Mechanics, 10.1146/annurev.fluid.31.1.301, 31:1, (301-346), Online publication date: 1-Jan-1999. Bridges T (1999) Toral-equivariant partial differential equations and quasiperiodic patterns, Nonlinearity, 10.1088/0951-7715/11/3/005, 11:3, (467-500), Online publication date: 1-May-1998. Knobloch E and Pierce R (2009) On mean flows associated with traveling water waves, Fluid Dynamics Research, 10.1016/S0169-5983(97)00030-0, 22:2, (61-71), Online publication date: 1-Feb-1998. Bridges T (1997) A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453:1962, (1365-1395), Online publication date: 8-Jul-1997. This Issue15 March 1996Volume 354Issue 1707 Article InformationDOI:https://doi.org/10.1098/rsta.1996.0019Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Manuscript received20/09/1994Manuscript accepted07/02/1995Published online01/01/1997Published in print15/03/1996 License:Scanned images copyright © 2017, Royal Society Citations and impact