arXiv subjects

M. A. Manna

Publications and source records attributed to M. A. Manna.

15 recordsLinked to original sources

arXiv2024

Wind-wave interaction in finite depth: linear and nonlinear approaches, blow-up and soliton breaking in finite time, integrability perspectives

This work is an analytical investigation of the evolution of surface water waves in Miles and Jeffreys theories of wind wave interaction in water of finite depth. The present review is divided into two major parts. The first corresponds to the surface water waves in a linear regime and its nonlinear extensions. In this part, Miles theory of wave amplification by wind is extended to the case of finite depth. The dispersion relation provides a wave growth rate depending on depth. Our theoretical results are in good agreement with the data from the Australian Shallow Water Experiment and the data from the Lake George experiment. In the second part of this study, Jeffreys theory of wave amplification by wind is extended to the case of finite depth, where the Serre-Green-Naghdi is derived. We find the solitary wave solution of the system, with an increasing amplitude under the action of the wind. This continuous increase in amplitude leads to the soliton breaking and blow-up of the surface wave in finite time. The theoretical blow-up time is calculated based on actual experimental data. By applying an appropriate perturbation method, the SGN equation yields Korteweg de Vries Burger equation (KdVB). We show that the continuous transfer of energy from wind to water results in the growth of the KdVB soliton amplitude, velocity, acceleration, and energy over time while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys approach due to finite depth. Again, blow-up and breaking occur in finite time. These times are calculated and expressed for soliton- and wind-appropriate parameters and values. These values are measurable in usual experimental facilities. The kinematics of the breaking is studied, and a detailed analysis of the breaking time is conducted using various criteria. Finally, some integrability perspectives are presented.

nlin.PS↗

arXiv2012

The growth of wind-waves in finite depth

In order to study the growth of wind waves in finite depth we extend Miles' theory to the finite depth domain. A depth-dependent wave growth rate is derived from the dispersion relation of the wind/water interface. A suitable dimensionless finite depth wave age parameter allows us to plot a family of wave growth curves, each family member characterized by the water depth. Two major results are that for small wave age, the wave growth rates are comparable to those of deep water and for large wave age, a finite-depth wave-age-limited growth is reached, with wave growth rates going to zero. The corresponding limiting wave length and limiting phase speed are explicitely calculated in the shallow and in the deep water cases. A qualitative agreement with well-known empirical results is established and shows the robust consistency of the linear theoretical approach.

physics.flu-dyn↗

arXiv2011

An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative methods, an asymptotic model for small-aspect-ratio waves is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical irrotational results is performed.

nlin.SI↗

arXiv2005

Theory of small aspect ratio waves in deep water

In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water) without potential flow assumption. The resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes, on the other side possess a threshold amplitude where wave crest peaking is achieved. An explicit expression of the crest angle at wave breaking is found in terms of the wave velocity. By numerical simulations, stable soliton-like solutions (experiencing elastic interactions) propagate in a given velocities range on the edge of which they tend to the peakon solution.

nlin.PS↗

arXiv2003

A singular integrable equation from short capillary-gravity waves

From a columnar approximation of the Euler equations of an incompressible fluid with surface tension, we derive in the short-wave approximation a new integrable classical 1+1 dimensional field theory for the motion of the surface. Together with a Lorentz invariance,this system has the novel feature of solutions which become multiple valued in finite time.

physics.flu-dyn↗

arXiv2000

Asymptotic Dynamics of Ripples

A new nonlinear equation governing asymptotic dynamics of ripples is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system. It admits peakon solutions with amplitude, velocity and width in interrelation and static compacton solutions with amplitude and width in interrelation. Short wave pattern formation is shown to result from a balance between linear dispersion and nonlinearity.

nlin.PS↗

arXiv1997

Asymptotic dynamics of short-waves in nonlinear dispersive models

The multiple-scale perturbation theory, well known for long-waves, is extended to the study of the far-field behaviour of short-waves, commonly called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation can propagates short-waves. This result contradict the Benjamin hypothesis that short-waves tends not to propagate in this model and close a part of the old controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon equations are understood as an all orders asymptotic behaviour of short-waves. It is proved that the antikink solution of phi-4 model which was never obtained perturbatively can be obtained by perturbation expansion in the wave-number k in the short-wave limit.

solv-int↗

arXiv1997

The Role of the Korteweg-de Vries Hierarchy in the N-Soliton Dynamics of the Shallow Water Wave Equation

We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the shallow water wave model equation. As a consequence of the requirement of a secularity-free perturbation theory, we show that the well known N-soliton dynamics of the shallow water wave equation, in the particular case of $α=2 β$, can be reduced to the N-soliton solution that satisfies simultaneously all equations of the Korteweg-de Vries hierarchy.

patt-sol↗

arXiv1997

Modified Korteweg-de Vries Hierachies in Multiple-Times Variables and the Solutions of Modified Boussinesq Equations

We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each order of the perturbative scheme. We show that the multiple-time variables needed to obtain a regular perturbative series are completely determined by the associated linear theory in the case of a solitary-wave solution, but requires the knowledge of each order of the perturbative series in the case of a kink-wave solution. These appropriate multiple-time variables allow us to show that the solitary-wave as well as the kink-wave solutions of the modified Botussinesq equation are actually respectively a solitary-wave and a kink-wave satisfying all the equations of suitable modified Korteweg-de Vries hierarchies.

solv-int↗

arXiv1996

Multiple-Time Higher-Order Perturbation Analysis of the Regularized Long-Wavelength Equation

By considering the long-wave limit of the regularized long wave (RLW) equation, we study its multiple-time higher-order evolution equations. As a first result, the equations of the Korteweg-de Vries hierarchy are shown to play a crucial role in providing a secularity-free perturbation theory in the specific case of a solitary-wave solution. Then, as a consequence, we show that the related perturbative series can be summed and gives exactly the solitary-wave solution of the RLW equation. Finally, some comments and considerations are made on the N-soliton solution, as well as on the limitations of applicability of the multiple scale method in obtaining uniform perturbative series.

patt-sol↗

arXiv1995

Reductive Perturbation Method, Multiple-Time Solutions and the KdV Hierarchy

We apply a multiple-time version of the reductive perturbation method to study long waves as governed by the Boussinesq model equation. By requiring the absence of secular producing terms in each order of the perturbative scheme, we show that the solitary-wave of the Boussinesq equation can be written as a solitary-wave satisfying simultaneously all equations of the KdV hierarchy, each one in a different slow time variable. We also show that the conditions for eliminating the secularities are such that they make the perturbation theory compatible with the linear theory coming from the Boussinesq equation.

patt-sol↗

arXiv1995

Boussinesq Solitary-Wave as a Multiple-Time Solution of the Korteweg-de Vries Hierarchy

We study the Boussinesq equation from the point of view of a multiple-time reductive perturbation method. As a consequence of the elimination of the secular producing terms through the use of the Korteweg--de Vries hierarchy, we show that the solitary--wave of the Boussinesq equation is a solitary--wave satisfying simultaneously all equations of the Korteweg--de Vries hierarchy, each one in an appropriate slow time variable.

patt-sol↗

arXiv1994

The Korteweg-de Vries Hierarchy and Long Water-Waves

By using the multiple scale method with the simultaneous introduction of multiple times, we study the propagation of long surface-waves in a shallow inviscid fluid. As a consequence of the requirements of scale invariance and absence of secular terms in each order of the perturbative expansion, we show that the Korteweg-de Vries hierarchy equations do appear in the description of such waves. Finally, we show that this procedure of eliminating secularities is closely related to the renormalization technique introduced by Kodama and Taniuti.

patt-sol↗

arXiv1994

Hydrothermal Surface-Wave Instability and the Kuramoto-Sivashinsky Equation

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrothermal surface-wave instabilities may give rise to disturbances governed by the Kuramoto-Sivashinsky equation. A possible connection to hot-wire experiments is also discussed.

patt-sol↗

arXiv1993

Dissipative Boussinesq System of Equations in the Bénard-Marangoni Phenomenon

By using the long-wave approximation, a system of coupled evolution equations for the bulk velocity and the surface perturbations of a Bénard-Marangoni system is obtained. It includes nonlinearity, dispersion and dissipation, and it can be interpreted as a dissipative generalization of the usual Boussinesq system of equations. As a particular case, a strictly dissipative version of the Boussinesq system is obtained. Finnaly, some speculations are made on the nature of the physical phenomena described by this system of equations.

patt-sol↗