On a particular form of stabilized capillary-gravitational waves of finite amplitude at the surface of fluid over an undulating bed : PMM vol. 37, n6, 1973, pp. 1020–1034

The problem of stabilized plane capillary-gravitational waves of finite amplitude at the surface of a stream of perfect incompressible fluid flowing over an undulating bed and subjected to pressure periodically distributed along the surface and defined by some infinite trigonometric series is considered. The intersection of the bed with a vertical plane is assumed to be a periodic curve, called the bed line, defined by some infinite trigonometric series. The problem is rigorously formulated and reduced to the solution of a system of nonlinear integral and transcendental equations. The solution is constructed in the form of series in powers of a small dimensionless parameter to which amplitudes of the first harmonics of the bed line and of the surface pressure wave are proportional. An approximate equation is derived for the wave profile.The particular case is considered, when the length of the bed line wave arc is equal to the length of the stabilized free wave line corresponding to the specified flow velocity over a horizontal flat bed and constant pressure along the surface. In such case the parameter of the integral equation is equal to one of the eigenvalues of the kernel of that equation and the solution is constructed in the form of series in powers of the cube root of the small parameter mentioned above.A similar problem but for constant pressure along the surface was considered by the author in [1, 2] and in his paper presented at the 13-th International Congress on Theoretical and Applied Mechanics (Moscow, 1972 [3]).Another similar problem of capillary-gravitational waves over an undulating bed was considered in [4], where besides the topological proof of the existence and uniqueness of solution the algorithm for constructing the latter is given, but the calculation of approximations is only outlined and the mechanical meaning of solution is not investigated in depth.Unlike in [4] the equation of the bed line and the expression for pressure at the surface are specified here in a form which makes it possible to express any approximations in the form of finite sums, and an analysis of the fundamental system of nonlinear integral and transcendental equations by the LiapunovSchmidt analytical methods and their developments is presented.     

General exact solutions for linear and nonlinear waves in a Thirring model

In the present paper, we construct exact solutions to a system of partial differential equations iu_x + v + u | v | ² = 0, iv_t + u + v | u | ² = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of u or v with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Rotational waves generated by current‐topography interaction

We study nonlinear free‐surface rotational waves generated through the interaction of a vertically sheared current with a topography. Equivalently, the waves may be generated by a pressure distribution along the free surface. A forced Korteweg–de Vries equation (fKdV) is deduced incorporating these features. The weakly nonlinear, weakly dispersive reduced model is valid for small amplitude topographies. To study the effect of gradually increasing the topography amplitude, the free surface Euler equations are formulated in the presence of a variable depth and a sheared current of constant vorticity. Under constant vorticity, the harmonic velocity component is formulated in a simplified canonical domain, through the use of a conformal mapping which flattens both the free surface as well as the bottom topography. Critical, supercritical, and subcritical Froude number regimes are considered, while the bottom amplitude is gradually increased in both the irrotational and rotational wave regimes. Solutions to the fKdV model are compared to those from the Euler equations. We show that for rotational waves the critical Froude number is shifted away from 1. New stationary solutions are found and their stability tested numerically.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Nonlinear modulation of SH waves in an incompressible hyperelastic plate

Propagation of nonlinear shear horizontal (SH) waves in a homogeneous, isotropic and incompressible elastic plate of uniform thickness is considered. The constituent material of the plate is assumed to be generalized neo-Hookean. By employing a perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the nonlinear modulation of waves is governed asymptotically by a nonlinear Schr?dinger (NLS) equation. Then the effect of nonlinearity on the propagation characteristics of asymptotic waves is discussed on the basis of this equation. It is found that, irrespective of the plate thickness, the wave number and the mode number, when the plate material is softening in shear then the nonlinear plane periodic waves are unstable under infinitesimal perturbations and therefore the bright (envelope) solitary SH waves will exist and propagate in such a plate. But if the plate material is hardening in shear in this case nonlinear plane periodic waves are stable and only the dark solitary SH waves may exist.     

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Permanent Wave Structures on an Open Two Layer Fluid

There is a significant nonlinear interaction between fast free surface waves and slow interface waves when the group velocity of the free surface waves is the same as the phase velocity of the interface waves. This interaction leads to permanent wave structures consisting of a wave group of permanent envelope on the free surface and a wave of permanent shape on the interface. A theory is developed for periodic permanent wave structures of this type, from which solutions are found numerically. The theory includes all significant quadratic nonlinear interactions between free surface harmonics and interface harmonics, as well as between the interface harmonics themselves. It is found that there is a sequence of forms of differing free surface group structure.     

A simple variational approximation of shallow water type

The free-surface hydrodynamical problem governing progressive surface waves of permanent type is approximated under assumptions that correspond to hydrostatic pressure.A variational formulation of the full nonlinear problem is used followed by a Ritz-Kantorovich expansion for the relevant functional. By taking the leading term in the expansion a single ordinary differential equation (nonlinear) for the free surface profile is obtained. Solutions are possible in terms of cnoidal functions.Some explicit results for infinitesimal amplitude waves and solitary waves within the model are presented.     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.     

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.     

Remark on peakon solution to the nonlinear surface wind waves equation

We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

一个二流体系统中非线性水波的Hamilton描述

讨论了一个二流体系统中非线性水波的Hamilton描述,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面.文中将速度势函数展开成垂向坐标的幂级数,在浅水长波的假定下,取下层流体的“动厚度”与上层流体的“折合动厚度”为广义位移、界面上和自由面上的速度势为广义动量,根据Hamilton原理并运用Legendre变换导出该系统的Hamilton正则方程,从而将单层流体情形的结果推广到分层流体的情形.