Stationary Periodic Waves Research Articles

Observation of modulation instability and rogue breathers on stationary periodic waves

We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by input random noise, as well as the formation of rogue breather solutions induced by a coherent perturbation. Our results confirm the generalization of modulation instability when more complex background waves are involved.

Periodic waves and their stability in competing cubic-quintic nonlinearity

Exact expressions are obtained for stationary periodic waves in cubic-quintic nonlinear media with opposite signs of nonlinearity. It is shown that defocussing/focussing quintic nonlinearity can lead to stabilization of snoidal/(cnoidal and dnoidal) periodic waves depending on whether the Kerr medium is focusing/defocusing. Direct numerical simulations confirm results of the linear stability analysis. Different stable patterns e.g. bright solitons and kinks are also obtained under certain conditions.

Stable periodic waves supported by competing cubic-quintic nonlinearity

We perform a linear stability analysis of stationary periodic waves in cubic–quintic nonlinear media and show that weak χ(5) nonlinearity can lead to stabilization of cnoidal and destabilization of snoidal periodic wave patterns existing in focusing and defocusing χ(3) media, respectively. Direct computer simulations confirm results of the linear stability analysis. The stabilization of periodic waves is expected to be a common phenomenon in physical systems where focusing–defocusing, attractive–repulsive, nonlinear self-actions compete with each other.

Temporal evolutions and stationary waves for dissipative Benjamin–Ono equation

Initial value problems as well as stationary solitary and periodic waves are investigated for dissipative Benjamin–Ono (DBO) equation. Multi-hump stationary waves and their structures are identified numerically and the stability regions of stationary periodic waves are also examined numerically. These results elucidate a close relation between irregular behaviours in the initial value problem and the multiplicity of stationary waves.

Stationary travelling-wave solutions of an unstable KdV–Burgers equation

Both “solitary” and “periodic” stationary travelling wave solutions are investigated numerically for an unstable Korteweg–de Vries–Burgers equation u t+uu x+u xxx−η(u+u xx)=0 (η>0) . A family of stationary solitary wave solutions whose members are distinguished by the number of “humps” is found for a given η. Corresponding to each solitary wave thus found, a family of stationary periodic waves with the same number of “humps” exists under periodic condition and ends up in the infinite periodicity to the corresponding solitary wave. The numerical results are consistent with the theoretical estimates based on the conservation properties.

Exact Evolution Equations for Surface Waves

This paper considers surface gravity-capillary waves in an ideal fluid of finite depth and generalizes exact evolution equations for free gravity waves obtained by Dyachenko et al. to those for forced gravity-capillary waves. The model derived here describes the time evolution of the free surface and the velocity potential evaluated at the free surface under external pressure forcing. Two integro-differential equations are written explicitly in terms of these two dependent variables, and no extra step is required to close the system. These equations are solved numerically for the particular case of stationary periodic waves, and the results compared with analogous ones available in the literature.

Laminarizing effects of dispersion in an active-dissipative nonlinear medium

The Kuramoto-Sivashinsky equation has become a popular prototype for systems which exhibit spatial-temporal chaos. We show here that a linear dispersion term δ h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures, as is consistent with prior numerical and experimental observations. The study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation. It is shown that by δ > 1.1, the infinite families of stationary periodic waves of the Kuramoto-Sivashinsky equation, each ending in a solitary wave, have been annihilated successively such that only a lone family of periodic waves, consisting of one-hump KdV pulses for δ > 3.7, remains as the only periodic stationary wave attractors of the system. These periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial-temporal chaos in an extended domain with significant dispersion.

Method of averaging and the generalized variational principle for nonsinusoidal waves: PMM vol. 36, n≗1, 1972. pp. 71–78

A method of asymptotic expansion is proposed for wave processes which are close to stationary periodic waves of arbitrary form. It is shown that equations of the first approximation can be written in the form of Lagrange equations of the second kind for averaged Lagrange and Rayleigh functions. At the present time most analytical results in the theory of nonlinear wave processes are obtained with the aid of approximate methods based on the smallness of one or another parameter in the initial equations or in the boundary (initial) conditions. Wave processes in media with small nonlinearity and strong dispersion, where the solution is close to one or the superposition of several quasiharmonic waves [1–4] have been studied with relatively great completeness. Furthermore, it is well known that in many problems related to waves on the surface of a liquid, in the plasma [5], in transmission lines for electromagnetic waves [6], and also in problems of nonlinear field theory [7] the necessity arises to examine essentially nonsinusoidal waves with arbitrary relationship of nonlinearity and dispersion parameters. Here also some methods exist for obtaining approximate solutions based on local similarity of the process to a stationary traveling wave [5, 8, 9]. From the point of view of generality and physical clearness, apparently, the Hamilton's variational principle in the averaged form as proposed by Whitham [8] is of greatest interest among these methods. In this connection the equations for envelopes (of amplitude, frequency, etc.) of the quasistationary wave are obtained in the form of differential equations of Euler in the corresponding variational problem with averaged Lagrangian. Such an approach, however, is directly applicable only to strictly conservative systems for which the Lagrangian is known (the determination of the latter frequently represents quite a complicated problem [10]). Furthermore, the independent value has the structure of the scheme for asymptotic expansion which permits to obtain averaged equations in any approximation with respect to the small parameter, as it was already done for one nonlinear second order equation [11] and an arbitrary system of first order equations [12]. This paper examines processes which can be described by partial differential equations of the Lagrangian type, including in particular the Rayleigh dissipation function. The asymptotic method permits to examine processes which are locally similar to a plane stationary wave. We succeed in showing that the equations of the first approximation can be derived from the generalized Hamilton's variational principle in the averaged form.