Cnoidal Wave Solutions Research Articles

The dark soliton on a cnoidal wave background

The method of Darboux transformation, which is applied on cnoidal wave solutions of the sine-Gordon equation, gives solitons moving on a cnoidal wave background. Interesting characteristics of the solution, i.e., the velocity of solitons and the shift of crests of cnoidal waves along a soliton, are calculated. Solutions are classified into three types (Type-1A, Type-1B, Type-2) according to their apparent distinct properties.

Coupled periodic waves with opposite dispersions in a nonlinear optical fiber

Using the Hirota’s method and elliptic θ-functions, we obtain three families of exact periodic (cnoidal) wave solutions for two nonlinear Schrödinger (NLS) equations coupled by XPM (cross-phase-modulation) terms, with a ratio σ of the XPM and SPM (self-phase-modulation) coefficients. Unlike the previous works, we obtain the solutions for the case when the coefficients of the group-velocity-dispersion (GVD) in the coupled equations have opposite signs. In the limit of the infinite period, the solutions with σ>1 carry over into inverted bound states of bright and dark solitons in the normal- and anomalous-GVD modes (known as “symbiotic solitons”), while the infinite-period solution with σ<1 is an uninverted bound state (also an unstable one). The case of σ=2 is of direct interest to fiber-optic telecommunications, as it corresponds to a scheme with a pulse stream in an anomalous-GVD payload channel stabilized by a concomitant strong periodic signal in a mate normal-GVD channel. The case of arbitrary σ may be implemented in a dual-core waveguide. To understand the stability of the coupled waves, we first analytically explore the modulational stability of CW (constant-amplitude) solutions, concluding that they may be completely stable for σ⩾1, provided that the absolute value of the GVD coefficient is smaller in the anomalous-GVD mode than in the normal-GVD one, and certain auxiliary conditions on the amplitudes are met. The stability of the exact cnoidal-wave solutions is tested in direct simulations. We infer that, while, strictly speaking, in the practically significant case of σ=2 all the solutions are unstable, in many cases the instability may be strongly attenuated, rendering the above-mentioned paired channels scheme usable. In particular, the instability is milder for a smaller period of the wave pattern, and/or if the anomalous GVD is weaker than the normal GVD in the mate channel. When the instability sets in, it first initiates quasi-reversible modulations, and only at a later stage the wave pattern decays into a “turbulent” state.

Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations

For general classes of nonlinear, dispersive wave equations, we discuss the existence of the analogue of the cnoidal-wave solutions of the Korteweg–de Vries equation. The solutions in question have travelling and periodic features. The topological-degree theory of positive operators on Banach spaces is applied.

Cnoidal-type surface waves in deep water

Two-dimensional potential flows due to progressive surface waves in deep water are considered. For periodic waves, only gravity is included in the dynamic boundary condition, but both gravity and surface tension are taken into account for solitary waves. The validity of the steady first-order cnoidal wave approximation, i.e. the periodic solution of KdV, is extended to infinite depth by renormalizations. This renormalized cnoidal wave (RCW) solution is expressed as a Fourier–Padé approximation. It is analytically simpler and more accurate than fifth-order Stokes approximations. It is also capable of describing the recently discovered sharp-crested wave. A sharp-crested wave is obtained when the fluid velocity at the crest is larger than the phase speed. When the wavelength is infinite, RCW yields an algebraic solitary wave. Depending on the surface tension, the solitary wave involves one or two interfaces: a wave of depression; a wave of depression with a pocket of air; a wave of elevation with a pocket of air. Solitary waves are found for all values of the surface tension. However, this does not necessarily mean that these waves are solutions of the exact equations. Moreover, RCW approximate solitary waves always present a dipole singularity. It is also shown that a cnoidal wave in deep water can be rewritten as a periodic distribution of dipoles, each dipole representing an algebraic solitary wave. This provides a new paradigm for descriptions of water wave phenomena.

Dynamics of the local entanglement on two vortex filaments described by the Korteweg–de Vries equation

An entanglement equation is found to reduce a Korteweg–de Vries (KdV) equation. An entanglement equation is the localized induction approximation of the Biot–Savart equation describing the dynamics of a co-rotating vortex pair. A soliton solution together with a cnoidal wave solution of the KdV equation shows that vortex filaments make a local entanglement when the distance between a pair of vortex filaments locally decreases.

SOLITON SOLUTIONS IN A MODIFIED ϕ4 MODEL

We consider a ϕ4 model in 1+1 dimensions modified by the addition of an extra kinetic term similar to the Skyrme term in higher dimensions and a potential term. We generalize the procedures of the leading order method to include elliptic functions and obtain cnoidal wave type solutions of the familiar ϕ4 model. As a limit case, we re-obtain the well-known solitary and travelling wave solutions of this model from the constructed cnoidal wave solutions. A further kink solution of the pure ϕ4 model is also obtained. We obtain soliton solutions of the modified ϕ4 model and show that it possesses two ϕ4 kink solutions, with the same velocity and total energy, as well as a double-kink solution. We study some properties of the modified model. In particular, we see that the two ϕ4 kinks have a definite velocity and this velocity is a critical velocity for the double-kink structures. Finally, we conclude the paper with some features and comments.

Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations

We construct Bäcklund transformations (BT) and several classes of exact cnoidal wave and travelling wave solutions for both the Sawada–Kotera (SK) and Kaup–Kupershmidt (KK) equations. AKNS-like system has been constructed for a unified equation of the SK and KK equations. The BT are derived from the Riccati form of inverse method. We generalise the leading order analysis method to include elliptic functions and obtain cnoidal wave solutions for both the SK and KK equations. As reduced cases, we obtain several classes of travelling wave solutions for these equations. Conclusions and some features and comments are given.

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found.

On cauchy problems for the RLW equation in two space dimensions

A cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and the some estimations the uniqueness and the stability of the periodic solution with both x, y to the Cauchy problem are proved by the priori estimations.

Abundant Similarity Reductions and Dromion-Like Solutions to the Generalized Kroteweg-de Vries Equation in the (2+1)-Dimensional Space**The project supported by National Natural Science Foundation of China under Grant No. 10072013, the National Key Basic Research Development Project Program of China under Grant No. G1998030600 and Doctoral Foundation of China under Grant No. 98014119

In this paper,eight types of (1+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators by using the direct method.In addition,the cnoidal wave solution and dromion-like solution are also derived by using the reduced nonlinear ordinary differential equations.The (1+1) dromion obtained by Lou [J.Phys.A28 (1995) 7227] and Zhang [Chin.Phys.9 (2000) 1] is only a special case of our results.Moreover,some properties of the dromion-like solutions are analyzed.

Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations

We consider quasiperiodic and periodic (cnoidal) wave solutions of a set of n-component vector nonlinear Schrödinger equations (VNLSEs). In a biased photorefractive crystal with a drift mechanism of nonlinear response and Kerr-type nonlinearity, n-component nonlinear Schrödinger equations can be used to model self-trapped mutually incoherent wave packets. These equations also model pulse–pulse interactions in wavelength-division-multiplexed channels of optical fiber transmission systems. Quasiperiodic wave solutions for the VNLSEs in terms of n-dimensional Kleinian functions are presented. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for n-component nonlinear Schrödinger equations are found.

Cnoidal Waves and Solitons in the Quantum Lattice Gas Model in a Ring and Effects of the Aharonov-Bohm Flux on Them

Cnoidal wave solutions in the quantum lattice gas model in the presence of the Aharonov-Bohm flux in a ring are found. The energy gap, the effective mass and the binding energy of cnoidal waves are given. It is pointed out that the spin-wave-like excitation and the soliton are special cases of the cnoidal wave. Effects of the Aharonov-Bohm flux on the soliton are considered. It is shown that the width, the peak, the binding energy and the effective mass of the soliton and the energy gap are affected and a Josephson-like current is induced by the Aharonov-Bohm flux.

Three‐dimensional Korteweg‐de Vries equation and traveling wave solutions

The three-dimensional power Korteweg-de Vries equation[ut+unux+uxxx]x+uyy+uzz=0, is considered. Solitary wave solutions for any positive integernand cnoidal wave solutions forn=1andn=2are obtained. The cnoidal wave solutions are shown to be represented as infinite sums of solitons by using Fourier series expansions and Poisson's summation formula.

The Stability and Existence Condition of Nonlinear Travelling Wave Solution in Basic Uniform Flow

Nonlinear waves in a stratified atmosphere with basic uniform flow are studied by adopting geostrophic momentum approximation (cf. Hoskins, 1975) and the method of travelling wave solution. A nonlinear equation containing a unique variable of vertical p velocity is derived from a complete system of equations considering frictionless and adiabatic fluid. The stability of the solution for the nonlinear equation is discussed then. Furthermore, the approximate cnoidal wave solution, solitary wave solution and their existence condition are obtained.

Nearly conconcentric Korteweg‐de Vries equation and periodic traveling wave solution

The generalized nearly concentric Korteweg‐de Vries equation is considered. The author converts the equation into the power Kadomtsev‐Petviashvili equation . Solitary wave solutions and cnoidal wave solutions are obtained. The cnoidal wave solutions are shown to be representable as infinite sums of solitons by using Fourier series expansions and Poisson′s summation formula.

Higher Order Solution of Nonlinear Waves. I. Cnoidal Wave in Unstable and Dissipative Media

An equilibrium solution is obtained analytically for a nonlinear and dispersive wave with weakly unstable and dissipative conditions. Based on a cnoidal wave solution to the Korteweg-de Vries (K-dV, for short) equation, the second order solution is obtained by taking into account the unstable and dissipative effects. These effects impose some restrictions on the cnoidal wave even when the effects are weak, that is, the relation between the wave-length and amplitude of the cnoidal wave appears which is not the case for the K-dV equation. The explicit second order solution brings about asymmetry in the wave form.

The growth of periodic waves in gas-fluidized beds

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham &amp; Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham &amp; Merkin (1986) and Anderson, Sundaresan &amp; Jackson (1995) is then carried out.

Generalized boussinesq model for periodic non-linear shallow-water waves

This paper presents the development of a generalized Boussinesq (gB) model for the periodic non-linear shallow-water waves. An incident cnoidal wave solution for the gB model is derived and applied to the wave simulation. A set of radiation boundary conditions is also established to transmit effectively the cnoidal waves out of the computational domain. The classical solutions of the second-order cnoidal waves are discussed within the content of the KdV equation and the generalized Boussinesq equations. An Euler's predictor-corrector finite-difference algorithm is used for numerical computation. The propagation of normally incident cnoidal waves in a channel is studied. The simulated wave profiles agree well with the analytical results. The temporal and spatial evolution of an obliquely incident cnoidal wave is also modelled. The phenomenon of Mach reflection is discussed.

Hydrodynamic interactions of cnoidal waves with a vertical cylinder

This paper describes a study of the diffraction of periodic nonlinear shallow water waves around a vertical cylinder. The generalized Boussinesq (GB) numerical model is used to effectively simulate the propagation of cnoidal waves and the associated three-dimensional nonlinear interactions. The numerical computation is based on an Euler predictor-corrector algorithm in a curvilinear coordinate system. An approximate second-order cnoidal wave solution for the GB equations is derived and used to provide the initial and incident wave conditions. The temporal and spatial evolution of an incident cnoidal wave after interacting with a cylinder is presented. The maximum force acting on the cylinder is also computed. The GB model simulation predicts strong nonlinear and three-dimensional effects on the forces which cannot be predicted by linear theories.

Obliquely propagating ion-acoustic nonlinear periodic waves in a magnetized plasma with two electron species

Obliquely propagating ion-acoustic nonlinear periodic waves in a magnetized plasma consisting of warm adiabatic ions and two Maxwellian electron species are studied. Using the reductive perturbation method, the Korteweg–de Vries (KdV) equation is derived and its cnoidal wave solution is discussed. It is found that as the amplitude of the cnoidal wave increases, so does its frequency. The effects of variations in the density and temperature ratios of the two electron species, the ion temperature, the angle of obliqueness and the magnetization on the characteristics of the cnoidal wave are discussed in detail. When the coefficient of the nonlinear term of the KdV equation, a1, vanishes, the modified Korteweg–de Vries equation is derived, and its periodic-wave solutions are discussed in detail. In the limiting case these periodic-wave solutions reduce to soliton or double-layer solutions.