Cnoidal Wave Solutions Research Articles

The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations

The Serre equations are a pair of strongly nonlinear, weakly dispersive, Boussinesq-type partial differential equations. They model the evolution of the surface elevation and the depth-averaged horizontal velocity of an inviscid, irrotational, incompressible, shallow fluid. They admit a three-parameter family of cnoidal wave solutions with improved kinematics when compared to KdV theory. We examine their linear stability and establish that waves with sufficiently small amplitude/steepness are stable while waves with sufficiently large amplitude/steepness are unstable.

A class of coupled nonlinear Schrödinger equations: Painlevé property, exact solutions, and application to atmospheric gravity waves

The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.

Ion-acoustic nonlinear periodic waves in electron-positron-ion plasma

Ion-acoustic nonlinear periodic waves, namely, ion-acoustic cnoidal waves have been studied in electron-positron-ion plasma. Using reductive perturbation method and appropriate boundary condition for nonlinear periodic waves, the Korteweg–de Vries (KdV) equation is derived for the system. The cnoidal wave solution of the KdV equation is discussed in detail. It is found that the frequency of the cnoidal wave is a function of its amplitude. It is also found that the positron concentration modifies the properties of the ion-acoustic cnoidal waves. The existence regions for ion-acoustic cnoidal wave in the parameters space (p,σ), where p and σ are the positron concentration and temperature ratio of electron to positron, are discussed in detail. In the limiting case these ion-acoustic cnoidal waves reduce to the ion-acoustic soliton solutions. The effect of other parameters on the characteristics of the nonlinear periodic waves is also discussed.

The orbital stability of the cnoidal waves of the Korteweg–de Vries equation

The cnoidal wave solution of the integrable Korteweg–de Vries equation is the most basic of its periodic solutions. Following earlier work where the linear stability of these solutions was established, we prove in this Letter that cnoidal waves are (nonlinearly) orbitally stable with respect to so-called subharmonic perturbations: perturbations that are periodic with period any integer multiple of the cnoidal-wave period. Our method of proof combines the construction of an appropriate Lyapunov function with the seminal results of Grillakis, Shatah and Strauss (1987, 1990) [17,18]. The integrability of the Korteweg–de Vries equation is used in that we need the presence of at least one extra conserved quantity in addition to those expected from the Lie point symmetries of the equation.

Self-similar cnoidal and solitary wave solutions of the (1+1)-dimensional generalized nonlinear Schrödinger equation

With the help of the similarity transformation and the solvable stationary nonlinear Schrodinger equation (NLSE), we obtain exact chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized NLSE exhibiting spatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. As an example, we investigate their propagation dynamics in a nonlinear optical system, and present a series of interesting properties of optical waves.

Lie Point Symmetries and Exact Solutions of the Coupled Volterra System

The coupled Volterra system, an integrable discrete form of a coupled Korteweg–de Vries (KdV) system applied widely in fluids, Bose–Einstein condensation and atmospheric dynamics, is studied with the help of the Lie point symmetries. Two types of delayed differential reduction systems are derived from the coupled Volterra system by means of the symmetry reduction approach and symbolic computation. Cnoidal wave and solitary wave solutions for a delayed differential reduction system and the coupled Volterra system are proposed, respectively.

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

We study the spectral (in)stability of one-dimensional solitary and cnoidal waves of various Boussinesq systems. These systems model three-dimensional water waves (i.e., the surface is two-dimensional) with or without surface tension. We present the results of numerous computations examining the spectra related to the linear stability problem for both stationary solitary and cnoidal waves with various amplitudes, as well as multipulse solutions. The one-dimensional nature of the wave forms allows us to separate the dependence of the perturbations on the spatial variables by transverse wave number. The compilation of these results gives a full view of the two-dimensional stability problem of these one-dimensional solutions. We demonstrate that line solitary waves with elevated profiles are spectrally stable with respect to one-dimensional perturbations and long transverse perturbations. We show that depression solitary waves are spectrally stable with respect to one-dimensional perturbations, but unstable with respect to transverse perturbations. We also discuss the instability of multipulse solitary waves and cnoidal-wave solutions of the Boussinesq system.

Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients

By means of the similarity transformation connecting with the solvable stationary cubic-quintic nonlinear Schrödinger equation (CQNLSE), we construct explicit chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized CQNLSE with spatially inhomogeneous group velocity dispersion (GVD) and amplification or attenuation. As an example, we investigate their propagation dynamics in a soliton control system.

Obliquely propagating cnoidal waves in a magnetized dusty plasma with variable dust charge

We have studied obliquely propagating dust-acoustic nonlinear periodic waves, namely, dust-acoustic cnoidal waves, in a magnetized dusty plasma consisting of electrons, ions, and dust grains with variable dust charge. Using reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, we have derived Korteweg–de Vries (KdV) equation for the plasma. It is found that the contribution to the dispersion due to the deviation from plasma approximation is dominant for small angles of obliqueness, while for large angles of obliqueness, the dispersion due to magnetic force becomes important. The cnoidal wave solution of the KdV equation is obtained. It is found that the frequency of the cnoidal wave depends on its amplitude. The effects of the magnetic field, the angle of obliqueness, the density of electrons, the dust-charge variation and the ion-temperature on the characteristics of the dust-acoustic cnoidal wave are also discussed. It is found that in the limiting case the cnoidal wave solution reduces to dust-acoustic soliton solution.

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2 T , the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.

Bright solitons on a cnoidal wave background for the inhomogeneous nonlinear Schrödinger equation

In this paper, we investigate the bright solitons on a cnoidal wave background train of the inhomogeneous nonlinear Schrödinger equation, which may be applicable to many physically realizable systems such as Bose–Einstein condensation media and plasma, etc. We use well-known methods to reduce the inhomogeneous nonlinear Schrödinger equation to a standard nonlinear Schrödinger equation by using the combination of Husimi's and Lens-type transformations. We study the superposed configuration of soliton with a cnoidal wave solution of the underlying equation. Finally, we discuss the dynamics of soliton on a cnoidal wave background in Bose–Einstein condensation trapped in linear density and harmonic density profiles separately.

Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves

AbstractWe determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero.

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor–corrector finite–difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.

Cnoidal wave solutions to Boussinesq systems

In this paper, two different techniques will be employed to study the cnoidal wave solutions of the Boussinesq systems. First, the existence of periodic travelling-wave solutions for a large family of systems is established by using a topological method. Although this result guarantees the existence of cnoidal wave solutions in a parameter region in the period and phase speed plane, it does not provide the uniqueness nor the non-existence of such solutions in other parameter regions. The explicit solutions are then found by using the Jacobi elliptic function series. Some of these explicit solutions fall in the parameter region where the cnoidal wave solutions are proved to exist, and others do not; so the method with Jacobi elliptic functions provides additional cnoidal wave solutions. In addition, the explicit solutions can be used in many ways, such as in testing numerical code and in testing the stability of these waves.

New Cnoidal and Solitary Wave Solutions of CoupledHigher-Order Nonlinear Schrödinger System in Nonlinear Optics

The coupled higher-order nonlinearSchrödinger system is a major subject in nonlinearoptics as one of the nonlinear partial differential equation whichdescribes the propagation of optical pulses in optic fibers. Byusing coupled amplitude-phase formulation, a series of new exactcnoidal and solitary wave solutions with different parameters areobtained, which may have potential application in optical communication.

Solitons in a one-dimensional ferromagnetic chain under the influence of single-ion anisotropy

Cnoidal wave solutions and soliton solutions in a one-dimensional ferromagnetic chain with anisotropy exchange interaction and single-ion anisotropy are obtained through travelling wave solution method and the influence of single-ion anisotropy on cnoidal waves and solitons is discussed. The results indicate that single-ion anisotropy makes the cnoidal waves and solitons easy to excite in an easy-plane ferromagnetic chain but difficult in an easy-axis one, and makes them more stable in the easy-axis ferromagnetic chain but less stable in the easy-plane one. The influence of single-ion anisotropy on soliton excitations in a one-dimensional isotropic ferromagnet is also discussed.

Parametrically controlling solitary wave dynamics in the modified Korteweg-de Vries equation

We demonstrate the control of solitary wave dynamics of modified Kortweg-de Vries (MKdV) equation through the temporal variations of the distributed coefficients. This is explicated through exact cnoidal wave and localized soliton solutions of the MKdV equation with variable coefficients. The solitons can be accelerated and their propagation can be manipulated by suitable variations of the above parameters. In sharp contrast with nonlinear Schr\"{o}dinger equation, the soliton amplitude and widths are time independent.

Optical Solitary Waves in Fourth-Order Dispersive Nonlinear SchrödingerEquation with Self-steepening and Self-frequency Shift

By making use of the generalized sine-Gordon equation expansionmethod, we find cnoidal periodic wave solutions and fundamentalbright and dark optical solitary wave solutions for thefourth-order dispersive and the quintic nonlinear Schrödingerequation with self-steepening, and self-frequency shift. Moreover,we discuss the formation conditions of the bright and dark solitary waves.

Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear optics

By using coupled amplitude-phase formulation, we construct the quartic anharmonic oscillator equation from the coupled higher order nonlinear Schrödinger equation. For arbitrary physical parameter values, we discuss the construction of new cnoidal wave solutions and the exact solutions of both bright (GVD < 0) and dark (GVD < 0) solitary waves. In addition, we investigate the pairs of dark–dark and bright–bright solitary waves.

Cnoidal wave solutions for a class of fifth-order KdV equations

A suitable ansatz and Jacobi elliptic function expansion method are used to construct new exact cnoidal wave solutions of the modified fifth-order Korteweg-de Varies (KdV) equation and the generalized fifth-order KdV equation which includes, as special cases, some well-known equations. When the modulus of the Jacobi elliptic function m → 1 , the corresponding solitary wave solutions are also obtained.