Cnoidal Wave Solutions Research Articles

Exact Jacobi elliptic solutions of some models for the interaction of long and short waves

<abstract><p>Some systems were recently put forth by Nguyen et al. as models for studying the interaction of long and short waves in dispersive media. These systems were shown to possess synchronized Jacobi elliptic solutions as well as synchronized solitary wave solutions under certain constraints, i.e., vector solutions, where the two components are proportional to one another. In this paper, the exact periodic traveling wave solutions to these systems in general were found to be given by Jacobi elliptic functions. Moreover, these cnoidal wave solutions are unique. Thus, the explicit synchronized solutions under some conditions obtained by Nguyen et al. are also indeed unique.</p></abstract>

Ion-acoustic cnoidal waves in a non-Maxwellian plasma with regularized κ-distributed electrons

This study examines how regularized kappa distributed (RKD) electrons affect the characteristics of the ion-acoustic cnoidal waves (IACWs) in normal plasma. The RKD is a generalized form of the standard kappa distribution that accounts for the impacts of thermal broadening and finite size effects on the plasma particles. By employing the reductive perturbation technique, the Korteweg–de Vries equation is derived, and its nonlinear cnoidal wave (CW) solution is obtained and analyzed both analytically and numerically. It is found that the amplitude of compressive IACWs increases with an enhanced cutoff parameter α and decreases with increasing superthermality κ. Rarefactive IACWs yield opposing results as compared to compressive IACWs under the impact of κ and α. The results provide insight into the behavior of CWs in normal plasmas with non-Maxwellian distributions and contribute to the understanding of wave-particle interactions in laboratory and space plasmas.

APPLICABLE RANGE OF PERIODICAL WAVE THEORIES UPDATING LE MEHAUTE'S CHART

Le Mehaute (1976)’s chart has been widely used in coastal engineering community because of its simplicity. However, there is a need to update this chart based on newly developed Stokes wave theory. This paper adopts Fenton (1999)’s 3rd order and 5th order cnoidal wave solution for updating the chart. Fenton’s 3rd order solution is applicable in the range between the two lines represented by m=0.5 and m=0.96. For higher m values (region above the red dot line), Fenton’s 5th order solution is applicable. Alternatively, Clamond (1999)’s solution can be considered.

Vector combined cnoidal wave and soliton solutions for a 3D partially nonlocal CNLSE

A 3D distributed-coefficient coupled nonlinear Schrödinger equation(CNLSE) with the partially nonlocal nonlinearity(PNN) for locality in two transverse directions and non-locality in the longitudinal direction becomes the center of attention in this paper. A one-to-one corresponding expression from the distributed-coefficient CNLSE with the PNN to the constant-coefficient CNLSE is confirmed. By way of the Darboux transformation, combined vector solutions of cnoidal wave and soliton with different structures for double components are found. Evolutional behaviors of dark-bright soliton pair, dark-bright soliton pair traveling parallelly along the top of cnoidal wave, and soliton break-up are revealed when parameters of solutions are selected as different values.

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N-solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

A new reductive perturbation formalism for ion acoustic cnoidal waves

A new formalism for the derivation of the cnoidal wave solution is presented by introducing a new set of initial conditions and a subacoustic moving frame. The resulting set of solutions is also constructed in a way that ensures the conservation of number density. The solutions are illustrated in a variety of graphical representations, and the effect of the amplitude's magnitude on the cnoidal wave form is presented. Interestingly, it is shown that the wavelength of the solutions decreases with amplitude. In addition, it is shown that the small-amplitude cnoidal wave solutions converge to linear waves in the small-amplitude limit.

Nonlinear index of a two-level model: The essential role of the population inversion

Optical wave propagation in an assembly of two-level atoms is considered. An asymptotic model of nonlinear Schr\"odinger (NLS) type is derived in a rigorous way by means of the perturbative expansion method. The results of the obtained model are then compared with both a numerical solution of the initial set of equations of the two-level model and an exact analytical cnoidal wave solution. It is seen that the evolution of the population difference is a key factor for the accurate determination of the nonlinear index, and that the NLS approximation allows one to determine it theoretically. The knowledge of the population difference allows one to correct the cnoidal wave solution, which then matches the numerical solution up to the limits of our numerical accuracy.

Consistent Riccati expansion solvability and soliton–cnoidal wave solutions of a coupled KdV system

The consistent Riccati expansion (CRE) solvability of a coupled KdV system is investigated with the aid of the Riccati equation. Differing from the complete Painlevé integrability for three branches, this coupled KdV system is shown to be CRE solvable for one branch and non-CRE solvable for two other branches. Based on the last consistent differential equation under the CRE solvable case, a kind of soliton–cnoidal wave interaction solution is derived explicitly.

Nonlinear periodic structures in magnetoplasma with nonthermal electrons and positrons

In the present study, we address the problem of cnoidal waves (CWs) in magnetized electron-positron-ion (e-p-i) plasma with nonthermal electrons and positrons. The Korteweg-de Vries equation (KdVE) is derived using the reductive perturbation technique (RPT) and its cnoidal wave (CW) solution is analyzed. The impact of relevant plasma parameters on the characteristics of the ion-acoustic (IA) cnoidal structures are discussed in detail. The application of the present investigation is discussed.

Steady periodic waves over a planar seabed: a global characterization

The problem of propagation of steady periodic waves over a planar seabed is faced through the definition of a suitable semi-analytical iterative scheme. The latter is capable of describing highly nonlinear waves in deep, intermediate and shallow water conditions. Comparisons with the existing fifth-order theories show that the proposed model is accurate in all the regimes of motion and that it does not present any of the limitations affecting the Stokes and cnoidal wave solutions. Further, it also provides a reliable approximation of the dynamics of maximum-amplitude waves. The definition of the iterative scheme is preceded by a detailed study of the geometrical wave parameters. This latter part is aimed at the definition of a global scaling for water waves valid in all the regimes of motion.

Numerical Experiments on Extreme Waves Through Oblique–Soliton Interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications, more precisely in physics, biology, chemistry and engineering fields. This domain serves as a point of interest to a large extent in the world’s mathematical community. Thus, this paper purveys an analytical study of a generalized extended (2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model. We further reduce the equation using the subalgebras obtained. Besides, more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique. Consequently, we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright, dark, periodic (cnoidal and snoidal), compact-type as well as singular solitons. The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time, are outlined with suitable diagrams. This can be of interest to oceanographers and ocean engineers for future analysis. Furthermore, to visualize the dynamics of the results found, we present the graphic display of each of the solutions. Conclusively, we construct conservation laws of the understudy equation via the application of Noether’s theorem.

New Localized and Periodic Solutions to a Korteweg–de Vries Equation with Power Law Nonlinearity: Applications to Some Plasma Models

Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on.

New Periodic and Localized Traveling Wave Solutions to a Kawahara‐Type Equation: Applications to Plasma Physics

In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions (localized and periodic solutions) to the generalized Kawahara equation (gKE). As a particular case, some traveling wave solutions to both Kawahara equation (KE) and modified Kawahara equation (mKE) are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions (WSEFs) and Jacobian elliptic functions (JEFs). For KE, some direct and indirect approaches are carried out to derive the periodic and localized solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma.

Kinetic Alfvénic cnoidal waves in Saturnian magnetospheric plasmas

The characteristics of kinetic Alfvénic cnoidal waves (KACWs) in Saturn's magnetosphere plasma composed of two temperature superthermal electrons and inertial ions are presented. The Korteweg–de Vries (KdV) equation and its cnoidal wave solution are derived by adopting reductive perturbation technique. Only positive potential kinetic Alfvénic cnoidal and solitary waves are evolved in Saturnian magnetospheric plasma. The influence of superthermality of cold electrons, concentration of hot electrons, plasma beta and angle of propagation of the wave with respect to the magnetic field has been analyzed on the characteristics of KACWs. The findings of this investigation may shed the light on the possible existence of KACWs and acceleration as well as energy transportation in space plasmas especially in Saturn's magnetosphere.

Cnoidal wave in a two-level medium: an alternative approach to nonlinear susceptibilities

We consider a cnoidal wave propagating in a two-level system, for which an exact analytic solution exists. Two effective components of the third-order nonlinearity , describing Kerr effect and third harmonics generation, can be derived from it. Some discrepancy has been found with respect to the usual formalism of nonlinear susceptibilities. Numerical computation shows that the exact cnoidal wave solution the most accurately describes the nonlinear propagation of a quasi-monochromatic wave. Especially, the third harmonic resonance of the third-order nonlinearity, predicted by the formalism of nonlinear susceptibilities, is not observed.

Stability analysis and novel solutions to the generalized Degasperis Procesi equation: An application to plasma physics.

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model.

Periodic and localized structures in dusty plasma with Kaniadakis distribution

Abstract The propagation of electrostatic dust-ion-acoustic nonlinear periodic waves is investigated in dusty plasma wherein electrons follow Kaniadakis distribution. The Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived by employing reductive perturbation method and their cnoidal wave solutions are analysed. The effect of relevant parameters (viz., κ-deformed parameter κ and dust concentration β) on the dynamics of cnoidal structures is discussed. Further it is found that amplitude of compressive cnoidal waves increases with increasing values of β, while reverse effect is observed in case of rarefactive cnoidal structures with rising values of β. Also κ-deformed parameter κ bears no effect on cnoidal waves associated with KdV equation, whereas κ-deformed parameter κ significantly affects the cnoidal waves associated with mKdV equation.

Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials

AbstractUnder compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. Finally, we show the evolution of the spacing between peaks for all solutions as a function of that parameter.

Bäcklund transformations, nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painlevé expansion. Three kinds of non-auto and auto Bäcklund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the nth Bäcklund transformation. The consistent Riccati expansion method is employed to derive a Bäcklund transformation. Furthermore, the soliton solutions, fusion-type N-solitary wave solutions and soliton–cnoidal wave solutions are gained through Bäcklund transformations.