Showing papers on "Cnoidal wave published in 2021"

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

Abstract: In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained. Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. Moreover, the standard tanh method is utilized to get a new set of solitary wave solutions for the evolution equation. Over and above, the hyperbolic ansatz method (a new ansatz in the form of squre-sech) is employed to get a new set of solitary wave solutions for the evolution equation. Furthermore, all obtained solutions of the planar Extended Kawahara equation cover the traveling wave solutions of the planar modified Kawahara equation. These solutions maybe useful to many researchers interested in studying the propagation of nonlinear waves in nonlinear dispersion mediums like plasma physics, optical fibers, fluid mechanics, and many different branches of science.

Nonlinear periodic structures in a magnetized plasma with Cairns distributed electrons

Abstract: The electrostatic ion acoustic (IA) nonlinear periodic (cnoidal) waves are studied in a magnetized plasma comprising of cold ions and Cairns distributed electrons. By employing reductive perturbation technique (RPT), the nonlinear Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are derived, and their cnoidal wave (CW) solutions are obtained. For a given set of plasma parameters, the present model supports both compressive and rarefactive nonlinear periodic structures. In the present work, the effects of plasma parameters like nonthermality of electrons ( $$\beta $$ ) and obliqueness ( $$ l_{z} $$ ) are investigated. It is found that increasing values of $$\beta $$ and $$l_{z}$$ lead to enhancement (reduction) in amplitude of IA compressive (rarefactive) nonlinear periodic structures. Our present study has some relevance to the CW structures observed in magnetosphere via POLAR and FAST spacecrafts.

Dynamics of mechanical metamaterials: A framework to connect phonons, nonlinear periodic waves and solitons

Abstract: Flexible mechanical metamaterials have been recently shown to support a rich nonlinear dynamic response. In particular, it has been demonstrated that the behavior of rotating-square architected systems in the continuum limit can be described by nonlinear Klein–Gordon equations. Here, we report on a general class of solutions of these nonlinear Klein–Gordon equations, namely cnoidal waves based on the Jacobi elliptic functions sn, cn and dn. By analyzing theoretically and numerically their validity and stability in the design- and wave-parameter space, we show that these cnoidal wave solutions extend from linear waves (or phonons) to solitons, while covering also a wide family of nonlinear periodic waves. The presented results thus reunite under the same framework different concepts of linear and non-linear waves and offer a fertile ground for extending the range of possible control strategies for nonlinear elastic waves and vibrations.

Long-wave equation for a confined ferrofluid interface: Periodic interfacial waves as dissipative solitons

Abstract: We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted nonuniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports traveling waves, governed by a novel modified Kuramoto--Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equations allows for the existence of dissipative solitons. These permanent traveling wave's propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The traveling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the traveling waves. Interestingly, multiperiodic waves, which are a non-integrable analog of the double cnoidal wave, also found to propagate under the model long-wave equation. These multiperiodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified above.

Ion Acoustic Cnoidal Waves in Ion-Beam Dense Plasma in the Presence of Quantizing Magnetic Field

Abstract: A study of propagation properties of ion-acoustic (IA) cnoidal waves in a magnetized degenerate quantum plasma comprising of inertial positive ions, weakly relativistic ion beam, and trapped electrons in the presence of quantizing magnetic field has been presented. The reductive perturbation technique is adopted to derive the Korteweg–de Vries (KdV)-type equation. Furthermore, the cnoidal wave solution of the KdV-type equation is obtained by using the Sagdeev pseudopotential method. Only positive potential IA cnoidal waves are evolved. The combined effects of ion-beam concentration, ion-beam velocity, quantizing magnetic field, and the temperature degeneracy of trapped electrons on the different characteristics of the IA cnoidal waves have been analyzed. The results of our present investigation may be useful to study the characteristics of IA cnoidal waves in dense astrophysical regions, such as white dwarfs.

Linear and Nonlinear Phenomena in a Flow of Surface Plasmon–Polaritons

Abstract: The excitation and propagation of plasmon–polariton modes are considered at a single interface of a dielectric medium and metal in linear and nonlinear regimes. Physical mechanisms of the emergence of a nonlinear response of free electrons in the metal based on the quantum hydrodynamic model are described. It is shown that the period and profile of the envelope of a cnoidal wave of surface plasmon–polaritons in the nonlinear regime change, depending on the conditions of excitation and the energy density of the exciting electromagnetic wave.

Non-local symmetry, interaction solutions and conservation lawsof the $$(1+1)$$ ( 1 + 1 ) -dimensional Wu–Zhang equation

Abstract: In this article, we take into account the $$(1+1)$$ -dimensional dispersive long-wave equation which is reduced from the Wu–Zhang equation. Firstly, the B $$\ddot{\text{ a }}$$ cklund transformation and the non-local symmetry are successfully acquired from the truncated Painlev $$\acute{\text{ e }}$$ expansion. At the same time, the non-local symmetry is transformed to Lie point symmetry by a suitable prolonged system. Then some solitary solutions are derived without a hitch via new B $$\ddot{\text{ a }}$$ cklund transformation which originates from Lie point symmetry. Secondly, by utilising the consistent Riccati expansion method and the consistent tanh-expansion method, we find the interaction solutions between soliton and cnoidal wave by using the Jacobi elliptic function. Lastly, the conservation laws which are related to symmetries of the equation are successfully obtained by Ibragimov’s method.

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

Abstract: 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.

Coupling Scaling for High-efficient Cnoidal Wave Generation in Microresonators

Abstract: We model the threshold and efficiency of microcombs by scaling the cavity coupling with the Lugiato-Lefever equation (LLE) and the traveling wave theory. Stable cnoidal waves with efficiency ≥ 40% are analytically identified at over-coupling.

Nonlinear Periodic Wave Propagation of Fast and Slow Magnetosonic Waves in the Presence of Oblique Magnetic Field

Abstract: Fast and slow magnetosonic cnoidal and solitary wave structures in electron–ion plasmas are studied in the presence of an oblique magnetic field. Under the low-frequency wave assumption, cold ions are taken to be dynamic, while inertialess electrons are assumed as isothermally heated. The two-fluid magnetohydrodynamic model is applied to present the basic set of dynamic equations for magnetosonic waves in the presence of oblique magnetic field. The linear analysis is performed for studying fast and slow modes of magnetosonic waves and also discussed its limiting cases with magnetic field angle. The nonlinear analysis is done via the reductive perturbation method, and the Korteweg–de Vries (KdV) equation is derived with appropriate periodic boundary condition. The cnoidal wave and soliton solutions for both fast and slow magnetosonic waves are presented. The effects of variations of magnetic field angle and plasma beta on both fast and slow magnetosonic cnoidal and soliton wave structures are discussed using the space plasma parameters that exist in the literature.

Magnetoacoustic Nonlinear Periodic Wave and Soliton Structures in Degenerate Pair Plasmas in the Presence of Exchange-Correlation Force

Abstract: By employing a two-fluid quantum magnetohydrodynamic (QMHD) model, linear and nonlinear aspects of magnetoacoustic wave propagation in dense magnetized electron–positron (e-p) plasma are examined in the presence of exchange-correlation potential as well. The degenerate plasma pressure and the Bohm potential effects are already taken in the model. The reductive perturbation method (RPM) is used, and the modified Korteweg–de Vries (modified KdV) equation (or KdV equation containing linear drift term) is derived for nonlinear magnetoacoustic wave propagation in degenerate pair plasma under periodic boundary conditions. The well-known Sagdeev potential approach is opted to obtain the analytical solution of magnetoacoustic cnoidal wave and soliton in dense pair plasmas. The numerical plots of magnetoacoustic cnoidal wave and soliton structures are also presented, and the condition of their formation in dense e-p plasma is discussed in detail. The parametric analysis in the absence and the presence of exchange-correlation potential effects of electrons and positrons is discussed. Moreover, the parametric analysis of the variations of plasma density and magnetic field intensity on the formation and propagation of nonlinear structures in dense e-p plasma in the presence of exchange-correlation force effects in the model is also done using the astrophysical plasma parameters mentioned in the literature for pulsars (or white dwarfs).

On Coupled Boussinesq Equations and Ostrovsky equations free from zero-mass contradiction

Abstract: Long weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations Previously, we considered the case when the materials of the layers have close mechanical properties, and the system supports radiating solitary waves Here we are concerned with a more challenging case, when the mechanical properties of the materials of the layers are significantly different, and the system supports wave packet solutions We construct a weakly-nonlinear solution of the Cauchy problem for this system, considering the problem in the class of periodic functions on an interval of finite length The solution is constructed for the deviation from the evolving mean value using a novel multiple-scales procedure involving fast characteristic variables and two slow time variables By construction, the Ostrovsky equations emerging within the scope of this procedure are solved for initial conditions with zero mean while initial conditions for the original system may have non-zero mean values Asymptotic validity of the solution is carefully examined numerically We also discuss the application of the solution to the study of co-propagating waves generated by the solitary or cnoidal wave initial conditions, as well as the case of counter-propagating waves and the resulting wave interactions One local and two nonlocal conservation laws ae obtained and used to control the accuracy of the numerical simulations