Nonlinear Wave Solutions Research Articles

Construction of a weakly nonlinear dispersion solitary wave solution for the Zakharov–Kuznetsov-modified equal width dynamical equation

By applying the modification form of extended auxiliary equation mapping method, we investigated the solitary wave solutions of Zakharov–Kuznetsov-modified equal width dynamical equation. We obtained these solitary wave solutions in the form of bright and dark solitons, periodic solitary wave, traveling wave, trigonometric functions, elliptic functions and hyperbolic functions solutions. The physical structure of these new solutions showed two and three dimensions graphically by using the computer software Mathematica. The obtained new solutions are the power and effectiveness of this new method. We can solve other nonlinear system of PDEs, which are involved in mathematical physics and other various branches of physical sciences with the help of this new method.

Madden–Julian Oscillations Seen in the Upper-Troposphere Vorticity Field: Interactions with Rossby Wave Trains

AbstractThe vorticity variability associated with the Madden–Julian oscillation (MJO) is examined. The analysis is focused on the 150-hPa pressure level, because a clear dipolar-vortex signal, reminiscent of the theoretically proposed strongly nonlinear solitary Rossby wave solution (albeit with the opposite sign), is seen in raw data at that level. A local empirical orthogonal function (EOF) analysis over the equatorial region of the Eastern Hemisphere (0°–180°E) identifies the two principal components representing an eastward propagation of a dipolar vortex trapped to the equator. Association of this propagation structure with the moist convective variability of the MJO is demonstrated by regressing the outgoing longwave radiation (OLR) against this EOF pair. The obtained evolution of the OLR field is similar to the one obtained by a direct application of the EOF to the OLR. A link of the local vorticity variability associated with the MJO to the global dynamics is further investigated by regressing the global vorticity field against the time series of the identified local EOF pair. The Rossby wave trains tend to propagate toward the Indian Ocean from higher latitudes, just prior to an initiation of the MJO, and in turn, they propagate back toward the higher latitudes from the MJO active region over the Indian Ocean. A three-dimensional regression reveals an equivalent barotropic structure of the MJO vortex pair with the signs opposite to those at 150 hPa underneath. A vertical normal mode analysis finds that this vertical structure is dominated by the equivalent height of about 10 km.

Spectral stability of nonlinear gravity waves in the atmosphere

AbstractWe apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

Dynamical Behavior of Supernonlinear Positron-Acoustic Periodic Waves and Chaos in Nonextensive Electron-Positron-Ion Plasmas

Abstract Propagation of nonlinear and supernonlinear positron-acoustic periodic waves is examined in an electron-positron-ion plasma composed of static positive ions, mobile cold positrons, and q-nonextensive electrons and hot positrons. Employing the phase plane theory of planar dynamical systems, all qualitatively different phase portraits that include nonlinear positron-acoustic homoclinic orbit, nonlinear positron-acoustic periodic orbit, supernonlinear positron-acoustic homoclinic orbit, and supernonlinear positron-acoustic periodic orbit are demonstrated subjected to the parameters q , μ 1 , μ 2 , σ 1 , σ 2 $q,{\mu_{1}},{\mu_{2}},{\sigma_{1}},{\sigma_{2}}$ , and V. The nonlinear and supernonlinear positron-acoustic periodic wave solutions are reported for different situations through numerical computations. It is observed that the nonextensive parameter (q) acts as a controlling parameter in the dynamic motion of nonlinear and supernonlinear positron-acoustic periodic waves. The dynamic motions for the positron-acoustic traveling waves with the influence of an extrinsic periodic force are investigated through distinct qualitative approaches, such as phase portrait analysis, sensitivity analysis, time series analysis, and Poincaré section. The results of this paper may be applicable in understanding nonlinear, supernonlinear positron-acoustic periodic waves, and their chaotic motion in space plasma environments.

State transition of lump-type waves for the (2+1)-dimensional generalized KdV equation

In this paper, we are mainly concerned with the (2+1)-dimensional generalized Korteweg–de Vries equation in fluid dynamics. Based on the translation transformation and Hirota bilinear method, we study the excitations of nonlinear lump-type waves on a constant background. A remarkable feature of these lump-type waves is that under some parameter conditions, these lump-type wave solutions can be converted into some amusing nonlinear wave structures, including the W-shaped solitary wave, double-peak solitary wave, parallel solitary wave, multi-peak solitary wave and periodic wave solutions. These results do not have an analog in the standard Kadomtsev–Petviashvili equation. The transition condition between the lump-type wave and other nonlinear wave solutions is presented. The dynamical behaviors of these nonlinear wave solutions are investigated analytically and illustrated graphically. Furthermore, the existence conditions for these nonlinear wave solutions are exhibited explicitly. Our results further enrich the nonlinear wave theories for the (2+1)-dimensional generalized Korteweg–de Vries equation.

Control of Nonlinear Wave Solutions to Neural Field Equations

Neural field equations offer a continuous description of the dynamics of large populations of synaptically coupled neurons. This makes them a convenient tool to describe various neural processes, s...

A class of exact nonlinear traveling wave solutions for shallow water with a non-stationary bottom surface

Abstract The GN nonlinear shallow water wave equations developed by Green and Naghdi (1976) [5,6] are valid for a non-stationary, non-uniform bottom surface and a non-uniform pressure on the top surface. In contrast, the S nonlinear shallow water wave equations developed by Serre (1953) [12,13] for uniform depth and later generalized by Seabra-Santos et al. (1987) for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops a class of exact nonlinear traveling wave solutions of the GN equations for a non-stationary, non-uniform bottom surface. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.

Physical flow properties for Pollard-like internal water waves

We present a study of the physical flow properties for a recently derived three-dimensional nonlinear geophysical internal wave solution. The Pollard-like internal wave solution is explicit in terms of Lagrangian labelling variables, enabling us to examine the mean flow velocities and mass flux in the three-dimensional setting. We show that the Pollard-like internal water wave does not have a net wave transport.

Nonlinear solitary waves (oscillitons) in dusty magnetized pair plasmas

Nonlinear theory is presented for solitary waves propagating parallel to the magnetic field in a cold plasma with charged dust grains or heavy ions. The particular case of an imbalanced electron-positron plasma with a portion of the charge residing on the dust particles is investigated. The charge imbalance plays a similar role to unequal ion and electron masses, in that an “oscilliton” type solution is found, using an approximate analytic solution, as well as numerical solutions of approximate equations and the full non-relativistic equations. Circularly polarized oscillitons occur for the electron-positron plasma for a Mach number above a critical value which is <1, in contrast to the pure pair plasma which has a linearly polarized non-oscillatory nonlinear wave solution for Mach number >1.

Space-time L 2 estimates, regularity and almost global existence for elastic waves

Abstract In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

Soliton wave-speed management: Slowing, stopping, or reversing a solitary wave

While dispersion management is a well-known tool to control soliton properties such as shape or amplitude, far less effort has been directed toward the theoretical control of the soliton wave speed. However, recent experiments concerning the stopping or slowing of light demonstrate that the control of the soliton wave speed is of experimental interest. Motivated by these and other studies, we propose a management approach for modifying the wave speed of a soliton (or of other nonlinear wave solutions, such as periodic cnoidal waves) under the nonlinear Schr\odinger equation. Making use of this approach, we are able to slow, stop, or even reverse a solitary wave, and we give several examples to bright solitons, dark solitons, and periodic wave trains, to demonstrate the method. An extension of the approach to spatially heterogeneous media, for which the wave may propagate differently at different spatial locations, is also discussed.

A comparative study of modified cubic B-spline differential quadrature methods for a class of nonlinear viscous wave equations

PurposeIn this study, a second-order standard wave equation extended to a two-dimensional viscous wave equation with timely differentiated advection-diffusion terms has been solved by differential quadrature methods (DQM) using a modification of cubic B-spline functions. Two numerical schemes are proposed and compared to achieve numerical approximations for the solutions of nonlinear viscous wave equations.Design/methodology/approachTwo schemes are adopted to reduce the given system into two systems of nonlinear first-order partial differential equations (PDE). For each scheme, modified cubic B-spline (MCB)-DQM is used for calculating the spatial variables and their derivatives that reduces the system of PDEs into a system of nonlinear ODEs. The solutions of these systems of ODEs are determined by SSP-RK43 scheme. The CPU time is also calculated and compared. Matrix stability analysis has been performed for each scheme and both are found to be unconditionally stable. The results of both schemes have been extensively discussed and compared. The accuracy and reliability of the methods have been successfully tested on several examples.FindingsA comparative study has been carried out for two different schemes. Results from both schemes are also compared with analytical solutions and the results available in literature. Experiments show that MCB-DQM with Scheme II yield more accurate and reliable results in solving viscous wave equations. But Scheme I is comparatively less expensive in terms of CPU time. For MCB-DQM, less depository requirements lead to less aggregation of approximation errors which in turn enhances the correctness and readiness of the numerical techniques. Approximate solutions to the two-dimensional nonlinear viscous wave equation have been found without linearizing the equation. Ease of implementation and low computation cost are the strengths of the method.Originality/valueFor the first time, a comparative study has been carried out for the solution of nonlinear viscous wave equation. Comparisons are done in terms of accuracy and CPU time. It is concluded that Scheme II is more suitable.

Nonsingular localized wave solutions for the nonlocal Davey–Stewartson I equation with zero background

In this paper, via the elementary Darboux transformation, we study the nonsingular localized wave solutions of the partially parity-time [Formula: see text] symmetric nonlocal Davey–Stewartson I equation with zero background. In addition to the common dromion and line-soliton solutions, we obtain some new localized wave solutions including the periodical-soliton, quasi-line-soliton and defected-line-soliton solutions. Meanwhile, we give the exact nonsingular parametric conditions for the derived solutions to display different localized wave structures. In addition, we discuss the dynamical behavior of the obtained nonlinear localized wave solutions with graphical illustration.

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Diffusive wave in the low Mach limit for compressible Navier–Stokes equations

The low Mach limit for 1D non-isentropic compressible Navier–Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and ill-prepared data. For the well-prepared data, the solutions of compressible Navier–Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at ±∞ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier–Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at ±∞ can be arbitrary large in this case.

Nonlinear electrostatic periodic waves and solitons in an inhomogeneous magnetized dusty plasma

Low frequency nonlinear electrostatic cnoidal and solitary waves are investigated in an inhomogeneous magnetized dusty plasma in the presence of shear flow. The Sagdeev potential approach is used to find the nonlinear wave solution. It is found that nonlinear electrostatic potential rarefactive (dip) structures are formed for cnoidal waves and solitons in a magnetized inhomogeneous dusty plasma instead of compressive (hump) nonlinear electrostatic wave structures formed in an inhomogeneous magnetized electron-ion plasma. The amplitude of the nonlinear electrostatic potential structure is found to be increased in the case of decreasing shear flow parameter values in comparison with the case of increasing shear flow parameter values. The D'Angelo instability in an inhomogeneous magnetized dusty plasma in the presence of shear flow is also discussed. The numerical plots are also presented for illustrations which are applicable to space dusty plasma regions containing shear flows. The Hamiltonian function of such a dynamical planar system is also discussed for its phase portrait analysis.

Finite-amplitude gravity waves in the atmosphere: travelling wave solutions

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

Nonlinear wave solutions for an integrable sixth-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is an integrable sixth-order nonlinear Schrödinger equation. Multi-soliton and higher-order breather solutions are obtained via the Darboux transformation. The higher-order rogue-wave solutions are derived via the generalized Darboux transformation. Effects of the higher-order terms on the interaction and propagation of the solitons, breathers and rogue waves are discussed graphically. Interactions between/among the solitons are elastic because the soliton amplitudes keep unchanged except for some phase shifts. In addition, the higher-order terms could enhance the steepness of the solitons. Periods of the Kuznetsov–Ma breathers are only related to the spectral parameter. Periods of the Akhmediev breathers are not only related to the spectral parameter, but also related to the coefficients of the higher-order terms. Akhmediev breathers have the phase shifts after the interaction, while Kuznetsov–Ma breathers have no phase shifts. The higher-order terms could enhance the steepness and symmetry of the rogue waves.

Solitary and cnoidal wave scattering by a submerged horizontal plate in shallow water

Solitary and cnoidal wave transformation over a submerged, fixed, horizontal rigid plate is studied by use of the nonlinear, shallow-water Level I Green-Naghdi (GN) equations. Reflection and transmission coefficients are defined for cnoidal and solitary waves to quantify the nonlinear wave scattering. Results of the GN equations are compared with the laboratory experiments and other theoretical solutions for linear and nonlinear waves in intermediate and deep waters. The GN equations are then used to study the nonlinear wave scattering by a plate in shallow water. It is shown that in deep and intermediate depths, the wave-scattering varies nonlinearly by both the wavelength over the plate length ratio, and the submergence depth. In shallow water, however, and for long-waves, only the submergence depth appear to play a significant role on wave scattering. It is possible to define the plate submergence depth and length such that certain wave conditions are optimized above, below, or downwave of the plate for different applications. A submerged plate in shallow water can be used as a means to attenuate energy, such as in wave breakers, or used for energy focusing, and in wave energy devices.

Nonlinear Propagation of Positron-Acoustic Periodic Travelling Waves in a Magnetoplasma with Superthermal Electrons and Positrons

The nonlinear propagation of positron acoustic periodic (PAP) travelling waves in a magnetoplasma composed of dynamic cold positrons, superthermal kappa distributed hot positrons and electrons, and stationary positive ions is examined. The reductive perturbation technique is employed to derive a nonlinear Zakharov–Kuznetsov equation that governs the essential features of nonlinear PAP travelling waves. Moreover, the bifurcation theory is used to investigate the propagation of nonlinear PAP periodic travelling wave solutions. It is found that kappa distributed hot positrons and electrons provide only the possibility of existence of nonlinear compressive PAP travelling waves. It is observed that the superthermality of hot positrons, the concentrations of superthermal electrons and positrons, the positron cyclotron frequency, the direction cosines of wave vector k along the z-axis, and the concentration of ions play pivotal roles in the nonlinear propagation of PAP travelling waves. The present investigation may be used to understand the formation of PAP structures in the space and laboratory plasmas with superthermal hot positrons and electrons.