Nonlinear Wave Solutions Research Articles

Dust ion acoustic waves for magnetized multi-component plasma

Abstract This theoretical investigation has been done to develop a mathematical model which could used to study such plasma system in the space where electrons follow non-extensive distribution, ions are positively and negatively charged, and dust particles are also charged like ions (both positive and negative) in characteristics in the presence of an magnetic field. Using the standard reductive perturbation method and stretching coordinates, at first Zakharov-Kuznetsov (ZK) equation was derived, then extended Tanh method was used for obtaining the nonlinear wave solution of ZK equation. Finally, a computational based numerical analysis has been done where many studied characteristics are found to be exited for the nonlinear propagation of different dense astrophysical compact objects in the space where magnetic field has been found to play an equal & important role like charged ions and dust gains.

Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

Nonlinear dynamical wave structures to the Date–Jimbo–Kashiwara–Miwa equation and its modulation instability analysis

A particular attention is paid to the nonlinear dynamical exact wave solutions to the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation (DJKME). A variety of solutions are extracted in different shapes like dark, singular, dark-singular by implementing [Formula: see text]-expansion function method and modified direct algebraic method. In addition, we also secure singular periodic and plane wave solutions with arbitrary parameters. We also discussed the modulation instability analysis of the governing model. The constraint conditions for the validity of existence of solutions are also reported. Moreover, three-dimensional and two-dimensional, and their corresponding contour graphs are sketched for a better understanding of the derived solutions with the values of arbitrary parameters.

Approximate nonlinear wave solutions of the coupled two-component Gross–Pitaevskii equations with spin–orbit interaction

Recent experimental observations of spin–orbit coupling (SOC) in Bose–Einstein condensates (BECs) open the way for investigating novel physics of nonlinear waves with promising applications in atomic physics and condensed matter physics. The interplay between atomic interactions and SOC are crucial for the understanding of the dynamics of nonlinear waves in BECs with SOC. Here, in the small linear coupling regime, an approach is presented which allows us to derive an infinite number of novel approximate solutions of the Gross–Pitaevskii equations (GPEs) in one and two dimensions including SOCs, time-dependent external potentials, and nonlinearities leading to breathers and periodic as well as quasiperiodic nonlinear waves. To verify the theoretical predictions we perform numerical simulations which show for several cases a very good agreement with the analytics. For the case of one spatial dimension, it is shown that functions describing the external potential and nonlinearities cannot be chosen independently. The management of the solutions is clarified along with some important physical properties such as Josephson oscillations and Rosen–Zener oscillations.

New solitonic and rogue wave solutions of a Klein–Gordon equation with quadratic nonlinearity

An analytical investigation is performed on soliton, lump wave solution, and rogue waves in the Klein–Gordon with quadratic nonlinearity through the extended tanh approach, which possesses complicated wave propagation arising in the field of nonlinear optics, theory of quantum field and solid state physics. As a result, an advanced form of interacting analytical solutions is achieved with some unrestricted parameters. Different conditions on the existing parameters of these solutions are found after analyzing its dynamic behavior. Based on the conditions, different type of rogue wave, bright bell and dark bell shape nature of the solutions are considered. The dynamics nonlinear wave solutions are showed in 3D plots with specific values of the existing parameters. Moreover, it is shown that nonlinear wave packets are localized in two dimensions with characteristics of rogue wave profiles.

Oscillatory and Stationary Patterns in a Diffusive Model with Delay Effect

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns.

Darboux-Bäcklund transformation, breather and rogue wave solutions for the discrete Hirota equation

Based on the Lax pair, the pseudopotential and Darboux-Bäcklund transformation of the discrete Hirota equation are given, from which a generalized form of nonlinear wave solutions is derived. By analyzing the properties of the pseudopotential, various types of localized wave solutions including n-direction, t-direction breather wave and rational solutions are obtained and the corresponding dynamical properties and evolutions are given. The obtained results may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, Bose-Einstein condensates and so on.

Bifurcation mechanism of interfacial electrohydrodynamic gravity-capillary waves near the minimum phase speed under a horizontal electric field

We investigate the evolution of interfacial gravity-capillary waves propagating along the interface between two dielectric fluids under the action of a horizontal electric field. There is a uniform background flow in each layer, and the relative motion tends to induce Kelvin–Helmholtz (KH) instability. The combined effects of gravity, surface tension and electrically induced forces are all taken into account. Under the short-wave assumption, the expansion and truncation method of Dirichlet-Neumann (DN) operators is applied to derive a reduced dynamical model. When KH instability is suppressed linearly by a considerably large electric field, our numerical results reveal that in certain regions of parameter space, nonlinear symmetric traveling wave solutions can be found near the minimum phase speed. Additionally, the detailed bifurcation structures are presented together with typical wave profiles.

Higher dimensional localized and periodic wave dynamics in an integrable (2+1)-dimensional deep water oceanic wave model

In this article, a new integrable (2+1)-dimensional Kundu–Mukherjee–Naskar model which is a variant of the well-known nonlinear Schrödinger equation is investigated. Bright–dark optical solitons along with periodic waves, complexiton, and rational solutions are constructed by employing a generalized traveling wave analysis, Jacobian-elliptic function, Riccati equation, and ansatz approach. Further, the dynamics of these bright/dark optical solitons and complexiton/periodic waves are studied by exploring the importance of arbitrary physical parameters with graphical demonstrations for a clear understanding and stability of the considered model is also investigated. The obtained higher dimensional nonlinear wave solutions of this integrable system shall have useful applications in different physical systems including the dynamics of beam propagation in optical fibers, ion-acoustic waves in magnetized plasma, and deep water oceanic rogue waves.

Bäcklund transformation and localized nonlinear wave solutions of the nonlocal defocusing coupled nonlinear Schrödinger equation

The nonlocal integrable defocusing coupled nonlinear Schrödinger system from a 3×3 spectral problem is investigated. The Bäcklund transformation of the nonlocal defocusing coupled nonlinear Schrödinger system is presented from the pseudopotentials derived from the Lax pair. The nonsingular localized wave solutions including the breather wave and exponential-rational solutions are obtained, whose evolutions and dynamical properties are discussed.

Nonlinear ultrasonic waves in a magneto-flexo-thermally actuated single walled armchair carbon nanotube embedded on polymer matrix

PurposeThe purpose of this paper is concerned with the study of nonlinear ultrasonic waves in a magneto-flexo-thermo (MFT) elastic armchair single-walled carbon nanotube (ASWCNT) resting on polymer matrix.Design/methodology/approachA mathematical model is developed for the analytical study of nonlinear ultrasonic waves in a MFT elastic armchair single walled carbon nanotube rested on polymer matrix using Euler beam theory. The analytical formulation is developed based on Eringen’s nonlocal elasticity theory to account small scale effect. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been analysed numerically by using the nonlinear foundations supported by Winkler-Pasternak model. The solution is obtained by ultrasonic wave dispersion relations.FindingsFrom the literature survey, it is evident that the analytical formulation of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix is not discussed by any researchers. So, in this paper the analytical solutions of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix are studied. Parametric studies is carried out to scrutinize the influence of the nonlocal scaling, magneto-electro-mechanical loadings, foundation parameters, various boundary condition and length on the dimensionless frequency of nanotube. It is noticed that the boundary conditions, nonlocal parameter and tube geometrical parameters have significant effects on dimensionless frequency of nanotubes.Originality/valueThis paper contributes the analytical model to find the solution of nonlinear ultrasonic waves in an MFT elastic ASWCNT embedded on polymer matrix. It is observed that the increase in the foundation constants raises the stiffness of the medium and the structure is able to attain higher frequency once the edge condition is C-C followed by S-S. Further, it is noticed that the natural frequency is arrived below 1% in both local and nonlocal boundary conditions in the presence of temperature coefficients. Also, it is found that the density and Poisson ratio variation affects the natural frequency with below 2%. The results presented in this study can provide mechanism for the study and design of the nano devices such as component of nano oscillators, micro wave absorbing, nano-electron technology and nano-electro--magneto-mechanical systems that make use of the wave propagation properties of ASWCNTs embedded on polymer matrix.

Dynamics of interaction between solitary and rogue wave of the space-time fractional Broer–Kaup models arising in shallow water of harbor and coastal zone

Under inquisition, we consider the space-time fractional Broer–Kaup model that replicate the real coastal profile to describe bidirectional wave transmission of long waves in shallow water. We integrate of the model via the unified scheme to derive exact solitary wave solutions. The achieved results present the hyperbolic, trigonometric, and rational functions solutions. All of these solutions convey the periodic and solitonic natures. In particular, we present combo waves that imply a curvy periodic waves in which rogue waves occur in both sides of each waves. The dynamics of obtained nonlinear wave solutions are demonstrated in 3-D and 2-D shapes with definite parametric values. It is shown that the obtained solutions of the model pact a very rich dynamical behavior and can be used to describe seismic type nonlinear wave motion in shallow water waves of the coastal and harbor region.

Finite difference solutions for nonlinear water waves using an immersed boundary method

AbstractIn this article, high‐order finite difference solutions of the exact potential‐flow problem for nonlinear water waves are developed which treat both the free surface and the uneven sea bottom boundary conditions using an immersed boundary method (IBM). The convergence, accuracy, and stability of this approach is first established for the linear problem, using various orders of scheme and different grid discretization strategies. The nonlinear wave problem on a flat bottom is then considered, and the solutions are compared with the highly accurate stream function theory solution. The convergence performance of the numerical solution for this nonlinear wave problem is established. Finally, linear wave shoaling and nonlinear wave propagation over a submerged bar are also tested. A preconditioned iterative solution strategy is also developed and shown to provide optimal scaling of the solution effort with increasing number of unknowns. Compared with existing free surface tracking methods, the IBM retains the same level of accuracy and stability when solving the linear problem. For the nonlinear problem, the IBM behaves reasonably in terms of accuracy, but generally with slightly higher computational effort. When it comes to the wave–body interaction problem, it is expected that the IBM will be advantageous compared with the σ‐transform method, since it removes the necessity of constructing an artificial continuous free‐surface inside the body.

New exact solitary wave solutions, bifurcation analysis and first order conserved quantities of resonance nonlinear Schrödinger’s equation with Kerr law nonlinearity

This paper anatomizes the exact solutions of the resonant non-linear Schrödinger’s equation (R-NLSE) with the Kerr law non-linearity with the assistance of the new extended direct algebraic technique. The secured soliton erections are newfangled and unreservedly invigorating for investigators. The graphically comprehensive report of some specific solutions is embellished with the well-judged values of parameters to illustrate their propagation. Then a planer dynamical system is introduced and the bifurcation analysis has been executed to figure out the bifurcation structures of the non-linear and super non-linear traveling wave solutions of the heeded model. All possible phase portraits are exhibited with specific values of parameters. Furthermore, a precise class of non-trivial and first-order conserved quantities is enumerated by the intervention of the multiplier approach.

Cnoidal wave solutions for the optical Benney-Luke equation

At proper scale of the asymptotic analysis, the optical Benney-Luke equation (OBLE) explains the propagation of a continuous wave (cw) laser beam in a nonlocal non-linear defocusing media. By means of an OBLE and appropriate boundary conditions, we have obtained non-linear periodic wave solutions. The corresponding analytical expressions and numerical plots of the cnoidal waves in the phase plane are presented using the Sagdeev pseudo-potential approach. Our results are written in terms of the media nonlocality through the optical surface tension parameter.

Periodic Wave Solutions and Their Asymptotic Property for a Modified Fornberg–Whitham Equation

Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter α and their asymptotic property for the modified Fornberg–Whitham equation. Furthermore, when α tends to given parametric values, the elliptic periodic wave solutions become the other three types of nonlinear wave solutions, which include the trigonometric periodic blow-up solution, the hyperbolic smooth solitary wave solution, and the hyperbolic blow-up solution.

Dynamical behavior of fractional Chen-Lee-Liu equation in optical fibers with beta derivatives

This paper studies the dynamical behaviors of nonlinear wave solutions of perturbed and unperturbed fractional Chen-Lee-Liu (CLL) equation in optical fibers with a newly defined beta derivative. The coupled amplitude-phase formulation is used for the derivation of a nonlinear differential equation which contains a fifth-degree nonlinear term describing the evolution of the wave amplitude in the nonlinear system. Variety of soliton solutions are found by using the new extended direct algebraic method. Then, discussed model is converted into the planer dynamical system with the help of Galilean transformation and the bifurcation behavior is reported. All possible forms of phase portraits with respect to the parameters of the considered problem are plotted. In addition, by applying an extrinsic periodic force the effect of physical parameters is investigated. Furthermore, sensitive analysis is applied for different initial value problems to analyze the quasiperiodic and quasiperiodic-chaotic behaviors.

Instabilities and bifurcations of liquid films flowing down a rotating fibre

Abstract

Marangoni instability of self-rewetting films modulated by chemical reactions flowing down a vertical fibre

This paper examines the Marangoni instability of a self-rewetting film coating on a vertical heated fibre and its modulation by chemical reactions. A single evolution equation based on the classic thin-film model is derived to study the flow dynamics. Linear stability analysis shows that without chemical reactions, wall heating plays a stabilizing role when Ti>T0 (Ti is the interfacial temperature and T0 is the temperature corresponding to the minimum of surface tension in self-rewetting fluids), and it plays a destabilizing role when Ti<T0. However, chemical reactions can strongly modulate how the wall heating affects the flow dynamics. For both situations (Ti>T0 or Ti<T0), the endothermic reaction further enhances respective Marangoni effects, and the exothermic reaction suppresses respective Marangoni effects. In particular, a strong exothermic reaction can totally neutralize and even reverse Marangoni effects. These findings are further confirmed by performing nonlinear traveling wave solutions and transient numerical simulations.

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.