Analytical Wave Research Articles

Diverse analytical wave solutions of plasma physics and water wave equations

In plasma physics and water waves, the Gilson-Pickering equation is an important unidirectional wave propagation model. A large number of analytic wave solutions have been established in the form of hyperbolic, trigonometric, exponential, and rational functions using the advanced auxiliary equation approach in this study. The solutions obtained have been compared with the solutions available in the literature, and it is observed that we have established further wave solutions than the solutions determined by the other methods, namely the (G′/G2)-expansion method, sinh-Gordon approach, etc. This method uses a homogeneous balance rule to estimate a polynomial-type solution and delivers an order. A traveling wave transformation has been used to convert the governing equation into a nonlinear differential equation. Each solution includes a variety of parameters related to the model and method. Using different parameter values, the nature of wave solutions is defined by three-dimensional (3D) and two-dimensional (2D) wave profiles. The solitons change their nature and location for the particular values of the dispersion coefficient α and free parameter κ, and shown in 2D figures to illustrate the different forms of solitons.

An analytical wave solution for the vibrational response and energy of an axially translating string in any propagation cycle

An axially traveling string system, which is a kind of traveling material, attracts considerable attention owing to its broad applications. In this paper, an analytical wave solution for the vibration and energy of an axially traveling string with fixed and viscous damper (dashpot) boundaries in any propagation cycle is considered. Firstly, a novel recursive and simplified technique is proposed to expand the analytical solution for a traveling string to any propagation cycle, which was limited to only one propagation cycle due to complexity in previous work. As a kind of analytical solution, the traveling wave method has more accuracy and efficiency compared to numerical methods. Secondly, different from the previous result, the modified Hamilton’s principle is applied to the derivation of the dashpot boundary condition for the mass changing of the traveling string. Following the pipeline hydrodynamics theory, the energy gradient for the ‘control volume’ and the ‘system’ of traveling string are accurately obtained, respectively. Thirdly, from the point of view of vibration suppression, the optimal damping at the right end of the string is defined and the optimal damping value is derived, which is of considerable practical interest in vibration suppression at boundaries for axially traveling materials.

Solitary and periodic waves in quadratic-cubic non-centrosymmetric waveguides

We present a wide class of solitary and periodic waves in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearities. We show the existence of bright, gray, and W-shaped solitary waves as well as periodic waves for the quadratic-cubic nonlinear Schrödinger equation. We also obtained the exact analytical algebraic-type solitary waves of this model, including bright and W-shaped waves. The results illustrate the propagation of potentially rich set of nonlinear structures through the optical system. Such privileged nonlinear waves characteristically exist due to a balance among diffraction, quadratic and cubic nonlinearities. The stability of the solutions is discussed numerically under finite initial perturbations. These results can be used in studying the propagation of shape-preserved and periodic waves in optical waveguides with quadratic-cubic nonlinearities.

A semi-analytical method for the dispersion analysis of orthotropic composite plates with periodically attached acoustic black hole resonators

A semi-analytical method is developed to calculate the wave propagation properties of thin orthotropic composite plate structures periodically attached to acoustic black hole (ABH) resonators. First, the free and forced vibration of the host orthotropic plate is described in terms of the analytical wave shapes obtained by the symplectic method. Second, a wave-based description is provided for the ABH resonators using the higher-order WKB method. Third, the equivalent dynamic stiffness of the resonator is obtained by considering the deformation-force relationship at the connections. Finally, an eigenvalue problem is established for the dispersion of the periodic plate based on wave propagation analysis, where the dynamic coupling between the host plate and resonator is innovatively described in terms of directly excited waves. Because the wave-based derivations in the proposed method are analytical, the calculations highly efficient and accurate, and were validated through several numerical examples.

Analytical behavior of soliton solutions to the couple type fractional-order nonlinear evolution equations utilizing a novel technique

Shallow water waves are one of the most prominent and widely used waves in coastal areas. The anomalous bi-directional transmission of long waves over shallow water is described by the nonlinear fractional partial differential equations (NLFPDEs), namely space-time-fractional coupled Whitham-Broer-Kaup (WBK) and coupled approximate long water (ALW) wave equations. To achieve new analytical solutions, the enhanced extended tanh-function technique for the mentioned fractional equations in the sense of conformable derivative is used. Some well-known wave shapes of solitons, namely, kink, bell-shape, and other types, through the suggested technique are achieved, and the 3D, contour, list point plots, and vector plots of the solutions are sketched to further illustrate the wave profile. The results achieved in this study have been validated using the computational software Maple and found to be accurate. It is observed that the method is realistic, and dependable to investigate more generalized analytical wave solutions.

Analytical field equation and wave function solutions of the Bianchi type I universe in vacuum f(R) gravity

It has been proposed that f(R) gravity is a viable proxy for General Relativity and a means for an alternative to dark energy. Within this context, we present the analytical solutions to the Bianchi type I universe. For this spacetime, the Killing and homothetic vector fields categorize the space into fourteen classes. Noether’s theorem is used to determine the first integrals associated with each isometry for every class and we use the geometry of the Noether point symmetries of the space to establish potential functions. We also quantize the Lagrangian of the spacetime to derive the Wheeler–DeWitt equation and successively determine invariant solutions for the wave function of the universe from the Lie point symmetries admitted by the equation. Finally, for all cases, exact classical solutions to the field equations are found.

Analytical study of nonlinear water wave equations for their fractional solution structures

This paper examines the three-dimensional nonlinear time-fractional water wave equations for their analytical wave solutions. These are the equations of the names [Formula: see text]-Zakharov–Kuznetsov–Burgers equation and [Formula: see text]-Yu–Toda–Sasa–Fukuyama equation. The obtained wave solutions are in the form of kink, periodic and singular waves by utilizing the [Formula: see text]-expansion approach. The aforesaid solutions are verified and demonstrated graphically via symbolic soft computations.

Nonlinear and supernonlinear ion-acoustic wave phenomena in an electron-positron-pair-ion quantum plasma

A four-component quantum plasma consisting of electrons, positrons, negative heavy ions and positive light ions is proposed in this work. The dynamics of nonlinear and supernonlinear ion-acoustic waves are studied in the framework of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations which are derived employing the reductive perturbation technique. Using Galilean transformation these two evolution equations are transformed into planar dynamical systems. All possible phase portraits and corresponding small-amplitude Sagdeev's pseudopotential of these dynamical systems are presented graphically. The unique topology of phase portrait and a maxima in between two minimas in pseudopotential curves clearly establish quantum ion-acoustic superperiodic waves. Solitary, periodic and superperiodic wave solutions corresponding respectively to homoclinic, periodic and superperiodic orbits in phase portraits are obtained numerically and the influence of different parameters on these waves is observed. Further, various kinds of analytical wave solutions for the two evolution equations are discussed.

DYNAMICAL BEHAVIOR OF THE LONG WAVES IN THE NONLINEAR DISPERSIVE MEDIA THROUGH ANALYTICAL AND NUMERICAL INVESTIGATION

This paper studies the well-known mathematical model’s analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques’ accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system’s features. The originality and novelty of this paper are discussed, along with previously published papers.

Modulational instability and position controllable discrete rogue waves with interaction phenomena in the semi-discrete complex coupled dispersionless system

The position controllable discrete analytical higher-order rogue wave and periodic wave solutions in the semi-discrete complex coupled dispersionless system, as well as their mixed interaction phenomena, are theoretically provided in this study. Our first priority is to study the modulation instability theory study for this system and deduce the formation mechanism from its plane wave solutions. Then, we use the generalized (n,N−n)-fold Darboux transformation to create three types of position-controlled localized wave solutions, including rogue waves, periodic waves and their mixed interaction solutions. In particular, we find three different kinds of rogue wave structures in the same discrete system, including dark–bright structure, bright–bright structure, and the structure with two peaks and two depressions, which is a rare phenomenon in the field of nonlinear waves. All these innovative structures are discussed analytically and shown graphically. Besides, the principal structures and locations of these novel solutions can be controlled by a set of unique parameters so that we can theoretically manage them to appear anywhere on the plane wave by changing these parameters. These results may play a potential role to describe the interaction between the current-fed string and the external magnetic field and other physical phenomena.

Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation

Abstract In the present study, the ion-acoustic solitary wave solutions for Kadomtsev–Petviashvili (KP) equation, potential KP equation, and Gardner KP equation are constructed. The nonlinear KP equations are studying the nonlinear process of waves without collisions plasma and having non-isothermal electrons and cold ions. Two-dimensional ion-acoustic solitary waves (IASWs) in magnetized plasma are consisting of electrons and ions. We obtained the ion-acoustic solitary wave solutions same as dark and bright, kink and anti-kink wave solitons. The physical phenomena of various structures for IASWs are represented graphically with symbolic computations. These results are more helpful in the development of soliton dynamics, quantum plasma, dynamic of adiabatic parameters, fluid dynamics, and industrial phenomena.

Signatures of misalignment in x-ray cavities of cavity-based x-ray free-electron lasers

Cavity-based x-ray free-electron lasers (CBXFEL) will allow use of optical cavity feedback to support generation of fully coherent x-rays of high brilliance and stability by electrons in undulators. CBXFEL optical cavities comprise Bragg-reflecting flat crystal mirrors, which ensure x-rays circulation on a closed orbit, and x-ray refractive lenses, which stabilize the orbit and refocus the x-rays back on the electrons in the undulator. Depending on the cavity design, there are tens of degrees of freedom of the optical elements, which can never be perfectly aligned. Here, we study signatures of misalignment of the optical components and of the undulator source with the purposes of understanding the effects of misalignment on x-ray beam dynamics, understanding misalignment tolerances, and developing cavity alignment procedures. Betatron oscillations of the x-ray beam trajectory (both symmetric and asymmetric) are one of the characteristic signatures of cavity misalignment. The oscillation period is in the general case a non-integer number of round-trip passes of x-rays in the cavity. This period (unlike the amplitude and offset of the oscillations) is independent of the type of misalignment and is defined by cavity parameters. The studies are performed on an example of a four-crystal rectangular cavity using analytical and numerical wave optics as well as ray-tracing techniques. Both confocal and generic stable cavity types are studied.

Stable abundant computational solitary wave structures of the perturbed time-fractional NLS equation

The conformable fractional derivatives Khater II (Khat. II) method and the Adomian decomposition (AD) approach are used to investigate the analytical and semi-analytical wave solutions to the perturbed time-fractional nonlinear Schrödinger (NLS) model. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. Several novel solitary wave solutions are constructed in distinct formats such as hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combination singular, and periodic wave solutions. Additionally, the Adomian decomposition technique determines the absolute error associated with analytical and semi-analytical wave solutions. The Hamiltonian system examines the stability of discovered solutions to determine their acceptability for usage in the model’s application.

Abundant stable wave structures for the nonlinear propagation of dislocations in crystals, phase differences across Josephson junctions

This study investigates the analytical solitary wave solutions of the nonlinear Sine–Gordon (SG) system by using two recent computational (Khater II and novel Kudryashov) schemes. The studied model is used as a model for various wave phenomena, including the propagation of dislocations in crystals, phase differences across Josephson junctions, torsion waves in strings and pendula, and waves along lipid membranes. It was already studied in the nineteenth century in connection with the theory of pseudospherical surfaces. The quantum version is used as a simple model for solid–state excitations. The solitonic property of the constructed solutions is also explained through some distinct graphs in two-, three-dimensional, contour, and polar plots. The stability property of the obtained soliton wave solutions is discussed based on the Hamiltonian system’s characterizations.

Novel Optical Soliton Waves in Metamaterials with Parabolic Law of Nonlinearity via the IEFM and ISEM

Here, the miscellaneous soliton solutions of the generalized nonlinear Schrödinger equation are considered that describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity. The novel analytical wave solutions to the mentioned nonlinear equation in the sense of the nonlinear ordinary differential transform equation are obtained. The techniques are the improved exp − Γ ϖ function method and the improved simple equation method. The nonlinear ordinary transform to concern the generalized Schrodinger equation to convert it for a solvable integer-order differential equation is used. After the successful implementation of the presented methods, the exact solitary wave solutions in the form of trigonometric, rational, and hyperbolic functions are obtained. Hence, the presented methods are relatable and efficient to solve nonlinear problems in mathematical physics.

Transcritical bifurcation of nonlinear electrostatic waves in a superthermal dusty plasma

The phase portraits and bifurcation properties of nonlinear dust ion-acoustic waves are studied in a complex plasma comprising cold dust particles, warm adiabatic ions, and κ-distributed electrons. A characteristic equation is obtained and solved numerically. The existence domain of fixed points and the stability of traveling wave solutions are discussed in the parameter space of Mach number M and dust concentration δ over a wide range of values of κ and ion temperature σ. It is shown that the motion dynamics of homoclinic orbits and nonlinear periodic orbits undergo a transcritical bifurcation at critical points of and , where two fixed points coalesce, and then switch their stabilities. We observe that at these bifurcation values with half-stable fixed points, the system supports a new analytical solitary wave solutions with much fatter tails than regular solitons. Furthermore, the existence domain of transcritical bifurcation parameters and transition between solitary and nonlinear periodic waves has been determined for different values of M and δ. It is found that the Mach number broadens the region prior to the bifurcation point, and enhances the onset of transition between modes, while the dust concentration has an opposite influence.

Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.

Abundant closed-form solutions of the (3+1)-dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering

This research article is a study of the (3+1)-dimensional Vakhnenko-Parkes equation that governs the propagation of high-frequency waves in a relaxing medium. To find a variety of new analytic exact solitary wave solutions, we apply three effective schemes, such as the generalized Kudryashov method (GKM), the generalized exponential rational function method (GERFM), and the generalized Riccati equation mapping method (GREMM). Obtained solutions are crucial to understand some existing, complicated physical phenomena. Many solutions are obtained, which are expressed by the exponential, trigonometric, and hyperbolic functions, including tanh, coth, sech, csch, tan, cot, sech, csch, and their combinations. These solutions include periodic-wave solutions, kink waves, combinations of kink and multi solitons, singular solitons, singular-kink structures, and periodic solitons. The 3D and 2D plots are shown by considering the possibility of arbitrary parameters occurring in the solutions better to understand the underlying mechanisms of the stated family. The computational results of this study assure us that the proposed approaches seem useful, efficient, and reliable tools for estimating the solutions attained in the study. Moreover, the method is expected to be frequently applied to examine various types of nonlinear PDEs, which emerge in mathematical physics, ocean engineering, and science.

Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Optical wave solutions of perturbed time-fractional nonlinear Schrödinger equation

The conformable fractional derivatives modified Khater (mKhat.) technique and the Adomian decomposition (AD) method is used to examine the perturbed time-fractional nonlinear Schrödinger (NLS) problem's analytical and semi-analytical wave solutions. This model describes the dynamics of optical solitons propagating via nonlinear optical fibers. For this model, we create a variety of different formulae for analytical wave solutions, including hyperbolic, trigonometric, rational, dark, brilliant, combined dark-bright, singular, combined singular, and periodic wave solutions. Additionally, the Adomian decomposition approach is utilized to evaluate the absolute error between analytical and semi-analytical wave solutions. The Hamiltonian system is used to analyze the stability of found solutions to demonstrate their suitability for implementation in the model's application.