Explicit Wave Research Articles

On abundant new solutions of two fractional complex models

We use an analytical scheme to construct distinct novel solutions of two well-known fractional complex models (the fractional Korteweg–de Vries equation (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation). A new fractional definition is used to covert the fractional formula of these equations into integer-order ordinary differential equations. We obtain solitons, rational functions, the trigonometric functions, the hyperbolic functions, and many other explicit wave solutions. We illustrate physical explanations of these solutions by different types of sketches.

Abundant new non-travelling wave solutions for the (3+1)-dimensional potential-YTSF equation

By combining the extended homoclinic test approach (EHTA) with the method of separation of variables explicit exact non-traveling wave solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation is constructed. With the aid of symbolic computation, abundant new explicit exact solutions of the (3+1)-dimensional potential-YTSF equation are obtained. The results obtained her e reveal the complex structure of the solutions of the (3+1) dimensional potential-YTSF equation. Many of the previous results obtained in other literature can be seen as special cases here. When some arbitrary functions included in these solutions are taken as some special functions, exact periodic solitary wave, periodic soliton, periodic cross-kink wave, periodic two-solitary wave are presented.

Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

Recent applications of discrete variational multi-electron method

The discrete variational multi-electron (DVME) method is a first-principles many-electron calculation method based on the discrete-variational Xα (DV-Xα) molecular orbital method. There is also the relativistic version of the DVME method, which is based on the relativistic DV-Xα method. Since the explicit many-electron wave functions of the multiplet states of transition-metal (TM) or rare-earth (RE) ions in crystals can be obtained, the DVME method has been applied to the analyses of various luminescent materials such as solid-state lasers and phosphors. In this paper, as examples of the recent applications of the DVME method, the following researches will be briefly presented: (1) Visualization of the energy-structure relationship of Cr3+ in oxides, and (2) Creation of the predictive model of the multiplet energy using machine learning.

Computational simulations of the couple Boiti–Leon–Pempinelli (BLP) system and the (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation

This research paper employs two different computational schemes to the couple Boiti–Leon–Pempinelli system and the (3+1)-dimensional Kadomtsev–Petviashvili equation to find novel explicit wave solutions for these models. Both models depict a generalized form of the dispersive long wave equation. The complex, exponential, hyperbolic, and trigonometric function solutions are some of the obtained solutions by using the modified Khater method and the Jacobi elliptical function method. Moreover, their stability properties are also analyzed, and for more interpretation of the physical features of the obtained solutions, some sketches are plotted. Additionally, the novelty of our paper is explained by displaying the similarity and difference between the obtained solutions and those obtained in a different research paper. The performance of both methods is tested to show their ability to be applied to several nonlinear evolution equations.

Optical rogue waves for a three-component coupled transient stimulated Raman scattering system

A three-component coupled transient stimulated Raman scattering system in nonlinear optics is investigated. Via applying the Lax-Darboux scheme, the explicit rogue wave solutions are constructed for the system. It is interesting that the rogue waves possess three types of geometric structures: one peak and two troughs, four troughs, and two peaks and two troughs, which exhibit perfect symmetry in space-time coordinate system. Furthermore, the extremum and asymptotic behaviors of the rogue wave are also analyzed.

An explicit plethora of solution for the fractional nonlinear model of the low–pass electrical transmission lines via Atangana–Baleanu derivative operator

Novel explicit wave solutions are constructed for the fractional nonlinear model of the low–pass electrical transmission lines. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types of this model. The stability property of the obtained solutions is tested to show the ability of our obtained solutions in using through the physical experiments. Moreover, the obtained analytical solutions are used to evaluate the initial and boundary conditions that allows applying the cubic & septic B–spline schemes to investigate the numerical solutions of this model. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical behavior of this model.

Solitary wave, M-lump and localized interaction solutions to the (4+1)-dimensional Fokas equation

At present, many researchers pay close attention to the solutions of nonlinear partial differential equations. Because these solutions can explain physical phenomena well. In this paper, soliton solutions, fissionable wave solutions, M-lump solutions and interaction solutions of the (4+1)-dimensional Fokas equation are explicitly generated. Taking advantage of the Hirota bilinear method, N-soliton solutions are constructed. Then, on bases of the obtained solutions, fissionable wave solutions are ascertained by appropriate selections of parameters. Applying the long wave limit method and restricting the conjugation conditions on the related solitons, M-lump solutions are derived. Afterwards, the interaction solutions which describe interactions of a lump and a soliton, a lump and two solitons, a lump and fissionable solitary waves are gained. Moreover, some graphs are given out to demonstrate the dynamic properties of the explicit analytical localized wave solutions.

Hall viscosity of composite fermions

Hall viscosity, also known as the Lorentz shear modulus, has been proposed as a topological property of a quantum Hall fluid. Using a recent formulation of the composite fermion theory on the torus, we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form $\nu=n/(2pn\pm 1)$, where $n$ and $p$ are integers, from the explicit wave functions for these states. The calculated Hall viscosities $\eta^A$ agree with the expression $\eta^A=(\hbar/4) {\cal S}\rho$, where $\rho$ is the density and ${\cal S}=2p\pm n$ is the "shift" in the spherical geometry. We discuss the role of modular invariance of the wave functions, of the center-of-mass momentum, and also of the lowest-Landau-level projection. Finally, we show that the Hall viscosity for $\nu={n\over 2pn+1}$ may be derived analytically from the microscopic wave functions, provided that the overall normalization factor satisfies a certain behavior in the thermodynamic limit. This derivation should be applicable to a class of states in the parton construction, which are products of integer quantum Hall states with magnetic fields pointing in the same direction.

The plethora of explicit solutions of the fractional KS equation through liquid\u2013gas bubbles mix under the thermodynamic conditions via Atangana\u2013Baleanu derivative operator

Novel explicit wave solutions are constructed for the Kudryashov–Sinelshchikov (KS) equation through liquid–gas bubbles mix under the thermodynamic conditions. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types of this model that is used to describe the phenomena of pressure waves through liquid–gas bubbles mix under the thermodynamic conditions. The stability property of the obtained solutions is tested to show the ability of our obtained solutions through the physical experiments. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical behavior of this model.

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

We study explicit model wave functions describing the fundamental quasiholes in a class of non-Abelian fractional quantum Hall states. This class is a family of paired spin-singlet states with internal degrees of freedom. We determine the braid statistics of the quasiholes by determining the monodromy of the explicit quasihole wave functions, that is how they transform under exchanges of quasihole coordinates. The statistics is shown to be the same as that of the quasiholes in the Read–Rezayi states, up to a phase. We also discuss the application of this result to a class of non-Abelian hierarchy wave functions.

Wave Digital Modeling of Nonlinear 3-terminal Devices for Virtual Analog Applications

We propose a novel modeling method for circuits containing arbitrary nonlinear 3-terminal devices, which operates in the wave digital (WD) domain. This approach leads to the definition of a general and flexible WD model for 3-terminal devices, whose number of ports varies from 1 to 6. The generality of the method is confirmed by the fact that the WD models of 3-terminal devices already discussed in the literature can be seen as particular cases of the model that we present here. As examples of applications of our method, we develop WD models of the three most widespread types of transistors in audio circuitry, i.e., the MOSFET, the JFET and the BJT. These models are here designed to be used in Virtual Analog audio applications; therefore, their derivation is aimed at minimizing computational complexity while avoiding implicit relations between port variables, as far as possible. Proposed MOSFET and JFET models are characterized by third-order polynomial equations; hence, explicit closed-form wave scattering relations are obtained. On the other hand, the Ebers–Moll model describing the BJT results in transcendental equations in the WD domain that cannot be solved analytically. In order to cope with this problem, we propose a modified Newton–Raphson (NR) method for solving the implicit Ebers–Moll equations in the WD domain. Such iterative method exhibits a significantly higher robustness and convergence rate with respect to the traditional NR method, without compromising its efficiency. Finally, WD implementations of some audio circuits containing transistors are discussed.

Exact explicit nonlinear wave solutions to a modified cKdV equation

In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.

BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS OF M-N-WANG EQUATION

By using the method of dynamical systems to Mikhailov-Novikov-Wang Equation, through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system of the derivative $\phi(\xi)$ of the wave function $\psi(\xi)$. Under different parameter conditions, for $\phi(\xi)$, exact explicit solitary wave solutions, periodic peakon and anti-peakon solutions are obtained. By integrating known $\phi(\xi)$, nine exact explicit traveling wave solutions of $\psi(\xi)$ are given.

Multiple residual symmetries and soliton-cnoidal wave interaction solution of the $$(2+1)$$-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff equation

The residual symmetry for the \((2+1)\)-dimensional negative-order modified Calogero–Bogoyavlenskii–Schiff (nmCBS) equation is derived from the truncated Painleve expansion, and is extended to the multiple residual symmetries, which can be transformed to Lie point symmetries by introducing a suitable prolonged system. The nth Backlund transformation (BT) related to multiple residual symmetries is given in terms of determinant. More importantly, we obtain the explicit soliton-cnoidal wave interaction solution from a consistent differential equation.

Analytical and Approximate Solutions for Complex Nonlinear Schrödinger Equation via Generalized Auxiliary Equation and Numerical Schemes

This article studies the performance of analytical, semi-analytical and numerical scheme on the complex nonlinear Schrödinger (NLS) equation. The generalized auxiliary equation method is surveyed to get the explicit wave solutions that are used to examine the semi-analytical and numerical solutions that are obtained by the Adomian decomposition method, and B-spline schemes (cubic, quantic, and septic). The complex NLS equation relates to many physical phenomena in different branches of science like a quantum state, fiber optics, and water waves. It describes the evolution of slowly varying packets of quasi-monochromatic waves, wave propagation, and the envelope of modulated wave groups, respectively. Moreover, it relates to Bose-Einstein condensates which is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Some of the obtained solutions are studied under specific conditions on the parameters to constitute and study the dynamical behavior of this model in two and three-dimensional.

Breaking wave solutions of a short wave model

In this paper, the breaking wave solutions of a short wave model is investigated by employing the bifurcation method of dynamical systems. All possible explicit breaking kink wave solutions and compacton solutions are obtained. Dynamical analysis reveals that the loop solutions actually contain two or more branches of breaking wave solutions. Some previous results are improved.

Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

We study two resonant Hamiltonian systems on the phase space L 2 ( R → C ) : the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation with harmonic trapping, a robust well-posedness theory, including global well-posedness in L 2 and all higher L 2 Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.

Baseband modulation instability, rogue waves and state transitions in a deformed Fokas–Lenells equation

We study a deformed Fokas–Lenells equation which is related to the integrable derivative nonlinear Schrodinger hierarchy with higher-order nonholonomic constraint. The baseband modulation instability as an origin of rogue waves is displayed. The explicit rogue wave solutions are obtained via the Darboux transformation. Typical rogue wave patterns such as the standard rogue wave, dark rogue wave and twisted rogue wave pair in three different components of the deformed Fokas–Lenells equation are presented. Besides, the state transitions between rogue waves and solitons are analytically found when the modulation instability growth rate tends to zero in the zero-frequency perturbation region. The explicit soliton solutions under the special parameter condition are given. The anti-dark and W-shaped solitons in their respective components are shown.

Exact Solutions for (2+1)-dimensional Nonlinear Schrödinger Equation Based on Modified Extended tanh Method

In this paper, modified extended tanh method is used to construct more general exact solutions of a (2+1)-dimensional nonlinear Schrödinger equation. With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and so on and their images.