Bifurcation Analysis and Single Traveling Wave Solutions of the Variable-Coefficient Davey–Stewartson System

This paper mainly studies the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system. By employing the traveling wave transformation, the variable-coefficient Davey–Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations. On the one hand, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. On the other hand, we use the polynomial complete discriminant method to obtain the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.

The Method for Evaluating Cross-Shore Migration of Sand Bar under the Influence of Nonlinear Waves Transformation

Sand bar migration on the gently sloping sandy bottom in the coastal zone as a result of nonlinear wave transformation and corresponding sediment transport is discussed. Wave transformation on the intermediate depth causes periodic exchange of energy in space between the first and the second wave harmonics, accompanied by changes in the wave profile asymmetry. This leads to the occurrence of periodical fluctuations in the wave-induced sediment transport. It is shown that the position of the second nonlinear wave harmonic maximum determines location of the divergence point of sediment transport on the inclined bottom profile, where it changes direction from the onshore to the offshore. Such sediment transport pattern leads to formation of an underwater sand bar. A method is proposed to predict the position of the bar on an underwater slope after a storm based on calculation of the position of the maximum amplitude of the second nonlinear harmonic. The method is validated on the base of field measurements and ERA 5 reanalysis wave data.

An Efficient Method for Analytically Solving (1+2)-dimensional Chiral Non-linear Schrödinger Equation

In this paper, we have successfully extracted novel analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger (NLS) equation by modified extended tanh expansion method combined with new Riccati solutions (METEM-cNRCS) as far as we know. When a wave transformation is applied to the considered Chiral NLS equation, a nonlinear ODE is obtained. Assuming the solutions of ODE have a form as the method suggests, and substituting the trial solutions to the ODE, we get a polynomial. Gathering the coefficients with the same power in the polynomial, we acquire an algebraic equation system. So, we may obtain the abundant solutions of the (1+2)-dimensional Chiral NLS equation by solving the system via Maple. The plots of some solutions are demonstrated to explain the dynamics of the solutions. It is expected that the results of the paper are a guide for future works in traveling wave theory.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

New Optical Soliton Solutions for Coupled Resonant Davey-Stewartson System with Conformable Operator

This paper investigates the novel soliton solutions of the coupled fractional system of the resonant Davey-Stewartson equations. The fractional derivatives are considered in terms of conformable sense. Accordingly, we utilize a complex traveling wave transformation to reduce the proposed system to an integer-order system of ordinary differential equations. The phase portrait and the equilibria of the obtained integer-order ordinary differential system will be studied. Using suitable mathematical assumptions, the new types of bright, singular, and dark soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing system. To achieve this, illustrative examples of the fractional Davey-Stewartson system are provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of conformable derivatives on behavior of acquired solutions for different fractional orders is also discussed. By comparing the proposed method with the other existing methods, the results show that the execute of this method is concise, simple, and straightforward. The results are useful for obtaining and explaining some new soliton phenomena.

Application of spectral analysis for determine the phase shift of signals with equal amplitudes using full-wave transformation when measuring the characteristics of weapons

The article defines the list of technical characteristics of armaments and military equipment (ARM), the value of which is measured using phase methods. An analysis of known methods that have found wide application in measuring technology, which is designed to determine the technical characteristics associated with the measurement of phase shift during the development, manufacture and operation of weapons. Based on this analysis, it was determined that the measuring systems are designed to determine the phase shift of two harmonic signals in their composition have two channels of information transmission. This architecture of the implementation of measuring systems leads to the fact that a significant impact on the accuracy of the proposed measurement problem, makes a component of the error due to the phase symmetry of the signal transmission channels, as well as internal and external noise. As an alternative approach to solving the measurement problem of determining the phase shift of two harmonic signals, which will significantly reduce the error component due to phase asymmetry of information transmission channels, it is proposed to use the signal obtained by summing harmonic signals after full-wave transformation followed by spectral analysis. In order to implement the above approach, a measurement problem was set to determine the phase shift of two harmonic signals, using spectral analysis of the signal obtained by summing the harmonic signals after their full-wave transformation. A list of assumptions required for the synthesis of analytical relations that establish the relationship between the spectra of phases and amplitudes (power) of the signal obtained by summing harmonic signals after their full-wave transformation and phase shift of two harmonic signals. Analytical relationships are proposed that establish the relationship between the above characteristics. It is shown that the values of the spectrum of phases and amplitudes, which are calculated using the proposed expressions, differ from the values obtained in the calculations using the Fourier series coefficients, not more than 0.1%.

The shock wave solutions of modified ZK Burgers equation in inhomogeneous dusty plasmas

This paper offers a shock wave solution to modified Zakharov–Kuznetsov (MZK) Burgers equation in inhomogeneous dusty plasmas with external magnetic field. For this purpose, the fluid equations are reduced to an MZK Burgers equation containing variable coefficients by reductive perturbation method. With the aid of travelling-wave transformation technique, we obtain the analytical oscillatory shock wave solution and monotonic shock wave solution for MZK Burgers equation. The effects of inhomogeneity, external magnetic field, dust charge variation on characteristics of two types of shock waves are examined in detail.

Analysis of light-wave nonstaticity in the coherent state

The characteristics of nonstatic quantum light waves in the coherent state in a static environment is investigated. It is shown that the shape of the wave varies periodically as a manifestation of its peculiar properties of nonstaticity like the case of the Fock-state analysis for a nonstatic wave. A belly occurs in the graphic of wave evolution whenever the wave is maximally displaced in the quadrature space, whereas a node takes place every time the wave passes the equilibrium point during its oscillation. In this way, a belly and a node appear in turn successively. Whereas this change of wave profile is accompanied by the periodic variation of electric and magnetic energies, the total energy is conserved. The fluctuations of quadratures also vary in a regular manner according to the wave transformation in time. While the resultant time-varying uncertainty product is always larger than (or, at least, equal to) its quantum-mechanically allowed minimal value ($$\hbar /2$$ ħ / 2 ), it is smallest whenever the wave constitutes a belly or a node. The mechanism underlying the abnormal features of nonstatic light waves demonstrated here can be interpreted by the rotation of the squeezed-shape contour of the Wigner distribution function in phase space.

Exact Solutions and Dynamic Properties of Perturbed Nonlinear Schrodinger Equation Arised From Nano Optical Fibers

The main idea of this paper is to investigate the exact solutions and dynamic properties in optical nanofibers, which is modeled by space-time fractional perturbed nonlinear schr\"odinger equation involving Kerr law nonlinearity with conformal fractional derivative. Firstly, by the complex fractional traveling wave transformation, the traveling wave system of the original equation is obtained, then a conserved quantity, namely the Hamiltonian is constructed, and the qualitative analysis of this system is conducted via this quantity by classifying the equilibrium points. Moreover, the prior estimate of the existence of the soliton and periodic solution is established via the bifurcation method. Furthermore, all exact traveling wave solutions are constructed to illustrate our results explicitly by the complete discrimination system for polynomial method.

Wave transformation around breakwater (case study: tourism harbour, Eastern Bali, Indonesia)

An essential aspect in the sustainable design of breakwater is the determination of the design wave condition. It is predicted by utilizing severe wave conditions of the past 10 to 20 years. The tourism harbor at eastern Bali, Indonesia, is located where extreme wave condition occurs. Therefore, this research studies the wave height before and after constructing a breakwater in the harbor area. The wave height was simulated using numerical modeling. The methodology was performed by using the coastal modeling software of the SMS-CGWAVE model. The result shows the highest design wave height value of 3.9 m in the direction from the southeast. The design breakwater can reduce wave height up to 0.9 m or a 75.5% reduction. Further study is needed to simulate the extension of breakwater length to meet the criterion design of wave height in the harbor basin.