Optical solitons and bifurcation analysis in fiber Bragg gratings with Lie symmetry and Kudryashov’s approach

A combination of Lie symmetry analysis and Kudryashov’s approach secures optical soliton solutions with fiber Bragg gratings. The bifurcation analysis is carried out, and the phase portrait is presented.

     

Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation

Nonlinear Dynamics - Multiple bright–dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schrödinger equation are constructed by using the...     

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

Nonlinear Dynamics - We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero...     

Optical solitons for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution

Nonlinear Dynamics - This paper is devoted in the study of the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution which...     

Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber

Nonlinear Dynamics - The multi-soliton solutions and breathers to the coupled higher-order nonlinear Schrödinger (CH-NLS) equation are derived in this work via the Riemann–Hilbert...     

Water hammer prediction and control: the Green’s function method

By Green’s function method we show that the water hammer (WH) can be analytically predicted for both laminar and turbulent flows (for the latter,with an eddy viscosity depending solely on the space coordinates),and thus its hazardous effect can be rationally controlled and minimized.To this end,we generalize a laminar water hammer equation of Wang et al.(J.Hydrodynamics,B2,51,1995) to include arbitrary initial condition and variable viscosity,and obtain its solution by Green’s function method.The predicted characteristic WH behaviors by the solutions are in excellent agreement with both direct numerical simulation of the original governing equations and,by adjusting the eddy viscosity coefficient,experimentally measured turbulent flow data.Optimal WH control principle is thereby constructed and demonstrated.     

Scalar and vector multipole and vortex solitons in the spatially modulated cubic–quintic nonlinear media

We derive scalar and vector multipole and vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media, which is governed by a (3+1)-dimensional N-coupled cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. If the modulation depth \(q=1\), the vortex soliton is constructed, and if \(q=0\), the multipole soliton, including dipole, quadrupole, hexapole, octopole and dodecagon solitons, is constructed, respectively, when the topological charge \(k=1\)–5. If the topological charge \(k=0\), scalar solitons can be obtained. Moreover, the number of layers for the scalar and vector multipole and vortex solitons is decided by the value of the soliton order number n.     

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

Meccanica - In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is...     

Spatiotemporal vector and scalar solitons of the coupled nonlinear Schrödinger equation with spatially modulated cubic–quintic–septimal nonlinearities

Nonlinear Dynamics - The spatially modulated cubic–quintic–septimal nonlinearities and transverse modulation are introduced to study the impact on a...     

Numerical solution of the Klein–Gordon equation via He’s variational iteration method

In this paper, we present the solution of the Klein--Gordon equation. Klein--Gordon equation is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. The He’s variational iteration method (VIM) is implemented to give approximate and analytical solutions for this equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique reduces the volume of calculations by avoiding discretization of the variables, linearization or small perturbations.     

One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation

Nonlinear Dynamics - The main concern of the present article is to study the fifth-order variable-coefficient Sawada–Kotera (VcSK) equation which describes the motion of long waves in shallow...     

Study of pure transverse motion in free cylinders and plates in flexural vibration by Ritz’s method

In this work the flexural vibration of a free cylinder of any aspect ratio is analysed. A general solution by powers series of the coordinates is proposed here to represent the displacements, with restrictions on the powers of the radial coordinates which prevent potential energy and stress singularities at the axis of the cylinder. By means of an analytic method, it is concluded that certain points of the cylinder have no axial motion. As a result of the pure transverse movement and of the fact that the cylinder bends, it is inferred that the axis is extended. Furthermore, in the symmetric modes, the points situated at the centres of the bases are displaced in the same direction and sense, and hence the distance between them does not vary in time. Flexural natural frequencies are numerically calculated by Ritz’s method with the general solution series proposed. Since the series used are more adequate, convergence is better than with classic series. The results are verified by FEM. Some consequences are extended to a rectangular plate, whose points of the middle surface vibrate transversally in the double-symmetric mode. In order to verify the theoretical results, a set of experiments with a laser interferometer is carried out. The experimental frequencies agree with the theoretical values.     

Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams

Dirac’s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems, leading very often to elegant and otherwise unworkable closed-form solutions. This is the case of cracked beams under static loads, whose theoretical and practical significance has attracted in recent years the interest of many researchers. Nevertheless, analytical formulations currently available for this problem are not completely satisfactory, either in terms of computational efficiency, when the continuity conditions must be enforced with auxiliary equations, or in terms of physical consistency, when the singularities in the beam’s flexural rigidity are represented with Dirac’s delta functions having a questionable negative sign. These considerations motivate the present study, which offers a novel and physically-based modelling of slender Euler–Bernoulli beams and short Timoshenko beams with any number and severity of cracks, conducing in both cases to exact closed-form solutions. For validation purposes, a standard finite element code is used, along with two nascent deltas (uniform and Gaussian density functions) to describe a smeared increase in the bending flexibility around the abscissa of the crack.