A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

Nonlinear Dynamics - We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrödinger equation for the complex...     

Singular Limits of the Klein–Gordon Equation

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, ${\hbar\to 0}$ , we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, ${\hbar\to 0, c=\hbar^{-\alpha}\to \infty}$ for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.     

A Higdon-like non-reflecting boundary condition for the Klein–Gordon equation with evanescent waves

The non-reflecting boundary condition developed by Higdon and automated by Givoli and Neta is highly effective at absorbing propagating waves in a finite difference setting, but it does not absorb evanescent waves. In this paper, we augment the Higdon scheme with additional terms to absorb these evanescent waves in the context of the two-dimensional Klein–Gordon equation. Numerical examples illustrate the performance of this technique.     

Revisiting the inhomogeneously driven sine–Gordon equation

The sine–Gordonequation is a semilinear wave equation used to model many physical phenomenon like seismic events that includes earthquakes, slow slip and after-slip processes, dislocation in solids etc. Solution of homogeneous sine–Gordon equation exhibit soliton like structure that propagates without change in its shape and structure. The question whether solution of sine–Gordon equation still exhibit soliton like behavior under an external forcing has been challenging as it is extremely difficult to obtain an exact solution even under simple forcing like constant. In this study solution to an inhomogeneous sine–Gordon equation with Heaviside forcing function is analyzed. Various one-dimensional test cases like kink and breather with no flux and non-reflecting boundary conditions are studied.     

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.     

Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities

This paper obtains the conservation laws of the Klein–Gordon equation with power law and log law nonlinearities. The multiplier approach with Lie symmetry analysis is employed to obtain the conserved densities. The 1-soliton solutions are subsequently used to compute the conserved quantities from the conserved densities. Later the perturbation terms are added and the conservation laws of the perturbed Klein–Gordon equation are studied.     

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.     

On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

Nonlinear Dynamics - There are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a...     

Numerical study of the exact solution of the Navier–Stokes equations describing free-boundary fluid flow

This paper presents a numerical study of the partially invariant solution of the Navier–Stokes equations for the plane case which describes unsteady flow in a layer bounded by a straight solid wall and a free boundary parallel to it. It is found that for different initial flow velocities, a steady state can be established with a decrease or an increase in the initial layer thickness or the layer thickness can be increased infinitely due to fluid inflow from infinity.     

Riemann–Hilbert approach of the Newell-type long-wave–short-wave equation via the temporal-part spectral analysis

Nonlinear Dynamics - In this paper, the Riemann–Hilbert approach is systematically established for the Newell-type long-wave–short-wave equation. Firstly, we start spectral analysis...     

Numerical solutions of Schrödinger wave equation and Transport equation through Sinc collocation method

Nonlinear Dynamics - This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for...     

Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation

Nonlinear Dynamics - Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable...     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

Numerical simulation of shock–vortex interaction in Schardin’s problem

Shock diffraction over a two-dimensional wedge and subsequent shock–vortex interaction have been numerically simulated using the AUSM $+$ + scheme. After the passage of the incident shock over the wedge, the generated tip vortex interacts with a reflected shock. The resulting shock pattern has been captured well. It matches the existing experimental and numerical results reported in the literature. We solve the Navier–Stokes equations using high accuracy schemes and extend the existing results by focussing on the Kelvin–Helmholtz instability generated vortices which follow a spiral path to the vortex core and on their way interact with shock waves embedded within the vortex. Vortex detection algorithms have been used to visualize the spiral structure of the initial vortex and its final breakdown into a turbulent state. Plotting the dilatation field we notice a new source of diverging acoustic waves and a lambda shock at the wedge tip.     

Moment Lyapunov exponents of the stochastic parametrical Hill’s equation

The Lyapunov exponent and moment Lyapunov exponents of Hill’s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time-varying damping which are small intensity stochastic excitations.     

Bounded states for breathers–soliton and breathers of sine–Gordon equation

Nonlinear Dynamics - The Wronskian solutions to the sine–Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. And a novel expression of N-soliton...