Self-similar Waves Research Articles

Spatial similaritons in the (2+1)-dimensional inhomogeneous cubic-quintic nonlinear Schrödinger equation

Along the idea of the similarity transformation, analytical spatial similaritons to a (2+1)-dimensional inhomogeneous cubic-quintic nonlinear Schrödinger equation with distributed diffraction and gain are derived when some certain compatibility conditions are satisfied. Based on these exact solutions, we investigate dynamic behaviors of self-similar cnoidal waves and chirped similaritons in the hyperbolically and Gaussian decreasing diffraction waveguides.

Numerical solutions to shock reflection and shock interaction problems for the self-similar transonic two-dimensional nonlinear wave systems

We discuss numerical results for two dimensional Riemann problems for the nonlinear wave system. We consider four sectorial Riemann data and present the numerical results for two configurations. The first configuration is that incident shocks create reflected shocks, and the state behind the reflected shocks becomes subsonic. The second configuration is an interaction between two transonic shocks; one is created by an interaction with the sonic circle, and the other is created by the shock reflection. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the reflected shock strength for the first configuration. Furthermore we present intriguing numerical results on interactions of two transonic shocks for the second configuration.

Controllable rogue waves in the nonautonomous nonlinear system with a linear potential

Based on the similarity transformation connected the nonautonomous nonlinear Schrodinger equation with the autonomous nonlinear Schrodinger equation, we firstly derive self-similar rogue wave solutions (rational solutions) for the nonautonomous nonlinear system with a linear potential. Then, we investigate the controllable behaviors of one-rogue wave, two-rogue wave and rogue wave triplets in a soliton control system. Our results demonstrate that the propagation behaviors of rogue waves, including postpone, sustainment, recurrence and annihilation, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z m and the parameter Z 0. Moreover, the excitation time of controllable rogue waves is decided by the parameter T 0.

The manipulation of optical rogue waves for the nonautonomous nonlinear Schrödinger equation

We obtain analytical solutions that describe self-similar rogue waves for the (1+1)-dimensional nonautonomous nonlinear Schrodinger equation with a linear potential. The explicit functions that describe the evolution of the width, peak, and phase are found exactly, from which it can be seen that the gain–loss parameter has no effect on the motion of rogue waves’ phases or their widths, and affects only the evolution of their peaks. We also find that the manipulation of rogue waves can be realized by modulating the values of the maximum accumulated time, Tm, and the accumulated time, T0, with the maximum amplitude of the rogue waves. The controllability for the type of excitation, such as sustainment, recurrence, and annihilation for rogue waves and the transverse position of excitation is exhibited.

An Interaction of a Rarefaction Wave and a Transonic Shock for the Self-Similar Two-Dimensional Nonlinear Wave System

We study a two dimensional Riemann problem for the self-similar nonlinear wave system which gives rise to an interaction of a transonic shock and a rarefaction wave. The interesting feature of this problem is that the governing equation changes its type from supersonic in the far field to subsonic near the origin. The subsonic region is then bounded above by the sonic line (degenerate) and below by the transonic shock (free boundary). Furthermore due to the rarefaction wave in the downstream, which interacts with the transonic shock, the problem becomes inhomogeneous and degenerate. We establish the existence result of the global solution to this configuration, and present analysis to understand the solution structure of this problem.

Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system

A similarity transformation connecting the variable coefficient nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation is constructed. The self-similar rogue wave triplet solutions (rational solutions) are analytically obtained for the nonautonomous nonlinear and dispersive system. The controllable behaviours of rogue wave triplets in two typical soliton management systems are discussed. In the exponential dispersion decreasing fibre, three kinds of rogue wave triplets with controllable behaviours are analysed. In the periodic distributed system, the rogue wave triplets recur periodically in the form of a cluster.

The Guderley problem revisited

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of considerable mathematical and physical interest. We investigate a novel application of the Guderley solution as a unique and challenging code verification test problem for compressible flow algorithms; this effort requires a unified understanding of the problem's mathematical and computational subtleties. Hence, we review the simplifications and group invariance properties that reduce the compressible flow equations for a polytropic gas to two coupled nonlinear eigenvalue problems: the first for the similarity exponent in the converging regime, and the second for a trajectory multiplier in the diverging regime. The information we provide, together with previously published material, gives a complete description of the computational steps required to construct a semi-analytic Guderley solution. We employ the problem in a quantitative code verification analysis of a cell-centred, finite volume, Eulerian compressible flow algorithm. Lastly, in appended material, we introduce a new approximation for the similarity exponent, which may prove useful in the future construction of certain semi-analytic Guderley solutions.

General relativistic self-similar waves that induce an anomalous acceleration into the standard model of cosmology

The authors prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical ($k=0$) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, they prove that the family reduces to an implicitly defined one-parameter family of distinct spacetimes determined by the value of a new acceleration parameter $a$, such that $a=1$ corresponds to the Standard Model. The authors prove that all of the self-similar spacetimes in the family are distinct from the non-critical $k\neq0$ Friedmann spacetimes, thereby characterizing the critical $k=0$ Friedmann universe as the unique spacetime lying at the intersection of these two one-parameter families. They then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, they use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center.

The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation

We present, analytically, self-similar rogue wave solutions (rational solutions) of the inhomogeneous nonlinear Schrödinger equation (NLSE) via a similarity transformation connected with the standard NLSE. Then we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated by the exponential parameter σ . Finally, we investigate the nonlinear tunneling effect for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or decreasing amplitudes via the modulation of the ratio of the amplitudes of rogue waves to the barrier or well height.

Exact self-similar wave solutions for the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schröinger equation with distributed coefficients

In this paper, a similarity transformation is presented to reduce the generalized (3 + 1)-dimensional cubic–quintic nonlinear Schrödinger equation with distributed coefficients to the related constant-coefficients one. Then a number of spatiotemporal self-similar wave solutions are constructed. Under the specific choice of the dispersion, cubic and quintic nonlinearities, phase modulation and the gain/loss, we investigate the dynamical behaviors of those spatiotemporal self-similar waves in an inhomogeneous optical fiber media.

Self-similar spatiotemporal solitary waves in Bessel—Hermite optical lattices

An exact self-similar solution to a (3+1)-dimensional nonlinear Schrödinger equation with gain in the Bessel-Hermite lattice is obtained analytically. The stability of the analytical solution is confirmed by using numerical simulation. It is shown that the light bullet has a stable ellipsoid or vortex profile and a linear spatiotemporal chirp.

Solitary Waves in Spherical Bessel Lattice

The analytical solitary waves solution to (3+1)-dimensional Gross-Pitaevskii equation with varying coefficients and an spherical Bessel lattice potential is obtained, which is in the form of self-similar solitary waves. The properties and the stability of the solitary waves family are discussed in detail.

Optical similaritons in a tapered graded-index nonlinear-fiber amplifier with an external source

We analytically explore a wide class of optical similariton solutions to the nonlinear Schr\"odinger equation appropriately modified to model beam propagation in a tapered, graded-index nonlinear-fiber amplifier with an external source. Under certain physical conditions, we reduce the coupled nonlinear Schr\"odinger equations to a single-wave equation that aptly describes similariton propagation through asymmetric twin-core fiber amplifiers. The asymmetric twin-core fiber is composed of two adjoining, closely spaced, single-mode fibers in which the active one is a tapered, graded-index nonlinear-fiber and the passive one is a step-index fiber. We obtain these self-similar waves for different choices of tapered index profile, using a M\"obius transformation. Our procedure is applicable for both self-focusing and self-defocusing nonlinearities.

Chirped Wave Solutions of a Generalized (3+1)-Dimensional Nonlinear Schr¨odinger Equation

The exact chirped soliton-like and quasi-periodic wave solutions of the (3+1)-dimensional generalized nonlinear Schrödinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detailedly in this paper. The form and the behaviour of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain (loss)

Spatiotemporal Similaritons in (3+1)-Dimensional Inhomogeneous Nonlinear Medium with Cubic-Quintic Nonlinearity

We obtain exact spatiotemporal similaritons to a (3+1)-dimensional inhomogeneous nonlinear Schrödinger equation, which describes the propagation of optical pulses in a cubic-quintic nonlinearity medium with distributed dispersion and gain. A one-to-one correspondence between such self-similar waves and solutions of the elliptic equation is found when a certain compatibility condition is satisfied. Based on exact solutions, we discuss evolutional behaviors of self-similar cnoidal waves and chirped similaritons in two kind of typical soliton control systems. Moreover, the comparison between chirped similaritons and chirp-free solitons is given.

Self-similar solitary wave family in Bessel lattice

We focus on the formation and propagation of self-similar solitary wave family in Kerr nonlinear media with external Bessel lattice. A novel analytical solitary wave solution to (3+1)-dimensional Gross-Pitaevskii equation with varying coefficients and an external potential is obtained. The components of solitary wave family are differentiated by three quantum parameters. The properties and the stability of the solitary wave family are discussed in detail.

Self-similar optical pulses tunneling in cubic-quintic nonlinear media with distributed coefficients

By means of universal similarity transformation, we derive the explicit spatial self-similar bright and dark soliton solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients and an external potential. An exact balance condition between the dispersion, nonlinearity, beam width and the gain/loss has been obtained. Under this parametric condition, we investigate the nonlinear tunneling effect of such solutions. We find that the amplitude and width of the self-similar waves are controllable while tunneling through the barriers and wells.

Chirped Waves for a Generalized (2 + 1)-Dimensional Nonlinear Schrödinger Equation

The exact chirped soliton-like and quasi-periodic wave solutions of (2 + 1)-dimensional generalized nonlinear Schrödinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detailedly in this paper. The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain (loss).

Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

In this note we propose a new set of coordinates to study the hyperbolic or nonelliptic cubic nonlinear Schrödinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.

Self-similar magnetic structures during the vortex-glass to vortex-liquid transition of type II superconductors

We examine the response to an external magnetic field by a multi-layer superconductor with an electrical resistance ρff(b)αbσ, where b is the dimensionless magnetic induction and σ is a parameter characterizing the ratio of the pinning activation energy to the energy of thermal fluctuations. When σ > 1 the sample is in the vortex glass phase, when 0 < σ < 1, it is in the vortex liquid phase, and a vortex glass to vortex liquid phase transition takes place at σ = 1. In the vortex glass phase, the magnetic field penetrates into the superconductor in the form of a self-similar wave. At all times it penetrates to a finite depth and its front moves at a finite velocity which depends on the parameters of the problem, such as the rate of pumping by the external magnetic field. In the vortex liquid phase the magnetic field penetrates to an infinite depth. Thus, the magnetic field penetrates to an infinite depth in the superconductor during a transition from the vortex glass phase into the vortex liquid phase.