Self-localized Solitons Research Articles

Self-localized solitons of the nonlinear wave blocking problem

In this paper, we propose a numerical framework to study the shapes, dynamics, and stabilities of the self-localized solutions of the nonlinear wave blocking problem. To our best knowledge, blocking of the solitons in the marine environment has not been studied before in the existing literature. With this motivation, we use the nonlinear Schrödinger equation (NLSE) derived by Smith as a model for the nonlinear wave blocking. We propose a spectral renormalization method (SRM) to find the self-localized solitons of this model. We show that for constant, linearly varying or sinusoidal current gradient, i.e. dU∕dx, the self-localized solitons of the Smith’s NLSE do exist. Additionally, we propose a spectral scheme with a 4th order Runge–Kutta time integrator to study the temporal dynamics and stabilities of such solitons. We observe that self-localized solitons are stable for the cases of constant or linearly varying current gradient however, they are unstable for sinusoidal current gradient, at least for the selected parameters. We comment on our findings and discuss the importance and applicability of the proposed approach.

Self-localized solitons of a q-deformed quantum system

Beyond a pure mathematical interest, q-deformation is promising for the modeling and interpretation of various physical phenomena. In this paper, we numerically investigate the existence and properties of the self-localized soliton solutions of the nonlinear Schrödinger equation (NLSE) with a q-deformed Rosen–Morse potential. By implementing a Petviashvili method (PM), we obtain the self-localized one and two soliton solutions of the NLSE with a q-deformed Rosen–Morse potential. In order to investigate the temporal behavior and stabilities of these solitons, we implement a Fourier spectral method with a 4th order Runge–Kutta time integrator. We observe that the self-localized one and two solitons are stable and remain bounded with a pulsating behavior and minor changes in the sidelobes of the soliton waveform. Additionally, we investigate the stability and robustness of these solitons under noisy perturbations. A sinusoidal monochromatic wave field modeled within the frame of the NLSE with a q-deformed Rosen–Morse potential turns into a chaotic wavefield and exhibits rogue oscillations due to modulation instability triggered by noise, however, the self-localized solitons of the NLSE with a q-deformed Rosen–Morse potential are stable and robust under the effect of noise. We also show that soliton profiles can be reconstructed after a denoising process performed using a Savitzky–Golay filter.

Self-localized solutions of the Kundu-Eckhaus equation in nonlinear waveguides

Abstract In this paper we numerically analyze the 1D self-localized solutions of the Kundu-Eckhaus equation (KEE) in nonlinear waveguides using the spectral renormalization method (SRM) and compare our findings with those solutions of the nonlinear Schrodinger equation (NLSE). For cubic-quintic nonlinearity with Raman effect, as a benchmark problem we numerically construct single, dual and N-soliton solutions for the zero optical potential, i.e. V = 0 , which are analytically derived before. We show that self-localized soliton solutions of the KEE with cubic-quintic nonlinearity and Raman effect do exist, at least for a range of parameters, for the photorefractive lattices with optical potentials in the form of V = I o cos 2 ( x ) . Additionally, we also show that self-localized soliton solutions of the KEE with saturable cubic-quintic nonlinearity and Raman effect do also exist for some range of parameters. However, for all of the cases considered, these self-localized solitons are found to be unstable. We compare our findings for the KEE with their NLSE analogs and discuss our results.

Controlling Bending and Twisting of Conjugated Polymers via Solitons

A strong generic coupling between self-localized solitons and conformations of a pi-conjugated polymer chain is demonstrated through ab initio calculations and the underlying mechanisms revealed by using an extended Hubbard model. We show that significant chain bending and twisting of trans- and cis-polyacetylene result from torque imbalances among atom-centered orbitals and the steric instability at single-bond cis-centers, respectively. The soliton-induced conformational effects create sufficient strains to provide an intrinsic high-strain-rate actuation mechanism in optical excitation processes.

Quantal properties of self-localized solitons in anharmonic oscillatory systems

We present a special unitary transformation in which the parameters are deduced from the classical amplitude pattern of the self-localized (solitary) mode (SLS). By means of this transformation the Hamiltonian is transcribed in such a way that the SLS is represented by a single anharmonic oscillator, and the quantal correspondence to the solitary motion appears to be the quasi-coherent state of this anharmonic oscillator.

Stepwise quantum decay of self-localized solitons

The two-phonon decay of self-localized solitons in a one-dimensional monatomic anharmonic lattice caused by cubic anharmonicity is considered. It is shown that the decay takes place with emission of phonon bursts. The average rate of emission of phonons is of the order of the vibrational quantum per vibrational period. The characteristic relaxation time is determined by the quantum anharmonicity parameter; this time may vary from a few (quantum lattices, large anharmonicity) to thousands (ordinary lattices, small anharmonicity) of vibrational periods.

Breathing self-localized solitons in the quartic Fermi-Pasta-Ulam chain.

If the fundamental self-localized soliton (SLS) of the Fermi-Pasta-Ulam chain is subjected to a perturbation of the same parity, a breathing behavior in space is observed. The time evolution then is characterized by two different frequencies. We show that the observed breathing behavior can be explained by means of a harmonic model with an effective spring hardening which is generated by the fundamental background SLS. The validity of this harmonic model is verified by means of numerical simulations. Improvements involving a parametric oscillator are mentioned but left to future study. \textcopyright{} 1996 The American Physical Society.

A Gauss procedure for the construction of localized solitons in anharmonic chains

Recently, in the Fermi-Pasta-Ulam chain stable self-localized solitons (SLS) have been found. In the present investigation a construction procedure for SLS solutions is presented. It is based on two ingredients: (a) the exploitation of the harmonic wing property and (b) a Gaussian step-by-step optimization starting from the wing. The procedure allows for highly accurate approximate solutions. The temporal stability of the solutions is verified.

Gauss procedure for the construction of self-localized solitons in discrete systems.

Gauss procedure for the construction of self-localized solitons in discrete systems.

Generation of supersonic and subsonic solitons in the Fermi-Pasta-Ulam chain

Abstract The dynamics of the Fermi-Pasta-Ulam chain with quartic anharmonicity are numerically investigated by means of a stimulating force of Gaussian time shape, which acts on a single atom. Depending on the strength and duration of this force, several phenomena have been observed. A pulse of originally continuous envelope disintegrates into solitary objects (‘multifurcative process’), which may be ultrasonic solitons of kink type, or subsonic and self-localized solitons of oscillating nature. In the oscillations of the latter two, frequencies are involved, above the Debye frequency, and the spatial envelope displays a ‘breathing’ behaviour. Some analytical background is also presented.

Microwave gap solitons and bistability in magnetostatic periodic structures

The existence of bistability, gap solitons, and self-localized solitons in a magnetostatic periodic structure are predicted. The very low threshold power necessary for the observation of such phenomenas at microwave frequencies makes these structures very attractive for microwave signal processing.