Breather Soliton Solutions Research Articles

Ultrashort light pulses generated from modulation instability: background removal and soliton content

Modulation instability can be used to convert a continuous light wave into a train of pulses on a constant background. It is a longstanding discussion whether these pulses can be converted into solitons. We clarify the situation by using a more general mathematical context, invoking the Akhmediev breather, Peregrine soliton and Kuznetsov-Ma soliton solutions of the wave equation, and suggest the use of a Mach–Zehnder interferometer to remove the background. Expressions for the pulse widths and peak powers thus obtained are presented, and their soliton content is determined. It turns out that more than 95 % of each pulse’s energy can be converted to a soliton.

The Multisoliton Solutions for the (2+1)-Dimensional Sawada-Kotera Equation

Applying bilinear form and extended three-wavetype of ansätz approach on the (2+1)-dimensional Sawada-Kotera equation, we obtain new multisoliton solutions, including the double periodic-type three-wave solutions, the breather two-soliton solutions, the double breather soliton solutions, and the three-solitary solutions. These results show that the high-dimensional nonlinear evolution equation has rich dynamical behavior.

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

With the aid of the known Backlund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained.

Klein-Gordon equation approach to nonlinear split-ring resonator based metamaterials: One-dimensional systems

The electrodynamics of a one-dimensional split-ring resonator (SRR) based nonlinear metamaterial is studied. The metamaterial is a one-dimensional periodic array of weakly coupled SRRs, with each SRR represented by a nonlinear resistor-inductor-capacitor (RLC) equivalent resonator circuit. Nonlinearity is introduced into the system by the addition of Kerr-type dielectric medium within the SRRs or by the introduction into the system of certain other nonlinear elements (e.g. diodes). In the continuum limit of the system, variations of the charge stored within the capacitive slits of the SRRs in both time and space are shown to be described along the array by a nonlinear Klein-Gordon equation. Analytical solutions of the nonlinear Klein-Gordon equation for various dark and bright envelope, breather, and pulse soliton solutions are obtained and studied. A discussion is given of the relationship between the Klein-Gordon solutions and the solutions of the nonlinear Schr\odinger equation approximation to the nonlinear Klein-Gordon equation. A comparison is made of the Klein-Gordon solutions with intrinsic localized mode (discrete breather) solutions of the discrete system and their continuum limits. An additional continuum limit differential equation for the breather modes of the system is obtained which is not bound by a weak coupling assumption, and its relation to the Klein-Gordon equation is studied. Analytic forms are given for the effects of dissipation in the system on the various bright and dark envelope, breather, and pulse solitons. Discussions are given of the effects of further than first neighbor couplings in the SRR system.

Extended Mixed Function Method and Its Application for Solving Two Classic Toda Lattice Equations

The mixed function method is extended from the (1 + 1)‐dimensional space to the (2 + 1)‐dimensional one, even those forms of exact solution do not exist in (1 + 1)‐dimensional NDDEs. By using this extended method, the Toda lattice and (2 + 1)‐dimensional Toda lattice equations are studied. Some new solutions such as discrete solitary wave solutions, discrete kink and antikink wave solutions, and discrete breather soliton solutions are obtained, and their dynamic properties are discussed.

Double periodic wave solutions and breather-soliton solutions of the (n+1)-dimensional sinh-Gordon equation

In this paper, using a new method named binary F-expansion method and a simple transformation technique, we discussed the (n+1)-dimensional sinh-Gordon equation. Many double periodic wave solutions and breather-soliton solutions are obtained.

Chaos, solitons and fractals in hidden symmetry models

Abstract A spontaneous symmetry breaking (or hidden symmetry) model is reduced to a system nonlinear evolution equations integrable via an appropriate change of variables, by means of the asymptotic perturbation (AP) method, based on spatio-temporal rescaling and Fourier expansion. It is demonstrated the existence of coherent solutions as well as chaotic and fractal patterns, due to the possibility of selecting appropriately some arbitrary functions. Dromion, lump, breather, instanton and ring soliton solutions are derived and the interaction between these coherent solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one can construct lower dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.

Nonlinear dynamics of solitary waves in an extensible rod

In 1991, Goldstein and Petrich showed that the dynamics of a curve in a plane is governed geometrically by the modified KdV (mKdV) hierarchy. In this paper, we show a way of utilizing their new tool of the soliton approach to analyse the nonlinear dynamics of elastic rods. Considering the stretching effect of a narrow rod, nonlinear dynamics is studied by using exact soliton solutions of the mKdV equation, such as one soliton and breather soliton solutions. The metric of the rod which represents the stretching effect is explicitly given by perturbation analyses, and it is shown that an initial tension of the rod plays an important role in forming solitons in the rod.