Higher-order Soliton Solutions Research Articles

Higher-Order Soliton Solutions for the Derivative Nonlinear Schrödinger Equation via Improved Riemann–Hilbert Method

In this paper we discuss an improved Riemann–Hilbert method, by which arbitrary higher-order soliton solutions for the derivative nonlinear Schrödinger equation can be directly obtained. The explicit determinant form of a higher-order soliton which corresponds to one pth order pole is given. Besides the interaction related to one simple pole and the other one double pole is considered.

Higher-order soliton solutions for the Sasa–Satsuma equation revisited via bar{partial } method

In optics, the Sasa–Satsuma equation can be used to model ultrashort optical pulses. In this paper higher-order soliton solutions for the Sasa–Satsuma equation with zero boundary condition at infinity are analyzed by bar{partial } method. The explicit determinant form of a soliton solution which corresponds to a single p_{l}-th order pole is given. Besides the interaction related to one simple pole and the other one double pole is considered.

Diverse higher-order soliton solutions and novel hybrid behaviours of the $$(2+1)$$-dimensional KP-BBM equation

Diverse higher-order soliton solutions and novel hybrid behaviours of the $$(2+1)$$-dimensional KP-BBM equation

Bright soliton solutions to the nonlocal Manakov equations of reverse-space type

Utilizing the bilinear method combined with the KP hierarchy reduction technique, bright multi-soliton solutions in terms of determinant to the nonlocal Manakov equations of reverse-space type are derived in the focusing case. Our results show that these solutions only allow an even number of solitons and each paired soliton exhibits the head-on collision with the same velocity. In the dynamic behaviors illustrated, the fundamental paired soliton supports valley and hump at the center of collision, as well as one degenerate soliton with position shifts. The higher-order soliton solutions exhibit the interaction of paired solitons, where the particular bound states include the mixed line-breathing patterns and the single breather with position shifts.

General multi-soliton and higher-order soliton solutions for a novel nonlocal Lakshmanan–Porsezian–Daniel equation

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, a novel integrable nonlocal LPD equation corresponding to a \(3\times 3\) Lax pair is proposed. Secondly, the inverse scattering process with a novel left-right \(3\times 3\) matrix Riemann–Hilbert(RH) problem is constructed. The analytical properties and symmetry relations for the Jost functions and scattering data are considerably different from the local ones. Due to the special symmetry properties for the nonlocal LPD equation, the zeros of the RHP problem are purely imaginary or occur in pairs. With different types and configuration of zeros, the soliton formula is provided and the rich dynamical behaviors for the three kinds of multi-solitons for the novel nonlocal LPD equation are demonstrated. Third, by a technique of adding perturbed parameters and limiting process, the formula of higher-order solitons for the nonlocal LPD equation is exhibited. Lastly, the plots of diverse higher-order solitons and various solutions corresponding to different combinations of the following zeros: purely imaginary higher-order zeros, purely imaginary simple zeros, pairs of non-purely imaginary simple zeros and pairs of non-purely imaginary higher-order zeros are displayed.

Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrödinger equation

Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrödinger equation

Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type

Two types of multiple bright soliton solutions for a focusing nonlocal reverse-time nonlinear Schrodinger equation are derived by using the Hirota’s bilinear method and the Kadomtsev–Petviashvili hierarchy reduction. Compared with the case of the local Manakov system, both types of soliton solutions are restricted to ones with even numbers. For the first type of solution, the fundamental paired soliton exhibits two paralleled line solitons along with the time axis and the special breathing soliton. For the second type of solution, the fundamental paired soliton only allows head-on collisions with the same velocity, in which the collision with spatial interferences and the degenerate soliton with position shifts can be realized with certain parameters. General higher-order soliton solutions of two types describe the superposition of the fundamental paired soliton, which show a few interesting dynamical behaviors such as the superposed breathing solitons, the interaction of two paired soliton bound states, the interaction of two degenerate solitons with position shifts and the breathing soliton with position shifts.

Multiple bright soliton solutions of a reverse-space nonlocal nonlinear Schrödinger equation

Multiple bright soliton solutions in determinant form for a focusing nonlocal nonlinear Schrödinger (NLS) equation are derived via the bilinear method and the KP hierarchy reduction. It is shown that only multi-soliton solutions with even numbers are allowed and each paired soliton exhibits the head-on collision with the same velocity in the interaction process. One paired soliton describes the different collision patterns including the centering-hump, the centering-valley, and the spatial interference as well as a degenerate soliton with position shifts. Higher-order soliton solutions depict the interactions among different types of the paired soliton, in which a reduced four-soliton solution exhibits an interesting breathing soliton with position shifts.

Excitations of nonlinear local waves described by the sinh-Gordon equation with a variable coefficient

We explore novel excitations in the form of nonlinear local waves, which are described by the sinh-Gordon (SHG) equation with a variable coefficient. With the aid of the self-similarity transformation, we establish the relationship between solutions of the SHG equation with a variable coefficient and those of the standard SHG equation. Then, using the Hirota bilinear method, we obtain a more general bilinear form for the standard SHG equation and find new one- and two-soliton waves whose forms involve two arbitrary self-similarity functions. By an appropriate choice of the smooth self-similarity functions, we determine and display novel localized waves, and discuss their properties. The method used here can be extended to the three- and higher order soliton solutions.

Impulse response of nonlinear Schrödinger equation and its implications for pre-dispersed fiber-optic communication systems.

In the presence of pre-dispersion, an exact solution of nonlinear Schrödinger equation (NLSE) is derived for impulse input. The phase factor of the exact solution is obtained in a closed form using the exponential integral. The nonlinear interaction among periodically placed impulses launched at the input is investigated, and the condition under which these pulses do not exchange energy is examined. It is found that if the complex weights of the impulses at the input have a secant-hyperbolic envelope and a proper chirp factor, they will propagate over long distances without exchanging energy. To describe their interaction, a discrete version of NLSE is derived. The derived equation is a form of discrete self-trapping (DST) equation, which is found to admit fundamental and higher order soliton solutions in the presence of high pre-dispersion. Nonlinear eigenmodes derived here may be useful for description of signal propagation and nonlinear interaction in highly pre-dispersion fiber-optic systems.

Exact soliton solutions of coupled nonlinear Schrödinger equations: shape-changing collisions, logic gates, and partially coherent solitons.

The different dynamical features underlying soliton interactions in coupled nonlinear Schrödinger equations, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, by explicitly constructing multisoliton solutions (up to four-soliton solutions) for two-coupled and arbitrary N-coupled nonlinear Schrödinger equations using the Hirota bilinearization method, we bring out clearly the various features underlying the fascinating shape changing (intensity redistribution) collisions of solitons, including changes in amplitudes, phases and relative separation distances, and the very many possibilities of energy redistributions among the modes of solitons. However, in this multisoliton collision process the pairwise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons. Detailed asymptotic analysis also shows that when solitons undergo multiple collisions, there exists the exciting possibility of shape restoration of at least one soliton during interactions of more than two solitons represented by three- and higher-order soliton solutions. From an application point of view, we have shown from the asymptotic expressions how the amplitude (intensity) redistribution can be written as a generalized linear fractional transformation for the N-component case. Also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters, leading to possible multistate logic. In addition, we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape-changing collisions, thereby explaining their variable profile and shape variation in collision process.

Nonlinear effects in optical fibers

Abstract The basic differential equations necessary to represent nonlinear propagation of short and ultrashort optical pulses in dielectric waveguides are derived. After introducing fundamental and higher-order soliton solutions of the nonlinear Schroedinger equations, the possibilities of realizing soliton-based communication systems are discussed. Higher-order nonlinear effects associated with selj-Raman gain and optical amplification in connection with the use of active fibers are also described.