Defocusing complex short-pulse equation and its multi-dark-soliton solution.

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

We showed recently [1] that a single soliton solution of the highly-nonlinear Schrödinger equation becomes multi-stable for a certain class of nonlinearities. This implies that more than one amplitude profile and propagation constant of a single soliton may exist for the same total power carried by the soliton. The multistable solitons exist only if the nonlinear component of the susceptibility as a function of intensity is either changing its sign or its derivative has a sufficiently sharp peak. This can result in such effects as bistable (or multistable, in general) self-trapping of light in nonlinear media as well as bistable propagation of soliton pulses in nonlinear optical fibers. Both of these processes may be described by the same nonlinear Schrödinger equation with a nonlinear term in the form Ef(|E|2) where f(|E|2) is an arbitrary function of the field intensity |E|2 with f(0) = 0. The so called cubic nonlinear Schrödinger equation with f(|E|2) proportional |E|2, corresponds to Kerr-nonlinearity in optical propagation and does not give rise to bistable solitons). By evaluating the propagation constant of a solitary solution δ as a function of its total power P for each given f(|E|)2), we found a class of nonlinear functions f(I) for which the dependence δ(P) becomes multivalued. We conjectured earlier [1] that stable (unstable) solitary solutions are those with dδ/dP > 0 (dδ/dP < 0). The first computer results by Enns and Rangnekar [2] supported this conjecture for a certain non-Kerr nonlinearity. In our further collaborative research [3], we discovered that depending on the nonlinear function f(I), the stable solitary solutions (dδ/dP > 0) fall into two classes: “weak” solitons (stable only against sufficiently small perturbation), and “robust” solitons (stable against any perturbation, in particular, against collisions with other solitons). It was shown [3] that while the condition dδ/dP > 0 is sufficient for the “weak” stability, some extra conditions imposed on the function f(I) warrant existence of “robust” solitons (one of them e.g. is f(I)/I2 = o(1) as I → ∞; stability against collapsing behavior). It was shown most recently by Enns [4] that at least for some of the nonlinear functions of (|E|2) giving rise to bistable solitons, the nonlinear Schrödinger equation passes the so called Painleve test and is therefore completely integrable. This confirms that the “robust” solitary solutions are indeed solitons. The propagation of single pulses in nonlinear fibers with an appropriate nonlinearity may provide the first known opportunity to obtain a temporal (dynamic) bistability as contrasted to the known kinds of optical bistability which are so far based on the equilibrium regimes.

The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schrödinger equations

The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation

In this paper, we utilize the first integral method with symbolic computation to constructing periodic and solitary wave solutions of complex nonlinear evolution equations. With the aid of Maple, new exact solutions of two nonlinear evolution equations in physics, namely, generalized Zakharov equations (GZ) and higher-order nonlinear Schrodinger equation (HONLS) describing propagation of ultrashort pulses in nonlinear optical fibers are obtained.

<abstract><p>This paper investigates soliton solutions to a two-component complex short pulse (c-SP) equation. Based on the known Lax pair representation of this equation, we verify the integrability of a two-component c-SP equation and find an equivalent convenient Lax pair through hodograph transformation. The classical Darboux transformation (DT) is utilized to construct multi-soliton solutions for the two-component c-SP equation as an ordinary determinant. Furthermore, the details of one-soliton and two-soliton solutions are presented and generalized for $ N $-fold soliton solutions. We also derive exact soliton solutions in explicit form using suitable reduction constraints from various "seed" solutions and explore them via graphs.</p></abstract>

Investigation of optical solitons and other solutions in optic fibers modeled by the improved perturbed nonlinear Schrödinger equation

We consider general principles of spectral compression of parabolic pulses in nonlinear optical fibers. It is known that the variance analysis provides a way to describe correctly the evolution of the pulse parameters under the spectral compression. We propose a computational model of multistage amplifier with spectral compression segments incorporated. The model provides the possibility to amplify middle-power input pulses to obtain pulse energies of hundreds of nanojoules with such output spectral characteristics that provide the possibility for further effective enhancement. We also discuss a modification of the model that provides the formation of a parabolic envelope and linear frequency modulation of the output pulse.

We propose a simple scheme to generate high-energy ultrashort pulses by combination and compression of multiple input pulses which share the same chirp profile. First, the multiple raised-cosine pulses in the input pulse train are modulated by a phase modulator, in which each modulation cycle covers two, three, four, or five pulses. Then, the modulated pulses are launched into a nonlinear fiber with the exponentially decreasing dispersion. We find that these pulses initially coalesce into a single pulse whose pulse profile is nearly hyperbolic secant, which then undergoes self-similar compression. Thus in the proposed method, first the combination of the multiple optical pulses occurs and then self-similar compression takes over. Besides, we also report the generation of ultrashort pulses by combination and compression of multiple hyperbolic secant pulses with the same chirp. The numerical results reveal that the resulting ultrashort pulse possesses a large portion of the input pulses for both raised-cosine and hyperbolic secant pulses. However, the compression factor and energy ratio are relatively higher for the hyperbolic secant pulses when compared with the raised-cosine pulses.

We consider propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

We consider the propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for the description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization of the defocusing mCSP equation is formulated via dependent variable and hodograph transformations. One- and two-dark soliton solutions are found by Hirota’s bilinear method and their properties are analyzed. It is shown that, depending on the parameters, the dark soliton solution can be either smoothed, cusponed or looped one. More specifically, the dark soliton tends to be evolved into a singular (cusponed or looped) one due to the increase of the spatial wave number in background plane waves and the increase of the depth of the trough. In the last part of the paper, we derive the defocusing mCSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the N-dark soliton solution in the form of determinants for the defocusing mCSP is provided.

The Sasa-Satsuma equation, a higher-order nonlinear Schr\"{o}dinger equation, is an important integrable equation, which displays the propagation of femtosecond pulses in optical fibers. In this paper, we investigate a generalized Sasa-Satsuma(gSS) equation. The Darboux transformation(DT) for the focusing and defocusing gSS equation is constructed. By using the DT, various of soliton solutions for the generalized Sasa-Satsuma equation are derived, including hump-type, breather-type and periodic soliton. Dynamics properties and asymptotic behavior of these soliton solutions are analyzed. Infinite number conservation laws and conserved quantities for the gSS equation are obtained.

Darboux transformation of two novel two-component generalized complex short pulse equations

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Linget al. 2016Physica D327, 13–29 (doi:10.1016/j.physd.2016.03.012); Fenget al. 2016Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).