Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms

We present the fifth-order equation of the nonlinear Schrodinger hierarchy. This integrable partial differential ¨ equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrodinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated ¨ flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.The authors acknowledge the support of the Australian Research Council (Discovery Project No. DP140100265). N.A. and A.A. acknowledge support from the Volkswagen Stiftung and A.C. acknowledges Endeavour Postgraduate Award support

Solutions of the higher-order Manakov-type continuous and discrete equations

We derive exact and approximate localized solutions for the Manakov-type continuous and discrete equations. We establish the correspondence between the solutions of the coupled Ablowitz-Ladik equations and the solutions of the coupled higher-order Manakov equations.The authors acknowledge the support of the ARC (Discovery Project DP140100265). N.A. and A.A. acknowledge the support of the Volkswagen Stiftung, while A.C. is grateful for support through an Endeavour Fellowship

Multisoliton complexes in a sea of radiation modes

We derive exact analytical solutions describing multi-soliton complexes and their interactions on top of a multi-component background in media with self-focusing or self-defocusing Kerr-like nonlinearities. These results are illustrated by numerical examples which demonstrate soliton collisions and field decomposition between localized and radiation modes.Comment: 7 pages, 7 figure

Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions

We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions

Extended nonlinear Schroeodinger equation with higher-order odd and even terms and its rogue wave solutions

We consider an extended nonlinear Schroeodinger equation with higher-order odd (third order) and even (fourth order) terms with variable coecients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specic constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota, and Lakshmanan - Porsezian - Daniel (LPD) equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue waves solutions of the corresponding equations

Comparison of measured and predicted bandwidth of graded-index multimode fibres

Measurements of pulse spreading in three graded index fibers have been performed over an extensive range of wavelengths, including regions in which the index profiles become optimal and the bandwidths correspondingly high. The refractive index distributions and profile dispersion parameter have been measured and used in a ray-tracing algorithm in order to predict bandwidths. Comparison reveals that averaging processes on the index data are usually necessary to account for noncircularity of the fiber and small variations in the deduced profile caused by the wavelength dependence of the near-field intensity distribution. Results obtained by this means usually tend to slightly underestimate the true fiber bandwidth, while alpha-profile predictions always result in overestimates by about one order of magnitude. Remaining discrepancies between measured and predicted bandwidths are attributed to small variations of the index profiles along the fiber length

Surface modification of Co-doped ZnO nanocrystals and its effects on the magnetic properties

A series of chemically prepared Co2+-doped ZnO colloids has been surface modified either by growing shells of ZnSe or by the in situ encapsulation in poly styrene . The surface modification effects using these two distinct chemical strategies on the magnetic properties of the nanocrystals were probed by electron paramagnetic resonance EPR . Structural characterization by means of x-ray diffraction and transmission electron microscopy gave no evidence of second phase formation within the detection limits of the used equipment. The EPR analysis was carried out by simulations of the powderlike EPR spectra. The results confirm that in the core of these nanocrystals Co was incorporated as Co2+, occupying the Zn2+ sites in the wurtzite structure of ZnO. Additionally we identify two Co signals stemming from the nanocrystals’ shell. The performed surface modifications clearly change the relative intensity of the EPR spectrum components, revealing the core and shell signals

Conservation Laws and Integral Relations for the Boussinesq Equation

We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering, which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying, and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger equation. For these rational solutions, the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations, which depend on the total degree of the associated polynomial

The nonabelian Liouville-Arnold integrability by quadratures problem: a symplectic approach

A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces

献辞

The ability to unveil growing rogue waves in the ocean is essential for safe marine travel in stormy conditions. This vital problem has not been adequately addressed so far. We show that the specific triangular spectra of rogue waves can be detected at early stages of their development in a chaotic wave field. Continuously measuring the spectra of various parts of the wave field allows us to find a rogue wave before the dangerous peak appears. This possibility of early detection is a necessary part of a rogue wave early-warning system.N.A. is a grateful recipient of the Alexander von Humboldt Prize. He acknowledges a grant on Extreme Events from the Volkswagen Foundation. J.M.S.C. acknowledges support from the Spanish Ministerio de Ciencia e Innovación under contract FIS2009-09895. A.A. and N.D. acknowledge the support of the Australian Research Council (Discovery Project DP110102068)