Higher-order integrable evolution equation and its soliton solutions

We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transformation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases: the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.A.A. and N.A. acknowledge the support of the Australian Research Council (Discovery Project DP110102068) and also thank the Volkswagen Foundation for financial support

Generalization of the Langmuir–Blodgett laws for a nonzero potential gradient

The Langmuir–Blodgett laws for cylindrical and spherical diodes and the Child–Langmuir law for planar diodes repose on the assumption that the electric field at the emission surface is zero. In the case of ion beam extraction from a plasma, the Langmuir–Blodgett relations are the typical tools of study, however, their use under the above assumption can lead to significant error in the beam distribution functions. This is because the potential gradient at the sheath/beam interface is nonzero and attains, in most practical ion beam extractors, some hundreds of kilovolts per meter. In this paper generalizations to the standard analysis of the spherical and cylindrical diodes to incorporate this difference in boundary condition are presented and the results are compared to the familiar Langmuir–Blodgett relation

Approach to first-order exact solutions of the Ablowitz-Ladik equation

We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solutions of the nonlinear Schrödinger equation (NLSE).Two of the authors (A.A. and N.A.) acknowledge the support of the Australian Research Council (Discovery Project No. DP0985394). N.A. is a grateful recipient of support from the Alexander von Humboldt Foundation (Germany)

Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation

We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L) equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer times. These periodic solutions comprise a family of two-parameter solutions with an arbitrary background field and a frequency of initial perturbation. The solutions are recurrent, since they return the field state to the original constant background solution after the process of nonlinear evolution has passed. These solutions can be considered as a complete resolution of the Fermi-Pasta-Ulam paradox for the A-L system. One remarkable consequence of the recurrent evolution is the nonlinear phase shift gained by the constant background wave after the process. A particular case of this family is the rational solution of the first-order or fundamental rogue wave.The authors acknowledge the support of the A.R.C. (Discovery Project DP110102068). One of the authors (N.A.) is a grateful recipient of support from the Alexander von Humboldt Foundation (Germany)

Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation

Finite-dimensional dynamical models for solitons of the cubic-quintic complex Ginzburg-Landau equation CGLE are derived. The models describe the evolution of the pulse parameters, such as the maximum amplitude, pulse width, and chirp. A clear correspondence between attractors of the finite-dimensional dynamical systems and localized waves of the continuous dissipative system is demonstrated. It is shown that stationary solitons of the CGLE correspond to fixed points, while pulsating solitons are associated with stable limit cycles. The models show that a transformation from a stationary soliton to a pulsating soliton is the result of a Hopf bifurcation in the reduced dynamical system. The appearance of moving fronts kinks in the CGLE is related to the loss of stability of the limit cycles. Bifurcation boundaries and pulse behavior in the regions between the boundaries, for a wide range of system parameters, are found from analysis of the reduced dynamical models. We also provide a comparison between various models and their correspondence to the exact results

Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits

We present an explicit analytic form for the two-breather solution of the nonlinear Schrödinger equation with imaginary eigenvalues. It describes various nonlinear combinations of Akhmediev breathers and Kuznetsov-Ma solitons. The degenerate case, when the two eigenvalues coincide, is quite involved. The standard inverse scattering technique does not generally provide an answer to this scenario. We show here that the solution can still be found as a special limit of the general second-order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions. A further restriction of this particular case, where the two eigenvalues are equal to i, produces the second-order rogue wave with two free parameters considered as differential shifts. The illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts. Thus we establish a hierarchy of second-order solutions, revealing the interrelated nature of the general case, the rogue wave, and the degenerate breathers

Creeping solitons in dissipative systems and their bifurcations

We present a detailed numerical study of creeping solitons in dissipative systems. A bifurcation diagram has been constructed for the region of transition between solitons and fronts. It shows a rich variety of transitions between various types of localized solutions. For the first time, we have found a sequence of period-doubling bifurcations of creeping solitons, and also a symmetry-breaking instability of creeping solitons. Creeping solitons may involve many frequencies in their dynamics, and this can result, in particular, in a multiplicity of zig-zag motions

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