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

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)

Soliton, rational, and periodic solutions for the infinite hierarchy of defocusing nonlinear Schrödinger equations

Analysis of short-pulse propagation in positive dispersion media, e.g., in optical fibers and in shallow water, requires assorted high-order derivative terms. We present an infinite-order “dark” hierarchy of equations, starting from the basic defocusing nonlinear Schrödinger equation. We present generalized soliton solutions, plane-wave solutions, and periodic solutions of all orders. We find that “even”-order equations in the set affect phase and “stretching factors” in the solutions, while “odd”-order equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are complex. There are various applications in optics and water waves

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)

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

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

Hamiltonian-versus-energy diagrams in soliton theory

Parametric curves featuring Hamiltonian versus energy are useful in the theory of solitons in conservative nonintegrable systems with local nonlinearities. These curves can be constructed in various ways. We show here that it is possible to find the Hamiltonian (H) and energy (Q) for solitons of non-Kerr-law media with local nonlinearities without specific knowledge of the functional form of the soliton itself. More importantly, we show that the stability criterion for solitons can be formulated in terms of H and Q only. This allows us to derive all the essential properties of solitons based only on the concavity of the curve H vs Q. We give examples of these curves for various nonlinearity laws and show that they confirm the general principle. We also show that solitons of an unstable branch can transform into solitons of a stable branch by emitting small amplitude waves. As a result, we show that simple dynamics like the transformation of a soliton of an unstable branch into a soliton of a stable branch can also be predicted from the H-Q diagram

Bright and dark rogue internal waves: The Gardner equation approach

We have found 'bright and dark' solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four 'bright' and 'dark' exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.The authors gratefully acknowledge the support of the Australian Research Council (Discovery Project DP150102057)

Generalized Sasa--Satsuma equation: Densities approach to new infinite hierarchy of integrable evolution equations

We derive the new infinite Sasa-Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 x 3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa-Satsuma hierarchy functionals. All these operators are parts of a generalized Sasa-Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multi-parameter dynamical system