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We prove that the nonlocality is a source of the internal mode generation in a bright soliton of the nonlocal nonlinear Schrodinger equation, at least in the weak nonlocality limit. The internal mode bifurcates from the edge of the continuous spectrum of the linearized eigenvalue problem into the gap of this spectrum. A dependence of the internal mode propagation constant position in the gap on the nonlocality rate is established. It is shown that the vibration amplitude of the soliton decays inversely proportional to the propagation distance, as in the local models.

The modulational instability of a plane wave for a discrete nonlinear Schrodinger equation with arbitrary nonlocality is analyzed. This model describes light propagation in a thin film planar waveguide arrays of nematic liquid crystals subjected to a periodic transverse modulation by a low frequency electric field. It is shown that nonlocality can both suppress and promote the growth rate and bandwidth of instability, depending on the type of a response function of a discrete medium. A solitary wave (breather-like) solution is built by the variational approximation and its stability is demonstrated.

A method is proposed to generate homoclinic solutions for an integrable nonlinear PDE with periodic boundaries. This approach resembles the dressing method known in the theory of solitons. The pole positions in the dressing factor are given by the complex double points of the Floquet spectrum associated with unstable modes of a seed solution of the nonlinear equation. As an example, we obtain the homoclinic orbit for the modified nonlinear Schrodinger equation solvable by the Wadati–Konno–Ichikawa spectral problem.

We study numerically the mutual propagation of two fractional topological charge Laguerre-Gaussian (LG) light beams in the moving atmosphere with the thermal nonlinearity. The fractional charge beam is constructed as a weighted superposition of a finite number of the standard LG modes with integer charges. Numerical simulations demonstrate an enhanced stability of the cross-section intensity distribution of the fractional charge LG beams against the thermal blooming, as compared with the case of integer charge beams. The dominant mechanism of such a stability results from the multimodal structure of the fractional charge beams.

We develop a perturbation theory for bright solitons of the F=1 integrable spinor Bose-Einstein condensate (BEC) model. The formalism is based on using the Riemann-Hilbert problem and provides the means to analytically calculate evolution of the soliton parameters. Both rank-one and rank-two soliton solutions of the model are obtained. We prove equivalence of the rank-one soliton and the ferromagnetic rank-two soliton. Taking into account a splitting of a perturbed polar rank-two soliton into two ferromagnetic solitons, it is sufficient to elaborate a perturbation theory for the rank-one solitons only. Treating a small deviation from the integrability condition as a perturbation, we describe the spinor BEC soliton dynamics in the adiabatic approximation. It is shown that the soliton is quite robust against such a perturbation and preserves its velocity, amplitude, and population of different spin components, only the soliton frequency acquires a small shift. Results of numerical simulations agree well with the analytical predictions, demonstrating only slight soliton profile deformation., Comment: 12 pages, 3 figures

The modulational instability of two interacting waves in a nonlocal Kerr-type medium is considered analytically and numerically. For a generic choice of wave amplitudes, we give a complete description of stable/unstable regimes for zero group-velocity mismatch. It is shown that nonlocality suppresses considerably the growth rate and bandwidth of instability. For nonzero group-velocity mismatch we perform a geometrical analysis of a nonlocality management which can provide stability of waves otherwise unstable in a local medium., 15 pages, 12 figures, to be published in Physica Scripta

The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient "linear experience" of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations.

We describe the full-time dynamics of modulational instability in F=1 spinor Bose-Einstein condensates for the case of the integrable three-component model associated with the matrix nonlinear Schroedinger equation. We obtain an exact homoclinic solution of this model by employing the dressing method which we generalize to the case of the higher-rank projectors. This homoclinic solution describes the development of modulational instability beyond the linear regime, and we show that the modulational instability demonstrates the reversal property when the growth of the modulation amplitude is changed by its exponential decay., Comment: 6 pages, 2 figures, text slightly extended, a reference added

A model is studied which describes a propagation of a subpicosecond optical pulse in dispersion-managed fibre links. In the limit of weak chromatic dispersion management, the model equation is reduced to a perturbed modified NLS equation having a nonlinearity dispersion term. By means of the Riemann--Hilbert problem, a perturbation theory for the soliton of the modified NLS equation is developed. It is shown in the adiabatic approximation that there exists a unique possibility to suppress the perturbation-induced shift of the soliton centre at the cost of proper matching of the soliton width and nonlinearity dispersion parameter. In the next-order approximation, the spectral density of the radiation power emitted by a soliton is calculated., Comment: 16 pages, 3 figures, to appear in J. Mod. Optics

Propagation of the Laguerre–Gaussian (LG) light beams with fractional topological charges in a moving defocusing nonlinear medium is studied. The fractional charge LG beam is constructed as a weighted superposition of the standard LG modes with integer charges. Our model is governed by a nonlocal nonlinear Schrodinger equation and describes in particular the beam propagation in the atmosphere along the earth surface with allowance for thermal nonlinearity and in the presence of a cross-wind. The numerical simulations have revealed the remarkable stability of the cross-section intensity distribution in the fractional charge LG beams during their propagation, as compared with the case of integer charge beams.

We analyze chainlike N-soliton dynamics in a weakly nonlocal, essentially nonintegrable system described by the cubic–quintic nonlinear Schrodinger equation. Quintic nonlinearity is not assumed to be small. This system is reduced to a generalized complex Toda chain model. Numerical simulations demonstrate adverse action of both cubic and quintic nonlocal responses, in their own right, on the quasi-equidistant train propagation, with a development of a chaotic regime. From the Toda chain model, we predict a possibility of mutually compensating both types of nonlocality-induced distortion, restoring thereby a deterministic mode of the train propagation in a weakly nonlocal medium. Analytical predictions corroborate well with numerical results.