We study the nonlinear stage of the modulation instability of a condensate in the framework of the focusing nonlinear Schrödinger equation (NLSE). We find a general N-solitonic solution of the ...focusing NLSE in the presence of a condensate by using the dressing method. We separate a special designated class of "regular solitonic solutions" that do not disturb phases of the condensate at infinity by coordinate. All regular solitonic solutions can be treated as localized perturbations of the condensate. We find an important class of "superregular solitonic solutions" which are small perturbations at a certain moment of time. They describe the nonlinear stage of the modulation instability of the condensate.
The one-dimensional focusing nonlinear Schrödinger equation (NLSE) on an unstable condensate background is the fundamental physical model that can be applied to study the development of modulation ...instability (MI) and formation of rogue waves. The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves. The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years. Here, we demonstrate that Kuznetsov-Ma and superregular NLSE breathers play the key role in the dynamics of a wide class of SLCP. During the nonlinear stage of MI development, collisions of these breathers lead to the formation of rogue waves. We present new scenarios of rogue wave formation for randomly distributed breathers as well as for artificially prepared initial conditions. For the latter case, we present an analytical description based on the exact expressions found for the space-phase shifts that breathers acquire after collisions with each other. Finally, the presence of Kuznetsov-Ma and superregular breathers in arbitrary-type condensate perturbations is demonstrated by solving the Zakharov-Shabat eigenvalue problem with high numerical accuracy.
Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the self-modulation of the continuous “envelope waves” with respect to small periodic perturbations that ...precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales.
Remarkable mathematical properties of the integrable nonlinear Schrödinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fiber links. ...Fundamental optical soliton, continuous, and discrete eigenvalues of the nonlinear spectrum have already been considered for the transmission of information in fiber-optic channels. Here, we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations that offers the advantage of a relatively simple decoder design. First, we describe an approach based on exploiting the general N-soliton solution of the NLSE for simultaneous coding of N symbols involving 4×N coding parameters. As a specific elegant subclass of the general schemes, we introduce a soliton orthogonal frequency division multiplexing (SOFDM) method. This method is based on the choice of identical imaginary parts of the N-soliton solution eigenvalues, corresponding to equidistant soliton frequencies, making it similar to the conventional OFDM scheme, thus, allowing for the use of the efficient fast Fourier transform algorithm to recover the data. Then, we demonstrate how to use this new approach to control signal parameters in the case of the continuous spectrum.
We describe in detail the construction of numerical fourth- and sixth-order schemes using the Magnus expansion for solving the Zakharov–Shabat system, which makes it possible to solve accurately the ...direct scattering problem for the nonlinear Schrödinger equation. To avoid numerical instabilities inherent in the procedure of solving the direct scattering problem, high-precision arithmetic is used. At present, application of the proposed schemes in combination with the highprecision arithmetic is a unique tool that can be used for analysis of complex wave fields, which contain a great number of solitons, and allows one to determine the complete discrete spectrum, including both eigenvalues and normalization constants. In this work, we study the errors in the proposed scheme using an example of the potential in the form of a hyperbolic secant. It is found that the time of calculation of the scattering matrix using the sixth-order algorithm is almost twice as long compared with that for the standard second-order Boffetta–Osborne algorithm, whereas the time gain resulting from the reduction of number of the wave-field discretization points with retaining the desired precision can reach an order of magnitude or more. Exact solution of the direct scattering problem requires that the discretization increment of high-amplitude wave fields should be comparable with the characteristic width of the largest solitons contained in such fields. In this case, the discretization increment can be sufficiently smaller than that required for reconstruction of the full Fourier spectrum of the wave field. Application of the proposed high-order approximation schemes can be of fundamental importance for successful operation with a great number of complex nonlinear wave fields, such as, e.g., that in the process of the statistical study of the scattering data.
We study vector solitons propagating on an unstable constant background (vector breathers) theoretically in the framework of the focusing two-component one-dimensional nonlinear Schrödinger equation. ...Based on the simplified inverse scattering transform technique called the dressing method, we find the exact solutions describing resonance interactions of the vector breathers. The resonance represents a three-breather process, i.e., a fusion of two breathers into one or decay of one breather into two, such that the characteristic wave vectors and frequencies of the breathers satisfy resonance conditions.
By means of the dressing technique, we build multipole solutions of the focusing Manakov system under a constant background. These solutions become degenerate when the poles of the dressing function ...merge. We find that with a special choice of the integration constants, such solutions describe the fusion or decay of the pulsing solitons—breathers—and their wave numbers and frequencies satisfy the typical resonance condition. We investigate the different cases of such resonance interactions.
Nonlinear wavefields governed by integrable models such as the Korteweg-De Vries (KdV) equation can be decomposed into the so-called scattering data playing the role of independent elementary ...harmonics evolving trivially in time. A typical scattering data portrait of a spatially localised wavefield represents nonlinear coherent wave structures (solitons) and incoherent radiation. In this work we present a fourth-order accurate algorithm to compute the scattering data within the KdV model. The method based on the Magnus expansion technique provides accurate information about soliton amplitudes, velocities and intensity of the radiation. Our tests performed using a box-shaped wavefield confirm that all components of the scattering data are computed correctly, while the test based on a single-soliton solution verifies the declared order of a numerical scheme.
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