Nikov System Research Articles

Resonant collisions among multi-breathers in the Mel’nikov system

We study the resonant collisions among multi-breathers in the Mel’nikov system, which is the Kadomtsev–Petviashvili (KP) equation coupled to the nonlinear Schrödinger equation. The two-breather resonant collisions are classified into two different types: (i) the emergence of a Y–shaped pattern when the breathers are oblique on each other; and (ii) the occurrence of fission of one breather into two breathers or fusion of two breathers into a single one when the involving breathers are parallel to each other. The resonant collision of three breathers generates more complicated collision scenarios. They are divided into separate and mutual resonant collisions depending on whether the whole breathers are involved in the collision process. On this basis, each collision mode demonstrates two distinct forms: (i) diverse web–shaped waveforms with three to five infinitely long breathers when three breathers are obliquely intersecting, where some waveforms display a polygonal pattern under certain parametric constraints; and (ii) the combination of plentiful fission or fusion processes when the breathers are parallel to each other. The parameter constraints are discussed in detail for the classification of resonant collision scenarios.

Novel bright-dark mixed $$\textit{N}$$-soliton for the ($$3+1$$)-component Mel’nikov system and its multi-component generalization

Novel bright-dark mixed $$\textit{N}$$-soliton for the ($$3+1$$)-component Mel’nikov system and its multi-component generalization

Construction of the soliton solutions and modulation instability analysis for the Mel’nikov system

Abstract This paper addresses the soliton solutions of the Mel’nikov system by the complex amplitude ansatz method and the He’s variational principle. The bright, anti-bright, dark, anti-dark, kink, anti-kink, bell-shaped, breather and homoclinic orbits are obtained as different possible solutions of the Mel’nikov system by the two methods. The solutions and their dynamical behaviour captured by the two methods are discussed. The solutions contain enough number of free physical parameters. These solutions give a better explanation of the associated physical phenomena. The modulation instability analysis of the steady-state solutions of the Mel’nikov system is investigated. This investigation has revealed that the Mel’nikov system is stable against small perturbation for a certain range of perturbation wave numbers. Apart from this, these solutions also help in validating the amount of accuracy in the numerical solutions and to perform the stability analysis further.

Soliton pairs in two-dimensional nonlocal media.

We study the interaction of optical beams of different wavelengths, described by a two-component, two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) model. Using a multiscale expansion method the NLS model is asymptotically reduced to the completely integrable 2D Mel'nikov system, the soliton solutions of which are used to construct approximate dark-bright and antidark-bright soliton solutions of the original NLS model; the latter being unique to the nonlocal NLS system with no relevant counterparts in the local case. Direct numerical simulations show that, for sufficiently small amplitudes, both these types of soliton stripes do exist and propagate undistorted, in excellent agreement with the analytical predictions. Larger amplitude of these soliton stripes, when perturbed along the transverse direction, disintegrate either to filled vortex structures (the dark-bright solitons) or to radiation (the antidark-bright solitons).

Lump and Mixed Rogue‐Soliton Solutions of the (2 + 1)‐Dimensional Mel’nikov System

Lump wave and line rogue wave of the (2 + 1)‐dimensional Mel’nikov system are derived by taking the ansatz as the rational function. By combining a rational function and different exponential functions, mixed solutions between the lump and soliton are derived. These solutions describe the interaction phenomena of the lump‐bright soliton with fission and fusion, the half‐line rogue wave with a bright soliton, and a rogue wave excited from the bright soliton pair, respectively. Some special concrete interaction solutions are depicted in both analytical and graphical ways.

Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

Bright-Dark Mixed N-Soliton Solutions of the Multi-Component Mel’nikov System

By virtue of the Kadomtsev–Petviashvili (KP) hierarchy reduction technique, we construct the general bright-dark mixed N-soliton solution to the multi-component Mel’nikov system. This multi-component system comprised of multiple (say M) short-wave components and one long-wave component with all possible combinations of nonlinearities including all-positive, all-negative and mixed types. Firstly, the two-bright-one-dark (2-b-1-d) and one-bright-two-dark (1-b-2-d) mixed N-soliton solutions in short-wave components of the three-component Mel’nikov system are derived in detail. Then we extend our analysis to the M-component Mel’nikov system to obtain its general mixed N-soliton solution. The formula obtained unifies the all-bright, all-dark and bright-dark mixed N-soliton solutions. For the collision of two solitons, an asymptotic analysis shows that for an M-component Mel’nikov system with M ≥ 3, inelastic collision takes place, resulting in energy exchange among the short-wave components supporting bright s...

GeneralN-Dark Soliton Solutions of the Multi-Component Mel’nikov System

A general form of $N$-dark soliton solutions of the multi-component Mel'nikov system is presented. Taking the coupled Mel'nikov system comprised of two-component short waves and one-component long wave as an example, its general $N$-dark-dark soliton solutions in Gram determinant form are constructed through the KP hierarchy reduction method. The dynamics of single dark-dark soliton and two dark-dark solitons are discussed in detail. It can be shown that the collisions of dark-dark solitons are elastic and energies of the solitons in different components completely transmit through. In addition, the dark-dark soliton bound states including both stationary and moving cases are also investigated. An interesting feature for the coupled Mel'nikov system is that the stationary dark-dark soliton bound states can exist for all possible combinations of nonlinearity coefficients including all-positive, all-negative and mixed types, while the moving case are possible when they take opposite signs or they are both negative. The dynamics and several interesting structures of the solutions are illustrated through some figures.

Vector nematicons: Coupled spatial solitons in nematic liquid crystals

Families of soliton pairs, namely vector solitons, are found within the context of a coupled nonlocal nonlinear Schr\odinger system of equations, as appropriate for modeling beam propagation in nematic liquid crystals. In the focusing case, bright soliton pairs have been found to exist provided their amplitudes satisfy a specific condition. In our analytical approach, focused on the defocusing regime, we rely on a multiscale expansion methods, which reveals the existence of dark-dark and antidark-antidark solitons, obeying an effective Korteweg-de Vries equation, as well as dark-bright solitons, obeying an effective Mel'nikov system. These pairs are discriminated by the sign of a constant that links all physical parameters of the system to the amplitude of the stable continuous wave solutions, and, much like the focusing case, the solitons' amplitudes are linked, leading to mutual guiding.

Matter-wave solitons in the counterflow of two immiscible superfluids

We study formation of solitons induced by counterflows of immiscible superfluids. Our setting is based on a quasi-one-dimensional binary Bose-Einstein condensate, composed of two immiscible components with large and small numbers of atoms in them. Assuming that the ``small'' component moves with constant velocity, either by itself, or being dragged by a moving trap, and intrudes into the ``large'' counterpart, the following results are obtained. Depending on the velocity, and on whether the small component moves in the absence or in the presence of the trap, two-component dark-bright solitons, scalar dark solitons, or multiple dark solitons may emerge, the last outcome taking place due to breakdown of the superfluidity. We present two sets of analytical results to describe this phenomenology. In an intermediate velocity regime, where dark-bright solitons form, a reduction of the two-component Gross-Pitaevskii system to an integrable Mel'nikov system is developed, demonstrating that solitary waves of the former are very accurately described by analytically available solitons of the latter. In the high-velocity regime, where the breakdown of the superfluidity induces the formation of dark solitons and multisoliton trains, an effective single-component description, in which a strongly localized wave packet of the ``small'' component acts as an effective potential for the ``large'' one, allows us to estimate the critical velocity beyond which the coherent structures emerge in good agreement with the numerical results.

Asymptotic reductions of two coupled (2 + 1)-dimensional nonlinear Schrödinger equations: application to Bose–Einstein condensates

We consider a system of two coupled (2 + 1)-dimensional nonlinear Schrodinger equations, describing two-component disc-shaped Bose–Einstein condensates. We present three different asymptotic reductions of this system. In particular, we derive the Mel'nikov system, the Yajima–Oikawa system as well as the Davey–Stewartson system (the latter is found as a special case of the Djordjevic–Redekopp system). Conditions for integrability of the reduced systems, their soliton solutions and the asymptotic relevance of such solutions to the original system are also discussed. Numerical results pertaining to the reduction to the Davey–Stewartson system are found to be in good agreement with the analytical predictions.

Exploding solitons and Shil'nikov's theorem

We have performed a detailed linear stability analysis of exploding solitons of the complex cubic–quintic Ginzburg–Landau (CGLE) equation. We have found, numerically, the whole set of perturbation eigenvalues for these solitons. We propose a scenario of soliton evolution based on this spectrum of eigenvalues. We relate exploding and self-restoring behavior of solitons to the Shil'nikov theorem, and point out common features and differences between our system, with an infinite number of degrees of freedom, and Shil'nikov's system with three degrees of freedom.

Vector solitons supported by the third-order dispersion

Novel vector solitons, composed by a bright soliton and a dark or an antidark soliton are derived for a system of two incoherently coupled nonlinear Schrödinger equations with the third-order dispersion. It is demonstrated that in the small-amplitude limit these solitons are described by the completely integrable Mel'nikov system.