Narrow Results

The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, $tan,sin,sec,cot,\text{csc},sinh,tanh,\mathrm{\text{sech}},cosh,$$coth,\mathrm{\text{csch}}$ and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences.

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of $100$ photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system.

In this paper, new exact traveling wave solutions are obtained by Hirota–Ramani equation. The many exact complex solutions of several types of nonlinear partial differential equations (NPDEs) are presented using the modified extended direct algebraic approach and new extended direct algebraic method, which is among the most effective mathematical techniques for finding a precise solution to NPDEs and put into a framework of algebraic computation. By selecting different bright and solitary soliton forms and by creating various analytical optical soliton solutions for the investigated equation, we hope to demonstrate how the analyzed model’s parameter impacts soliton behavior. It is possible to obtain new, complex solutions for nonlinear equations like the $(1+1)$-dimensional Hirota–Ramani equation.

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

New double Wronskian solutions to the Kadomtset–Petviashvili (KP) equation are derived. Solitons, rational solutions, Matveev solutions, complexitons and mixed solutions are given.

The authors theoretically investigate the formation of ultraslow dark and bright solitons via four-wave mixing (FWM) in a crystal of molecular magnets in the presence of a uniform d.c. magnetic field, where two strong continuous wave pump electromagnetic fields and a weak-pulsed probe electromagnetic field produce a pulsed FWM electromagnetic field. By solving the Maxwell–Schrödinger equations under the slowly varying envelope approximation and rotating-wave approximations, we demonstrate that both the weak-pulsed probe and FWM electromagnetic fields can evolve into dark and bright solitons with the same shape and the same ultraslow group velocity.

Nonlinear Schrödinger equations and corresponding quantum hydrodynamic (QHD) equations are widely used in studying of ultracold boson–fermion mixtures and superconductors. In this article, we show that more exact account of interaction in Bose–Einstein condensate (BEC), in comparison with the Gross–Pitaevskii (GP) approximation, leads to the existence of a new type of solitons. We use a set of QHD equations in the third order by the interaction radius (TOIR), which corresponds to the GP equation in the first order by the interaction radius. We analytically obtain a soliton solution which is an area of increased atom concentration. The conditions for existence of the soliton are studied. It is shown what solution exists if the interaction between the particles is repulsive. Particle concentration has been achieved experimentally for the BEC is of order of 10¹²–10¹⁴ cm^-3. In this case the solution exists if the scattering length is of the order of 1 μm, which can be reached using the Feshbach resonance. It is one of the limit case of existence of the solution. The corresponding scattering length decrease with the increasing of concentration of particles. We have shown that account of interaction up to TOIR approximation leads to new effects. The investigation of effects in the TOIR approximation gives a more detail information on interaction potentials between the atoms and can be used for a more detail investigation of the interatomic potential structure.

Theoretically, with the aid of a soliton model, the evolution of a new-phase nucleus near the first-order spin-reorientation phase transition in magnets has been investigated in an external magnetic field. The influence of an external field and one-dimensional defects of magnetic anisotropy on the dynamics of such nucleus has been demonstrated. The conditions for the localization of the new-phase nucleus in the region of the magnetic anisotropy defect and for its escape from the defect have been determined. The values of the critical fields which bring about the sample magnetization reversal have been identified and estimated.

The soliton solution for the nonlinear waves has been investigated in a composite magnetic-semiconducting medium. By using a hydrodynamic model of an infinite medium magnetized along the direction of propagation, a set of coupled nonlinear Zakharov equations has been derived. In the absence of carriers or magnetization, two extreme cases for the two independent decoupled nonlinear modes have been discussed. The propagation regions have also been numerically analyzed for the soliton solutions.

A microscopic theory of integer and fractional quantum Hall effects is presented here. In quantum density wave representation of charged particles, it is shown that, in a two-dimensional electron gas coherent structures form under the low temperature and high density conditions. With a sufficiently high applied magnetic field, the combined $N$ particle quantum density wave exhibits collective periodic oscillations. As a result the corresponding quantum Hall voltage function shows a step-wise change in multiples of the ratio $h/e^2$. At lower temperatures further subdivisions emerge in the Hall resistance, exhibiting the fractional quantum Hall effect.

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the $N$-soliton solutions in the Wronskian form are constructed, and the $(N-1)$- and $N$-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on $d(t)$, $h(t)$, $f(t)$ and $R(t)$, the soliton amplitude is merely related to $f(t)$, and the background depends on $R(t)$ and $f(t)$, where $d(t)$, $h(t)$, $q(t)$ and $f(t)$ are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and $R(t)$ is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.

Under investigation in this letter is a (2+1)-dimensional Sawada-Kotera equation. With the aid of the bilinear forms derived from the Bell polynomials, the Nth-order soliton solutions are obtained via the Pffafian method, and breather solutions are derived with the ansätz method. Analytic solutions obtained via the Pffafian method are the bell-type solitons. Two different kinds of the homoclinic breathers are seen, one of which is real and the other of which is complex, with two breathers interacting with each other. Homoclinic breather wave can evolve periodically along a straight line with a certain angle with the x axis and y axis, and its velocity, amplitude and width remain unchanged during the propagation. Homoclinic breather wave is not only space-periodic but also time-periodic. Interaction between the two breathers is elastic, which is similar to that of the solitons.

This paper studies the dynamics of solitons to the nonlinear Schrödinger’s equation (NLSE) with spatio-temporal dispersion (STD). The integration algorithm that is employed in this paper is the Riccati–Bernoulli sub-ODE method. This leads to dark and singular soliton solutions that are important in the field of optoelectronics and fiber optics. The soliton solutions appear with all necessary constraint conditions that are necessary for them to exist. There are four types of nonlinear media studied in this paper. They are Kerr law, power law, parabolic law and dual law. The conservation laws (Cls) for the Kerr law and parabolic law nonlinear media are constructed using the conservation theorem presented by Ibragimov.

In this letter, a (2+1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation is investigated, which describes the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Under two different constraints of the time-dependent coefficients in this equation, two different bilinear forms are derived by virtue of the binary Bell polynomials. Multiple solitary waves are constructed via the Hirota method, whose propagation properties and interaction characteristics are investigated graphically as well. Propagation and interaction of the solitons are illustrated graphically: (i) time-dependent coefficients affect the shape of the solitons; (ii) interaction of the solitons is elastic, i.e., amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

Very recently, a mechanism to the formation of rogue waves (RWs) has been proposed by the authors. In this paper, the formation of RWs in case of the complex Sharma–Tasso–Olver (STO) equation is studied. In the STO equation, one, two and three-soliton solutions are obtained. Due to the inelastic collisions, these soliton waves are fused to one. Under the free parameters constraint this behavior do occurs. The mechanism of formation of RWs is due to the collisions of solitons and multi-periodic waves (like spectral band). These RWs as giant waves, which may be very sharp or chaotic are similar to RWs in laser. The work is done here by using the generalized unified method (GUM).

In this paper, we consider a coupled nonlinear Maccari’s system (CNMS) which describes the motion of isolated waves localized in a small part of space. There are some integration tools that are adopted to retrieve the solitary wave solutions. They are the modified F-Expansion and the generalized projective Riccati equation methods. Topological, non-topological, complexiton, singular and trigonometric function solutions are derived. A comparison between the results in this paper and the well-known results in the literature is also given. The derived structures of the obtained solutions offer a rich platform to study the nonlinear CNMS. Numerical simulation of the obtained solutions are presented with interesting figures showing the physical meaning of the solutions.

In this paper, the N-fold Darboux transformation of the Suris system is established by gauge transformation of the Lax pair. As a result, the N-fold exact solutions of the Suris system are derived in terms of the determinant. It is shown that this system can support certain abundant and peculiar nonlinear structures, which may explain some interesting physical phenomena. Moreover, the infinitely many conservation laws of the Suris system are given.

The propagations are generally described through nonlinear Schrödinger equation (NLSE) in the optical solitons. In the NLSEs, the higher order NLSE with derivative non-Kerr nonlinear terms is a model that depicts propagation of pulses beyond ultra-short range in optical communication system. Several novel exact solutions of different kinds such as solitons, solitary waves and Jacobi elliptic function solutions are achieved via using modified extended mapping technique. Different kinds of exact results have prestigious exertions in engineering and physics. Structures of solitons different kinds are shown graphically by giving suitable values to parameters. The physical interpretations of solutions can be understand through structures. Several exact solutions and computing work confirm the supremacy and usefulness of the current technique.