Line Solitons Research Articles

The rational and semi-rational solutions to the Hirota Maccari system

In this paper, based on the KP hierarchy reduction method, we get the concrete form of semi-rational solutions to the Hirota–Maccari (HM) system. The semi-rational solutions describe a variety of solution types, such as lumps, breathers and line solitons. In addition, we simplify the representation of the higher-order semi-rational solutions of the HM equation and focus on the lump classification as well as the dynamics of mixed solutions about lumps and line solitons. Simultaneously, the modulation instability of the HM equation is also analyzed. The kinetics of semi-rational solutions for the HM system are subsequently discussed.

Twisted lump, lumpoff and rogue wave of the (2+1)-dimensional Kaup–Kupershmidt equation

Exact explicit rational and resonant rational–exponential solutions of the (2+1)-dimensional Kaup–Kupershmidt equation are derived by using the bilinear method. In addition to single-peak and double-peak lumps, the fundamental rational solution is shown to depict a twisted lump structure including two interactional single-peak lumps. The first resonant solution is able to describe a twisted lumpoff in which the twisted lump is cut by one line soliton and the moving path and velocities of lump, as well as the approximate starting time of interaction can be calculated. The second resonant solution allows a special interaction that the twisted lump is cut by two line solitons in opposite directions. This fascinating interaction is identified as a twisted rogue wave with the predictability, because the emerging time, location and moving route of lump can be caught exactly. Moreover, the condition of the existence of rogue wave indicates that it contains regular and reduced cases.

Dynamical structures of multi-soliton solutions to the Bogoyavlenskii’s breaking soliton equations

We derive a multi-soliton solution for the Bogoyavlenskii’s breaking soliton equation by utilizing the simplified Hirota’s approach. From this multi-soliton solution, we investigate various forms of single kinky–lump-type breather solitons, double kinky–lump-type breather solitons, collision of a kink line soliton with a kinky-type breather soliton, and collision of a pair of double kinky–lump breather solitons by the appropriate selection of the involved parameters. These breathers hold unlike features in various planes even in various times. Elastic and non-elastic collisions for double kinky-type lump breather are experienced in various planes and in various times. The effect and control of the propagation direction, energies, phase shifts and shape of waves by the parameters are also analyzed. Some figures are given to illustrate the dynamics of the achieved solutions. The acquired results can enrich the dynamical properties of the higher-dimensional nonlinear scenarios in the engineering fields.

Localized solitary waves and their dynamical stabilities in magnetized dusty plasma

We consider a magnetized dusty plasma, which composed of low-temperature and high-temperature ions, electrons, and dust particles. The dynamical behaviors can be described by a (3+1)-dimensional Zakharov–Kuznetsov equation (ZKE). Interestingly, a type of completely localized solitary waves, which are different from the line solitons, of ZKE are obtained analytically and approximately for the first time. This kind of solitary wave is also confirmed numerically by the Petviashvili method. Both the analytical and numerical results indicate that the amplitude of the localized wave is proportional to its velocity and inverse proportional to the nonlinear interaction strength. A finite difference scheme with second-order accuracy is presented to make the long-time nonlinear evolution of ZKE. The numerical results indicate that the localized solitons are always dynamically stable. Moreover, the collision between two solitary waves is investigated numerically. The results show that both elastic and inelastic collision exist when two localized solitary waves colliding.

Breathers, lumps and hybrid solutions of the $(2{+}1)$-dimensional Hirota–Satsuma–Ito equation

With the Hirota bilinear method and symbolic computation, explicit forms of N-soliton of the (2+1)-dimensional Hirota–Satsuma–Ito equation are derived. General high-order breather solutions are constructed through appropriate parameter restrictions. By performing an appropriate limiting procedures on soliton solutions and then making further parameter constraints, general lump solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation would be succinctly constructed. Furthermore, we provide the hybrid solutions which means different types of combinations in breathers, lumps and line solitons. In order to better understand the dynamical behaviors of the equation, the novel interaction and propagation characteristics are discussed graphically.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: General lump-soliton solutions on a background of periodic line waves

We study the general lump-type soliton solutions on a background of periodic line waves in the PT-symmetric nonlocal Davey–Stewartson I (DS I) equation. By using the Kadomtsev–Petviashvili hierarchy reduction method, new families of semi-rational solutions termed as lump-soliton solutions to the nonlocal DS I equation are constructed. Under particular parameters restrictions we obtain: (a) different types of lumps sitting on a background of periodic line waves and (b) different types of lumps interacting with line solitons on a background of periodic line waves. The interactions of lumps and line solitons are inelastic, generating three generic dynamical scenarios: (i) lumps fusing into line solitons, (ii) lumps fissioning from line solitons, and (iii) a combined process consisting of lumps fusing into and fissioning from line solitons.

Coupled sine-Gordon systems in living cellular structure

In this paper, we present a coupled sine-Gordon (CSG) system, which describes the dynamics of two deoxyribonucleic acid (DNA) twisting strands in open configuration. We investigate the associated linear eigenvalue problem of the CSG system and obtain generalized formulae for multiple soliton solutions by employing the Darboux transformation (DT). Explicit expressions of one- and two-kink and line soliton solutions, single breather and first-order degenerate solutions are obtained by using particular solutions to the associated linear eigenvalue problem. We also present the dynamics of the different solutions.

LUMP AND MIXED ROGUE-SOLITON SOLUTIONS TO THE 2+1 DIMENSIONAL ABLOWITZ-KAUP-NEWELL-SEGUR EQUATION

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rationalexponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.

Conservation laws and line soliton solutions of a family of modified KP equations

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

Lie symmetry reductions and group invariant solutions of (2 + 1)-dimensional modified Veronese web equation

The Lie symmetry method is successfully applied to compute group invariant solutions for (2 + 1)-dimensional modified Veronese web equation. The purpose of this present article is to study the modified Veronese web (mVw) equation and to obtain its infinitesimals, commutation table of Lie algebra, symmetry reductions and closed form analytical solutions. The obtained results are explicitly in the form of the functions $$f_1(y),f_2(t),f_3(x)$$ and $$f_4(x)$$ and hold numerous solitary wave solutions that are more helpful to describe dynamical phenomena through their evolution profile. The solutions are analysed physically via numerical simulation. Consequently, elastic behaviour multisolitons, line soliton, doubly soliton, parabolic wave profile, nonlinear behaviour of wave profile and elastic interaction soliton profile of solutions are demonstrated in the analysis and discussion section to make this study more praiseworthy.

Rational and semi-rational solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

Nonlinear waves are seen in nature, such as the water waves and plasma waves. Investigated in this paper is a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Based on the bilinear method, we get the rational solutions, which are different from the published ones, semi-rational solutions and breather-type kink soliton solutions. Through the rational solutions, we observe two types of waves: the lump waves and line rogue waves. The semi-rational solutions depict two types of interactions: (1) The fusion or fission between the lump wave and soliton; (2) The interaction between the line rogue wave and soliton. During the interaction between the line rogue wave and soliton, the line rogue wave evolves with three different shapes: the bright rogue waves, bright–dark rogue waves and dark rogue waves. Via the breather-type kink soliton solutions, we observe the breather-soliton mixture.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: Soliton solutions with nonzero background

Various solutions to the PT-symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2N-line solitons and 2N-lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrödinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.

Novel characteristics of lump and lump–soliton interaction solutions to the generalized variable-coefficient Kadomtsev–Petviashvili equation

With the inhomogeneities of media taken into account, a generalized variable-coefficient Kadomtsev–Petviashvili (vcKP) equation is proposed to model nonlinear waves in fluid mechanics and plasma physics. Based on Hirota bilinear method and symbolic computation, we present lump and lump–soliton interaction solutions of the vcKP equation. These local solutions are derived by taking the auxiliary function as the positive quadratic function or the linear combination of the positive quadratic function and the exponential function. Compared with the results allowed by the constant-coefficient KP equation, lump and lump–soliton solutions for the vcKP equation possess more abundant properties. It is shown that the velocity, moving path, and maximum height of the lump are completely characterized by the time functions rather than the constant parameters. The interaction between a lump and one line soliton are still nonelastic, but the track of the lump obeys the controllable function of time. The lump interacting with resonance soliton pairs exhibits a kind of special rogue wave in which the peak emerges and evolves with the varying path. The detailed analysis and discussion of these solutions are provided and illustrated.

Majorana Corner Modes with Solitons in an Attractive Hubbard-Hofstadter Model of Cold Atom Optical Lattices.

Higher-order topological superconductors hosting Majorana-Kramers pairs (MKPs) as corner modes have recently been proposed in a two-dimensional quantum spin Hall insulator proximity-coupled to unconventional cuprate or iron-based superconductors. Here, we show that such MKPs can be realized using a conventional s-wave superfluid with a soliton in cold atom systems governed by the Hubbard-Hofstadter model. The MKPs emerge in the presence of interaction at the "corners" defined by the intersections of line solitons and the one-dimensional edges of the system. Our scheme is based on the recently realized cold atom Hubbard-Hofstadter lattice and will pave the way for observing possible higher-order topological superfluidity with conventional s-wave superfluids or superconductors.

Patterns of water in light

The intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev–Petviashvilli-II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics.

Diverse solitons and interaction solutions for the (2+1)-dimensional CDGKS equation

In this paper, the (2[Formula: see text]+[Formula: see text]1)-dimensional CDGKS equation is studied and its diverse soliton solutions consisting of line soliton, periodic soliton and lump soliton with different parameters are derived based on the Hirota bilinear method and long-wave limit method. Based on exact solution formulae with different parameters, the interaction between line soliton and periodic soliton, the interaction between line soliton and lump soliton, as well as the interaction between periodic soliton and lump soliton are illustrated. According to the dynamical behaviors, it can be found that the effects of different parameters are on the propagation direction and shapes. Novel soliton interaction phenomena are also observed.

High-order breathers, lumps, and semi-rational solutions to the (2 + 1)-dimensional Hirota–Satsuma–Ito equation

Under investigation in this work is the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) equation. By employing Bell’s polynomials, bilinear formalism of the HSI equation is succinctly obtained. With the aid of the obtained bilinear formalism, we first construct general high-order soliton solutions by using Hirota’s bilinear method combined with the perturbation expansion. By taking particular complex conjugate conditions of the high-order soliton solutions, high-order breather solutions and the mixed solutions consisting of breathers and line solitons are succinctly derived. We further generate rational solutions termed high-order lumps by taking a long wave limit. Finally, we investigate two types of semi-rational solutions, which describe interaction between lumps and line solitons, or between lumps and breathers. These collisions are elastic, which do not lead to any changes of amplitudes and velocities, and shapes of the line solitons, breathers and lumps after interaction.

Fission and fusion collision of high-order lumps and solitons in a $$(3+1)$$ ( 3 + 1 ) -dimensional nonlinear evolution equation

Multiple dark soliton solutions and semi-rational solutions to a $$(3+1)$$ -dimensional nonlinear evolution equation are obtained by a combination of the Kadomtsev–Petviashvili hierarchy reduction method and the Hirota’s bilinear method. The collision phenomena of fission and fusion of high-order lumps and solitons in the $$(3+1)$$ -dimensional nonlinear evolution equation are described by these semi-rational solutions. After the collision of higher-order lumps and solitons, the lumps would fuse into or fissure from the line solitons. As lumps are created or annihilated, the exchange of energy occurs between the lumps and the line solitons.

Rational solutions of the (2+1)-dimensional Kaup–Kupershmidt equation

Rational solutions of the (2+1)-dimensional Kaup–Kupershmidt equation are presented in determinant form by taking the long wave limit of soliton solutions. With the appropriate parameters’ form, the soliton solution is firstly changed into the mixed solution consisting of solitons and periodic waves. Then taking the limit of the fixed parameter gives rise to the rational solution, which represents the interaction between the rational line soliton and the lump. It is shown that the rational line soliton exhibits the W-shaped soliton, while the lump is classified into the double-peak and the single-peak patterns. The interactions between two types of rational waves are discussed in different cases.