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Investigation is conducted on a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics. By virtue of the homoclinic-test, ansatz and polynomial-expansio...

In this paper, we investigate the dark solitons for the three-component Gross–Pitaevskii system, which describes the $$F=1$$ spinor Bose–Einstein condensate, with F denoting the atom’s spin. We construct a Lax pair and a binary Darboux transformation for such a system. Based on a nonzero seed solution, we obtain the kink and dark solitons corresponding to the ferromagnetic and polar states, respectively. Moreover, we obtain the W-shaped dark soliton. Interactions between the two dark solitons are discussed. We find that the interaction is inelastic if a kink soliton is included.

The Bose–Einstein condensates (BECs) are seen during the studies in atomic optics, cavity opto-mechanics, cavity quantum electrodynamics, black-hole astrophysics, laser optics and atomtronics. Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential which describes the dynamics of a (2+1)-dimensional BEC. Based on the Kadomtsev–Petviashvili hierarchy reduction, we construct the bilinear forms and the N th order rogue-wave solutions in terms of the Gramian. With the help of the analytic and graphic analysis, we exhibit the first- and second-order rogue waves under the influence of the strength of the interatomic interaction, α ( t ) , and of Ω ( t ) = ω R ∕ ω Z , where t is the scaled time, ω R and ω Z are the confinement frequencies in the radial and axial directions: When Ω ( t ) = 0 , the first-order rogue wave exhibits as an eye-shaped distribution; When Ω ( t ) is a periodic function, the rogue wave periodically raises; When Ω ( t ) is an exponential function, the rogue wave appears on the exponentially increasing background; When α ( t ) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when Ω ( t ) is a constant. With Ω ( t ) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with t increasing.

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a variable-coefficient fourth-order nonlinear Schrodinger system, which describes the simultaneous propagation of the optical pulses in an inhomogeneous optical fiber. The first-order breathers are constructed. Breathers are converted into several types of the nonlinear waves, i.e., multi-peak solitons, antidark solitons, periodic waves and W-shaped solitons, under the transition condition. With the decrease of | λ r + a 2 | , the peak number of a multi-peak soliton increases, where λr represents the real part of the spectral parameter and a is a real constant. Peak number reaches the maximum since the multi-peak soliton is transformed into a set of the periodic waves with λ r + a 2 = 0 . Widths and velocities of the multi-peak solitons are related to the group velocity dispersion and fourth-order dispersion coefficients. We find that the condition of baseband modulational instability coincides with the transition condition when we use the rogue-wave eigenvalues, i.e., the transition between the rogue waves and multi-peak solitons can occur in the baseband modulational instability.

Spin waves, the collective excitations of the electron spin systems in the ferromagnetic metals, are used in the telecommunication systems and radars. Under investigation is a (2+1)-dimensional nonlinear Schrodinger equation which describes the spin dynamics of a Heisenberg ferromagnetic spin chain. We construct its Lax pair which is different from the published ones. With respect to the coherent magnetism amplitude for the bosonic operators at the spin lattice sites, we derive the n-th-order breather solutions with n as a positive integer. We convert, under certain conversion conditions, the breathers to the lumps, rogue waves and two types of periodic waves which are named as the periodic-I and periodic-II waves in this paper. The breathers and periodic-I waves are affected by the uniaxial crystal field anisotropy parameter A as well as the bilinear exchange interactions. Periods of the periodic-II waves are affected by A and the lattice parameter. Via the second-order breather solutions, interactions of the two breathers, of the two periodic waves and of a breather and a periodic-I wave are graphically discussed. Through the theoretical and graphical analyses, it is found that the lumps and rogue waves are the long-wave limits of the breathers and periodic waves, respectively.

The Korteweg-de Vries (KdV)-type equations appear during the studies on the cosmic plasmas, planetary atmospheres and oceans. In this paper, via the Lie symmetry method, we investigate a three-coupled KdV system. Lie point symmetry generators and Lie symmetry groups are given. Based on the Lie point symmetry generators, the system is reduced to some ordinary differential systems. Applying the power-series, polynomial, Jacobi elliptic function and (G′/G) expansion methods on those reduced systems, we derive some group-invariant solutions including the explicit power-series, soliton, snoidal-wave and cnoidal-wave solutions. We graphically analyze the solitons and snoidal waves. Besides, we demonstrate the nonlinear self-adjointness of the system, based on which we present the conservation laws according to the Lie point symmetries.

For describing the propagation of ultrashort pulses in a high-speed, long-distance optical fiber transmission system with the fourth-order dispersion, cubic–quintic nonlinearity, self-steepening and self-frequency shift, a higher-order generalized nonlinear Schrodinger equation is investigated. We get the rogue-wave solutions. Effects of the modulation instability on the optical rogue waves are studied: Increasing the growth rate of the modulation instability makes the existence time of the optical rogue wave shorter. We numerically derive the optical breathers in the chaotic wave fields via the modulation instability. Spectrum of the optical chaotic wave field can be used to indicate the appearance of the optical breather in the chaotic wave field. Optical rogue waves in the chaotic wave fields are also gotten via the modulation instability.

The B-type Kadomtsev-Petviashvili (BKP) equations can describe certain nonlinear phenomena in the fluids. In this paper, a BKP equation is investigated. Lump-wave, mixed lump-kink wave, breather-wa...

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

Electromagnetic-wave/ferromagnetic-material investigations with magneto-optics are carried out in magneto-optic recording, computer data-storage, waveguides, etc. In this paper, we investigate a variable-coefficient modified Kadomtsev–Petviashvili system for certain electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film with the potential application in magneto-optic recording. With respect to the electromagnetic waves in the ferromagnetic film, symbolic computation helps us work out (1) two sets of the variable-coefficient-dependent auto-Backlund transformations via the generalized Laurent series, along with some solitonic features, (2) one set of the variable-coefficient-dependent bilinear forms and (3) two sets of the variable-coefficient-dependent N -solitonic solutions with N being a positive integer. We note that the electromagnetic field can excite the magnetization of a ferromagnetic medium in the form of solitons with the applications in magneto-optic recording. Relevant constraints on the variable coefficients are presented. We hope that the results could help the studies on magneto-optic recording, computer data-storage and waveguides.

The (2 $$+$$ 1)-dimensional Davey–Stewartson equations concerning the evolution of surface water waves with finite depth are studied. We derive the periodic-wave solutions through the Kadomtsev–Petviashvili hierarchy reduction. We obtain the growing-decaying periodic wave and three kinds of breathers via those solutions. We obtain the periodic wave takes on the growing and decaying property. Taking the long-wave limit on the periodic-wave solutions, we derive the semirational solutions describing the interaction of the rogue wave, lump, breather and periodic wave. We illustrate the lump and rogue wave and find that the rogue wave (lump) is the long-wave limit of the periodic wave (breather).

Lattices and liquids are common in physics, engineering, science, nature and life. The Yu-Toda-Sasa-Fukuyama (YTSF) equation describes the elastic quasiplane wave in a lattice or interfacial wave in a two-layer liquid. A (2 + 1)-dimensional reduced YTSF equation is studied in this paper. With symbolic computation, we get the lump solutions more general than those in the existing literature, which orient in all directions of space and time with more parameters. Via the lump solutions, we get the moving path of lump waves, lumpoff solutions and moving path of the lumpoff waves, which describe interactions between one stripe soliton and a lump wave. When the lump solutions produce the one stripe solitons, the lumpoff solutions are constructed. When a pair of stripe solitons cut lump wave, the rogue wave emerges. Time and place for the rogue wave to appear can be obtained from the coordinates of the interaction points between a pair of stripe solitons and the lump wave.

In this paper, we consider the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium, which is the isotropic Lax integrable extension of the Korteweg-de Vries equation. Infinitely-many conservation laws are constructed. N-soliton solutions in terms of the Wronskian are obtained via the Hirota method. Velocity and shape of the soliton can be influenced by those variable coefficients, while the amplitude of the soliton can not be affected. Collision between the two solitons is shown to be elastic. Breather wave solutions are constructed via the trilinear method. Such phenomena as the deceleration and compression of the breather waves are studied graphically. Rogue wave solutions are derived when the periods of the breather wave solutions go to the infinity. Separated and united composite rogue waves are discussed graphically.

In this paper, we investigate the nonlinear localized waves on the plane wave backgrounds of the coupled Fokas–Lenells system, which models the ultrashort optical pulses in a birefringent optical fiber. By a given Darboux transformation, we obtain some nonlinear localized waves on the plane wave backgrounds, and provide the conversion conditions between the vector breathers and solitons. Velocities of the conversion solitons are only related to the dispersion condition and perturbation parameter. We exhibit the interaction mechanism, which results in the shape changed of the conversion solitons. The interactions of the vector breathers and rogue waves are presented through the semi-rational solutions.

Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a variable-coefficient derivative nonlinear Schrodinger equation describing the propagation of the nonlinear Alfven waves in an inhomogeneous plasma. Based on the existing Lax pair, with respect to the left polarized Alfven wave, the N -fold generalized Darboux transformation, rational one/two Alfven soliton solutions and mixed two Alfven soliton solutions are derived, where N = 1 , 2 , 3 … . For those Alfven solitons, we find that (1) the solitonic width cannot be affected by the dispersion coefficient h ( τ ) and the loss/gain coefficient f ( τ ) , where τ is the stretched space variable; the solitonic amplitude grows (or decays) in the exponential rate exp [ − ∫ f ( τ ) d τ ] ; the solitonic amplitude cannot be affected by h ( τ ) ; the solitonic velocity and trajectory are related to h ( τ ) , while they cannot be affected by f ( τ ) ; (2) the rational two Alfven solitons experience no phase shift after the interaction; (3) the modulus square of mixed two Alfven solitons can be decomposed into two single Alfven solitons with the ∫ h ( τ ) d τ -dependent phase shifts; (4) the collapse of the rational one Alfven soliton is displayed.

Spin excitations for the magnetic materials are used in the nonlinear signal processing devices and microwave communication systems. Under consideration in this paper is a [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which describes the spin dynamics for a Heisenberg ferromagnetic spin chain. Through a reduced transformation, we convert such an equation into the [Formula: see text]-dimensional focusing NLS equation. Via the rogue-periodic solutions associated with two types of the Lie symmetry transformations of the NLS equation, we present the lump- and rogue-periodic solutions. Besides, the lump and mixed lump-soliton solutions are deduced. We graphically investigate the lump- and rogue-periodic waves and find that the amplitudes of the lumps and rogue waves are negatively related to [Formula: see text] and [Formula: see text]; the distances between two valleys of the lumps and widths of the rogue waves are affected by [Formula: see text] and [Formula: see text], where [Formula: see text] is the uniaxial crystal field anisotropy parameter, [Formula: see text] and [Formula: see text] are related to the bilinear exchange interaction, [Formula: see text] is the lattice parameter.

We focus on the spin dynamics of the nonlinear localized waves for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Based on the technique for the nonlinearization of spectral problem, we derive the values of the spectral parameter and periodic eigenfunction of the Lax pair associated with dn which is the Jacobian elliptic function of the third kind. According to the Darboux transformation, we take the seed solutions expressed by the Jacobian elliptic function and construct the semirational solutions. Such solutions are used for describing the eye-shaped lumps and line rogue waves on the dn-periodic backgrounds. Besides, the lump and mixed lump-soliton solutions are deduced through the semirational solutions.

In this paper, we investigate a (3+1)-dimensional modified Zakharov-Kuznetsov equation, which describes the nonlinear plasma-acoustic waves in a multicomponent magnetised plasma. With the aid of the Hirota method and symbolic computation, bilinear forms and one-, two- and three-soliton solutions are derived. The characteristics and interaction of the solitons are discussed graphically. We present the effects on the soliton’s amplitude by the nonlinear coefficients which are related to the ratio of the positive-ion mass to negative-ion mass, number densities, initial densities of the lower- and higher-temperature electrons and ratio of the lower temperature to the higher temperature for electrons, as well as by the dispersion coefficient, which is related to the ratio of the positive-ion mass to the negative-ion mass and number densities. Moreover, using the Lie symmetry group theory, we derive the Lie point symmetry generators and the corresponding symmetry reductions, through which certain analytic solutions are obtained via the power series expansion method and the (G′/G) expansion method. We demonstrate that such an equation is strictly self-adjoint, and the conservation laws associated with the Lie point symmetry generators are derived.

In the optical communication, people use the optical fibers to achieve the high bit-rate data transmission. In this paper, the AB system for the ultra-short pulses in a nonlinear optical fiber is investigated via the Lie symmetry analysis. Lie symmetries and symmetry reductions are derived via the Lie algorithm. Periodic- and solitary-wave solutions are obtained via the qualitative consideration. For the magnitude of the electric field in the optical fiber and the function associated with the occupation number which gives a measure of the atomic inversion in the nonlinear optical fiber, we can adjust the amplitudes, widths, and velocities of the solitary waves via the Lie symmetry transformations. The results would help the engineers select the ultra-short pulses in the optical communication.

In this paper, the AB system with time-dependent coefficients for the ultrashort pulses in an inhomogeneous optical fibre or the marginally unstable baroclinic wave packets in an atmospheric or oceanic system is investigated via the Lie symmetry analysis. We obtain the Lie symmetries, reduced equations and group-invariant solutions. The nonlinear self-adjointness of the AB system is proved, and the conservation laws associated with the Lie symmetries are constructed. For the amplitude of the electric field in the inhomogeneous optical fibre or the amplitude of the wave packet in the atmospheric or oceanic system, and for the quantity associated with the occupation number which gives a measure of the atomic inversion in the inhomogeneous optical fibre or the quantity measuring the correction of the basic flow in the atmospheric or oceanic system, we get some solitons through the Lie symmetry transformations, whose amplitudes, widths, velocities and backgrounds are different from those of the given ones and can be adjusted via the Lie group parameters. We find a family of the ultrashort pulses propagating in the inhomogeneous optical fibre or a family of the marginally unstable baroclinic wave packets propagating in the atmospheric or oceanic system.

In this paper, we study a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber for two orthogonally polarized optical pulses. With respect to the slowly-varying envelopes of two interacting optical modes and based on the existing binary Darboux transformation, we obtain four types of the bound-state solitons: degenerate-I, degenerate-II, degenerate–non-degenerate, and non-degenerate–non-degenerate bound-state solitons. We graphically analyze the interactions between the degenerate or non-degenerate solitons and four types of the bound-state solitons. When the degenerate solitons interact with the bound-state solitons, amplitudes and widths of the degenerate solitons remain unchanged. When the non-degenerate solitons interact with the bound-state solitons, amplitudes and widths of the bound-state solitons remain unchanged.

Electron-positron-ion plasmas are found in the primordial Universe, active galactic nuclei, surroundings of black holes and peripheries of neutron stars. We focus our attention on a modified Zakharov-Kuznetsov (mZK) equation which describes the ion acoustic drift solitary waves in an electron-positron-ion magnetoplasma. Lie symmetry generators and groups are presented by virtue of the Lie symmetry method. Optimal system of the one-dimensional subalgebras is presented, which is influenced via the ratio of the unperturbed ion density to electron density nio/neo, the ratio of the unperturbed positron density to electron density npo/neo, the ratio of the electron temperature to positron temperature Te/Tp and the normalized ion drift velocity v o * . Based on the optimal system, we construct the power-series, multi-soliton, breather-like and periodic-wave solutions. Two types of the elastic interactions, including the overtaking and head-on interactions between (among) two (three) solitons are discussed. We find that the amplitudes of the solitons and periodic waves are positively related to the electron Debye length λDe and negatively related to |ρi| with ρi as the ion Larmor radius. Besides, we find that the mZK equation is not only strictly self-adjoint but also nonlinearly self-adjoint. Condition for the nonlinear self-adjointness is related to nio/neo, npo/neo, Te/Tp and v o * . Based on the nonlinear self-adjointness of the mZK equation, conservation laws, which are related to nio/neo, npo/neo, Te/Tp, v o * , λDe, ρi and may be associated with the conservation of momentum and energy, are obtained.

We work on the (3+1)-dimensional Zakharov-Kuznetsov-Burgers equation for the ion-acoustic waves in a collisionless magnetized electron-positron-ion plasma. We derive the Lie point symmetry generators and Lie symmetry groups. We construct certain solutions which are related to the known solutions. Using the symmetry generators, we obtain the reduction equations, through one of which we derive some power-series solutions and travelling-wave solutions including the shock solutions via the power-series and polynomial expansion methods. Shock waves are pictured out. Effects of the normalized ion gyrofrequency, $\Omega_{i}$ , the normalized kinematic viscosity, $\eta$ , the real parameter measuring the deviation from the Maxwellian equilibrium, $ \kappa$ , the ratio of the ion temperature to electron temperature, $\sigma_{1}$ , and the ratio of the electron temperature to positron temperature, $ \sigma_{2}$ , on the amplitude of the shock wave, |a|, are found: i) |a| is a hyperbolic function of $\eta$ ; ii) $\vert a\vert \rightarrow 0$ when $\Omega_{i}$ , $\kappa$ or $\eta \rightarrow 0$ ; iii) | a| goes to a constant when $ \Omega_i \rightarrow \pm \infty$ ; iv) |a| becomes a constant when $\kappa \rightarrow \pm \infty$ ; v) $\vert a\vert \rightarrow 0$ when $\sigma_{1} \rightarrow \pm \infty$ ; vi) |a| keeps unchanged when $\sigma_{2}$ varies and $\sigma_{1}= \frac{-3(2\kappa + \lambda^{2} - 2\kappa\lambda^{2})}{5(-1+2\kappa)}$ where $\lambda$ denotes the ion-acoustic wave propagation speed; vii) $\vert a\vert \rightarrow 0$ when $\sigma_{2} \rightarrow \pm \infty$ and $\sigma_{1} \neq\frac{-3(2\kappa +\lambda^{2}-2\kappa\lambda^{2})}{5(-1+2\kappa)}$ . We present the conditions of the nonlinear self-adjointness. Based on the nonlinear self-adjointness, we construct the conservation laws which are related to $\Omega_{i}$ , $\eta$ , $\kappa$ , $\sigma_{1}$ and $\sigma_{2}$ .

Lattices are used in such fields as electricity, optics and magnetism. Under investigation in this paper is an inhomogeneous discrete nonlinear Schrödinger equation, which models the wave propagation in a lattice. Employing the Kadomtsev–Petviashvili (KP) hierarchy reduction, we obtain the rogue-wave solutions, and see that the rogue waves are affected by the coefficient of the on-site external potential. We see (1) the first-order rogue wave with one peak and two hollows; (2) the second-order rogue waves, each of which is with one peak or three humps; (3) the third-order rogue waves, each of which is with one peak or six humps, and those humps exhibit the triangular pattern, anti-triangular pattern and circular pattern. When the coefficient of the on-site external potential is a constant, the rogue waves periodically appear. When the coefficient of the on-site external potential monotonously changes, oscillations emerge on the constant background.

A set of the coherently-coupled nonlinear Schrodinger equations with the positive coherent coupling terms, which are related to the optical fiber communication, are studied through the binary Darboux transformation with the dimensional reduction. Formalisms of the solutions appear as the mixtures of the polynomial functions with exponential functions. When the spectral parameter is real, we obtain different kinds of the solutions, such as the soliton, degenerate-soliton, periodic, and soliton-like rational solutions. When the spectral parameter is complex with a non-zero imaginary part, we obtain the rogue waves and twisted rogue-wave pairs, and show that an eye-shaped rogue wave splits into a twisted rogue-wave pair.