Narrow Results

We study the bilinearization of $N=1$ Supersymmetric Coupled Dispersionless (SUSY-CD) integrable system by introducing transformations defined in terms of bosonic and fermionic superfields. From the bosonic and fermionic superfield bilinear equations, we obtain the superfield soliton solutions of SUSY-CD integrable system.

In this paper, we derive nonisospectral AKNS equations from the AKNS spectral problem. By using a new Loop algebra , their integrable couplings are constructed. Moreover, bilinear forms of the nonisospectral AKNS equation are given. New exact solutions are obtained through the Hirota method. Some nonisospectral characteristics of the obtained solutions are discussed.

In this paper we investigate the constant- and variable-coefficient Zakharov–Kuznetsov (ZK) equations respectively for the electrostatic solitons and two-dimensional ion-acoustic waves obliquely propagating in the inhomogeneous magnetized two-ion-temperature dusty plasmas. By virtue of the symbolic computation and Hirota method, new bilinear forms and N-soliton solutions are both derived. Asymptotic analysis on two-soliton solutions indicates that the soliton interaction is elastic. Propagation characteristics and interaction behavior of the solitons are discussed via graphical analysis. Effects of the dispersive and disturbed coefficients are analyzed. For the constant-coefficient ZK equation, amplitude of the one soliton becomes larger when the absolute value of dispersive coefficient B increases, while interaction between the two solitons varies with the product of B and disturbed coefficient C: when BC>0, two solitons are always parallel, or they interact with each other that way. For the variable-coefficient ZK equation, periodical soliton arises when the disturbed coefficient γ(t) is a periodical function, and periods of the solitons are inversely correlated to the period of γ(t).

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

In this paper, we investigate a Bogoyavlenskii–Kadomtsev–Petviashili equation, which can be used to describe the propagation of nonlinear waves in physics, biology and electrical networks. We find that the equation is Painlevé integrable. With symbolic computation, Hirota bilinear forms, solitary waves and multi-front waves are derived. Elastic collisions between/among the two and three solitary waves are graphically discussed, where the waves maintain their shapes, amplitudes and velocities after the collision only with some phase shifts. Inelastic collisions among the multi-front waves are discussed, where the front waves coalesce into one larger front wave in their collision region.

Under investigation in this paper is the $(3+1)$-dimensional coupled nonlinear Schrödinger system for an optical fiber with birefringence. With the Hirota method, bilinear forms of the system are derived via an auxiliary function, and the bright one- and two-soliton solutions are constructed. Based on those soliton solutions, soliton propagation and interaction are investigated analytically and graphically. Non-singular cases of the bright one-soliton solutions are presented, from which the single-peak and two-peak solitons can arise, respectively. Through the analysis on the bright two-soliton solutions, the elastic and inelastic interactions are investigated. Three kinds of the elastic interactions are presented, between the two one-peak solitons, a one-peak soliton and a two-peak soliton, and the two two-peak solitons.

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

Under investigation in this paper is a (3 + 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation, which describes the nonlinear behaviors of ion-acoustic waves in a magnetized plasma where the cooler ions are treated as a fluid with adiabatic pressure and the hot isothermal electrons are described by a Boltzmann distribution. With the Hirota method and symbolic computation, we obtain the one-, two- and three-soliton solutions for such an equation. We graphically study the solitons related with the coefficient of the cubic nonlinearity $M$. Amplitude of the one soliton increases with increasing $M$, but the width of one soliton keeps unchanged as $M$ increases. The two solitons and three solitons are parallel, and the amplitudes of the solitons increase with increasing $M$, but the widths of the solitons are unchanged. It is shown that the interactions between the two solitons and among the three solitons are elastic.

Under investigation in this paper is a (3 + 1)-dimensional generalized nonlinear Schrödinger equation with the distributed coefficients for the spatiotemporal optical solitons or light bullets. Through the symbolic computation and Hirota method, one- and two-soliton solutions are derived. We also present the Bäcklund transformation, through which we derive the soliton solutions. When the gain/loss coefficient is the monotonically decreasing function for the propagation coordinate $z$, amplitude for the spatiotemporal optical soliton or light bullet decreases along $z$, while when the gain/loss coefficient is the monotonically increasing function for $z$, amplitude for the spatiotemporal optical soliton or light bullet increases along $z$. Directions of the solitons are different because the signs of imaginary parts of the frequencies are adverse. Based on the two-soliton solutions, elastic and inelastic collisions between the two spatiotemporal optical solitons or light bullets are derived under different conditions presented in the paper.

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

Under investigation are the soliton interactions for a (2+1)-dimensional nonlinear Schrödinger equation, which can describe the dynamics of a nonlinear photonic quasi-crystal or vortex Airy beam in a Kerr medium. With the symbolic computation and Hirota method, analytic bright N-soliton and dark two-soliton solutions are derived. Graphic description of the soliton properties and interactions in a nonlinear photonic quasicrystal or Kerr medium is done. Through the analysis on bright and dark one solitons, effects of the optical wavenumber/linear opposite wavenumber and nonlinear coefficient on the soliton amplitude and width are studied: when the absolute value of the optical wavenumber or linear opposite wavenumber increases, bright soliton amplitude and dark soliton width become smaller; nonlinear coefficient has the same influence on the bright soliton as that of the optical wavenumber or linear opposite wavenumber, but does not affect the dark soliton amplitude or width. Overtaking/periodic interactions between the bright two solitons and overtaking interactions between the dark two solitons are illustrated. Overtaking interactions show that the bright soliton with a larger amplitude moves faster and overtakes the smaller, while the dark soliton with a smaller amplitude moves faster and overtakes the larger. When the absolute value of the optical wavenumber or linear opposite wavenumber increases, the periodic-interaction period becomes longer. All the above interactions are elastic. Through the interactions, soliton amplitudes and shapes keep invariant except for some phase shifts.

In this paper, we investigate the dynamics of a generalized Davydov model derived from an infinite chain of alpha helical protein molecules which contain three hydrogen bonding spines running almost parallel to the helical axis. Through the introduction of the auxiliary function, the bilinear form, one-, two- and three-soliton solutions for the generalized Davydov model are obtained firstly. Propagation and interactions of solitons have been investigated analytically and graphically. The amplitude of the soliton is only related to the complex parameter $\mu$ and real parameter $𝜃$ with a range of $[0,2\pi ]$. The velocity of the soliton is only related to the complex parameter $\mu$, real parameter $𝜃$, lattice parameter $𝜀$, and physical parameters ${\beta }_1$, ${\beta }_3$ and ${\beta }_4$. Overtaking and head-on interactions of two and three solitons are presented. The common in the interactions of three solitons is the directions of the solitons change after the interactions. The soliton derived in this paper is expected to have potential applications in the alpha helical proteins.

Analytic wave solutions including homoclinic wave, kink wave and soliton solutions for the 2D coupled complex Ginzburg–Landau equations are obtained using the auxiliary function method, Hirota method and the ansatz function technique under certain constraint conditions of coefficients in equations, respectively. The result shows that there exists a kink-wave solution which tends to one and the same periodic wave solution as time tends to infinite.

Geophysical flows consist of the large-scale motions of the ocean and/or atmosphere. Researches on the geophysical flows reveal the mechanisms for the transport and redistribution of energy and matter. Investigated in this paper is a variable-coefficient three-component AB system for the baroclinic instability processes in geophysical flows. With respect to the three wave packets as well as the correction to the mean flow, bilinear forms are obtained, and one-, two- and $N$-soliton solutions are derived under some coefficient constraints via the Hirota method. Soliton interaction is graphically investigated: (1) Velocities of the $A_j$ and $B$ components and amplitude of the $B$ component are proportional to the parameter measuring the state of the basic flow, where $A_j$ is the $j$th wave packet with $j=1,2,3$ and $B$ is related to the mean flow; Amplitudes of the $A_j$ components decrease with the group velocity increasing; Parabolic-type solitons, sine-type solitons and quasi-periodic-type two solitons are obtained; For the $B$ component, solitons with the varying amplitudes and dromion-like two solitons are shown; (2) Three types of the breathers with different interaction periods and numbers of the wave branches in a wave packet are analyzed; (3) Bound states are depicted; (4) Compression of the soliton is presented; (5) Interactions between/among the solitons and breathers are also illustrated.

In this paper, a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma is investigated. Via the Hirota method and symbolic computation, we work out some two-resonance Y-type soliton solutions as well as some hybrid solutions composed of the two-resonance Y-type solitons and solitons/breathers. Graphically, we display some two-resonance Y-type solitons. We present the interactions between the two-resonance Y-type soliton and one soliton, among the two-resonance Y-type soliton and two solitons, between the two-resonance Y-type soliton and first-order breather as well as among the two-resonance Y-type soliton and second-order breathers.

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation which consists of the KdV equation and the SK equation is the subject of investigation in this study. The studied equation has rich physical meaning in nonlinear waves. The KdV- type equations hold great importance as a prototypical representation of an infinite-dimensional system that is completely integrable and exactly solvable in the context of nonlinearity. The KdV equation is utilized to describe shallow water waves in a density-stratified ocean, which exhibit weak and nonlinear interactions with long internal waves. The Hirota bilinear method has been used with the support of various test functions. For the purpose of analyzing the governing equation, numerous solutions are secured, including breathers and two-wave solutions. Breather waves refer to solitary waves that exhibit both partial localization and periodic structure in either space or time. Breathers serve crucial functions in nonlinear physics and have been observed in various physical domains, including optics, hydrodynamics, and quantized superfluidity. To visually represent the results, a range of graphs with unique shapes are generated in accordance with the specified parameter values. The computational intricacies and outcomes underscore the technique’s efficacy, simplicity and transparency, demonstrating its suitability for numerous types of static and dynamic nonlinear equations pertaining to evolutionary phenomena in computational physics, in addition to other research and practical domains. The physical properties of solutions and the collision-related components of various nonlinear physical processes are illustrated with these results.

The exploration of nonlinear phenomena entails the representation of intricate systems with space-time variables, and across this line, Itô calculus, as the stochastic calculus version of the change pertaining to the variables formula and chain rules, involves the second derivative of f, coming from the property that Brownian motion has non-zero quadratic variation. To this end, the extended ($2+1$)-dimensional two-component Itô equation, as an applicable mathematical tool employed in this study for enhancing our understanding of the complex dynamics inherent in multidimensional physical systems, serves the purpose of modeling and understanding dynamic phenomena pervading various disciplines. For modeling complex phenomena, fractional differential equations (FDEs), ordinary differential equations (ODEs), partial differential equations (PDEs) as well as the other ones provide benefits in terms of accuracy and tractability. Accordingly, our study provides the analysis of the two-component nonlinear extended ($2+1$)-dimensional Itô equation using the Hirota bilinear method to derive multiple soliton solutions, including novel variations along with their dispersion coefficients, which shed light into the intriguing attributes of the Itô equation. The investigation further encompasses diverse soliton types, such as the general first-order soliton, second-order soliton with fission and bifurcation, third-order soliton, and fourth-order soliton with fission and bifurcation. Besides these, the study also explores the rogue wave and lump solutions by varying parameters across distinct planes. Consequently, these results validate the characteristics and utility of the two-component nonlinear extended ($2+1$)-dimensional Itô equation and its relevance to related systems. The novel findings based on the Itô calculus systems revealed, through the analyses, theoretical and experimental aspects in combination with the graphical presentation of the parameter effects on solitons in line with the analyses obtained. These have enhanced the understanding of the dynamics of intricate attributes governed by the two-component nonlinear extended ($2+1$)-dimensional Itô equation, particularly concerning chaotic patterns, fission and bifurcation soliton nonlinear complexities.