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Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

Based on inverse scattering transformation, a variable-coefficient fifth-order nonlinear Schrödinger equation is studied through the Riemann–Hilbert (RH) approach with zero boundary conditions at infinity, and its multi-soliton solutions with [Formula: see text] distinct arbitrary-order poles are successfully derived. By deriving the eigenfunction and scattering matrix, and revealing their properties, a RH problem (RHP) is constructed based on inverse scattering transformation. Via solving the RHP, the formulae of multi-soliton solutions are displayed when the reflection coefficient possesses [Formula: see text] distinct arbitrary-order poles. Finally, some appropriate parameters are selected to analyze the interaction of multi-soliton solutions graphically.

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.

In this letter, we study a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient nonlinear Schrödinger system, which describes an optical beam inside the two-dimensional graded-index waveguide with polarization effects. Through the Kadomtsev–Petviashvili hierarchy reduction, the [Formula: see text] dark–dark soliton and [Formula: see text] bright-dark soliton solutions in terms of the Gramian are obtained, where [Formula: see text] is a positive integer. We analyze the interaction and propagation of the dark–dark solitons graphically. With the different values of the diffraction coefficient [Formula: see text], periodic-, cubic- and parabolic-shaped dark–dark solitons are derived. With the different values of the gain/loss coefficient [Formula: see text], periodic- and arctangent-profile background waves are obtained. Moreover, we discuss the effects from the dimensionless beam width [Formula: see text], [Formula: see text] and [Formula: see text] on the solitons and background waves: Shapes of the solitons are affected by [Formula: see text] and [Formula: see text], while profiles of the background waves are affected by [Formula: see text] and [Formula: see text].

This paper deals with [Formula: see text]-soliton solution of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation by virtue of the Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, periodic and cross-kink solutions, which have been investigated by the approach of the bilinear method. Mainly, by choosing specific parameter constraints in the [Formula: see text]-soliton solutions, all cases of the periodic and cross-kink solutions can be captured from the one-, two- and three-soliton solutions. The obtained solutions are extended with numerical simulation to analyze graphically, which results into one-, two- and three-soliton solutions and also periodic and cross-kink solutions profiles. That will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Under investigation in this paper is the variable coefficient three-coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of solitonic excitations along three-spine [Formula: see text]-helical protein with inhomogeneous effect. Via the Hirota method and symbolic computation, the exact two-bright-one-dark (TBD) and one-bright-two-dark (BTD) soliton solutions are constructed analytically. The propagation properties are discussed for TBD and BTD solitons when the variable coefficient is a hyperbolic secant function. Figures are plotted to reveal the following interactions of TBD and BTD two solitons: (1) Evolution without interactions of double-parabola-shaped solitons, of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (2) Evolution with periodic interaction of double-parabola-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (3) Collision of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons.

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 [Formula: see text]-soliton solutions are derived under some coefficient constraints via the Hirota method. Soliton interaction is graphically investigated: (1) Velocities of the [Formula: see text] and [Formula: see text] components and amplitude of the [Formula: see text] component are proportional to the parameter measuring the state of the basic flow, where [Formula: see text] is the [Formula: see text]th wave packet with [Formula: see text] and [Formula: see text] is related to the mean flow; Amplitudes of the [Formula: see text] components decrease with the group velocity increasing; Parabolic-type solitons, sine-type solitons and quasi-periodic-type two solitons are obtained; For the [Formula: see text] 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.

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

In this paper, we can obtain periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries (gvc-KdV) equation for the first time by using the multivariate transformation technique. Periodic solutions are doubly and triply periodic solutions, and the decay mode solutions include the 1-decay solution and the 2-decay mode solution. Furthermore, we will also study the effect of variation of parameters for solutions systematically, relating to different forms of expression, through graphic display.

Long-wave–short-wave resonance interaction (LSRI) equations have been studied in the plasmas, gravity waves, nonlinear electron-plasma and ion-acoustic waves. By virtue of the bilinear method, two soliton solutions of the variable-coefficient LSRI equations are attained. Interaction of the solitons are studied when the coefficients are taken as the generalized Gauss functions. New types of the soliton interaction are exhibited. Position and width of the disturbances can be controlled.

In this paper, we study a Kundu–Eckhaus equation with variable coefficients, which describes the ultra-short optical pulses in an inhomogeneous optical fiber. We construct the Lax pair under certain variable-coefficient constraints. With the gauge transformation, one/N-fold binary Darboux transformations and limit forms of the one-fold binary Darboux transformation are obtained. Based on such transformations, one/N-dark (N = 2,3, [Formula: see text]) soliton solutions under those constraints are derived. Linear, periodic and parabolic dark solitons are presented, and numerical simulations are used to investigate the influence of the group velocity dispersion on the structures of the one-dark solitons. Based on the two-dark soliton solutions under certain variable-coefficient constraints, we also discuss the influence of the group velocity dispersion on the structures of the two-dark solitons. Head-on and overtaking collisions between the two linear, parabolic and cubic-type dark solitons are presented.

In this paper, we investigate a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of [Formula: see text] and [Formula: see text] can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where [Formula: see text] and [Formula: see text] are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

The generalized nonlinear Schrödinger equation with the variable coefficient is discussed, and the exact dark N-soliton solution is presented by using the Hirota bilinear method, from finding the 1-soliton to 2-soliton, and then we obtain the N-soliton solution. With the aid of Maple, a few figures of solutions under several different cases are shown when aleatoric constants and variables are given exact values.

Under investigation in this paper is a variable-coefficient generalized AB system, which is used for modeling the baroclinic instability in the asymptotic reduction of certain classes of geophysical flows. Bilinear forms are obtained, and one-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

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 [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-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 [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] 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 paper is a generalized variable-coefficient Boussinesq system, which describes the propagation of the shallow water waves in the two-layered fluid flow. Bilinear forms, Bäcklund transformation and Lax pair are derived by virtue of the Bell polynomials. Hirota method is applied to construct the one- and two-soliton solutions. Propagation and interaction of the solitons are illustrated graphically: kink- and bell-shape solitons are obtained; shapes of the solitons are affected by the variable coefficients [Formula: see text], [Formula: see text] and [Formula: see text] during the propagation, kink- and anti-bell-shape solitons are obtained when [Formula: see text], anti-kink- and bell-shape solitons are obtained when [Formula: see text]; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic.

Liquids with gas bubbles are commonly seen in medical science, natural science, daily life and engineering. Nonlinear-wave symbolic computation on the (3[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Kudryashov–Sinelshchikov model for a bubbly liquid is hereby performed. An auto-Bäcklund transformation and some solitonic solutions are obtained. With respect to the density fluctuation of the bubble-liquid mixture, both the auto-Bäcklund transformation and solitonic solutions depend on the bubble-liquid-viscosity, transverse-perturbation, bubble-liquid-nonlinearity and bubble-liquid-dispersion coefficient functions. We note that some shock waves given by our solutions have been observed by the gas-bubble/liquid-mixture experiments. Effects on a bubbly liquid with respect to the bubble-liquid-viscosity, transverse-perturbation, bubble-liquid-nonlinearity and bubble-liquid-dispersion coefficient functions might be detected by the future gas-bubble/liquid-mixture experiments.