Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation

In this paper, we obtain the 1-soliton solutions of the (3?+?1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and the generalized Benjamin equation. By using two solitary wave ansatz in terms of sech ^p and $\tanh^{p}$ functions, we obtain exact analytical bright and dark soliton solutions for the considered model. These solutions may be useful and desirable for explaining some nonlinear physical phenomena in genuinely nonlinear dynamical systems.     

Invariant solutions and conservation laws of the generalized Kaup–Boussinesq equation

The generalized Kaup–Boussinesq equation is a model which is used to describe the water wave. In this paper, Lie group analysis method is used to perform detailed analysis on the generalized Kaup–Boussinesq equation. Some invariant solutions are obtained under the transformation groups. The conservation laws of the generalized Kaup–Boussinesq equation are constructed using two methods: multiplier method and Ibragimov theorem.     

The generalized KaupߝBoussinesq equation: multiple soliton solutions

In this work, we investigate the generalized two-field Kaup–Boussinesq (KB) equation. The KB equation possesses the cubic nonlinearity that distinguishes it from the Boussinesq equation that contains quadratic nonlinearity. We use the simplified form of Hirota’s direct method to determine multiple soliton solutions and multiple singular soliton solutions for this equation. The study exhibits physical structures for a generalized water–wave model.     

New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation

In this paper, new exact solutions, including soliton, rational and elliptic integral function solutions, for the generalized Zakharov–Kuznetsov modified equal-width equation are obtained using a new approach called the extended trial equation method. In this discussion, a new version of the trial equation method for the generalized nonlinear partial differential equations is offered.     

New soliton solutions for Sasa–Satsuma equation

In this work, we survey exact solutions of Sasa–Satsuma equation (SSE). We utilize extended trial equation method (ETEM) and generalized Kudryashov method to acquire exact solutions of SSE. First of all, we gain some exact solutions such as soliton solutions, rational, Jacobi elliptic, and hyperbolic function solutions of SSE by means of ETEM. Furthermore, we procure dark soliton solution of this equation by the help of generalized Kudryashov method. Lastly, for certain parameter values, we draw two- and three-dimensional graphics of imaginary and real values of some exact solutions that we achieved using these methods.     

Dark solitons of the Biswas–Milovic equation by the first integral method

This paper studies the Biswas–Milovic equation that is a generalized version of the familiar nonlinear Schrodinger's equation describing the propagation of solitons through optical fibers for trans-continental and trans-oceanic distances with Kerr law nonlinearity by the aid of the first integral method. The dark 1-soliton solution is retrieved by the aid of this method and a couple of other singular periodic solutions are also obtained.     

Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel’nikov equation and the multicomponent Schrödinger–Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel’nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrödinger–Boussinesq system are generated.     

Four-soliton solution and soliton interactions of the generalized coupled nonlinear Schr?dinger equation

Based on the generalized coupled nonlinear Schr ¨odinger equation, we obtain the analytic four-bright–bright soliton solution by using the Hirota bilinear method. The interactions among four solitons are also studied in detail. The results show that the interaction among four solitons mainly depends on the values of solution parameters; k1 and k2 mainly affect the two inboard solitons while k3 and k4 mainly affect the two outboard solitons; the pulse velocity and width mainly depend on the imaginary part of ki(i = 1, 2, 3, 4), while the pulse amplitude mainly depends on the real part of ki(i = 1, 2,3, 4).     

New soliton solutions of Davey–Stewartson equation with power-law nonlinearity

This study is related to new soliton solutions of Davey–Stewartson equation (DSE) with power-law nonlinearity. The generalized Kudryashov method which is one of the analytical methods has been used for finding exact solutions of this equation. By using this method, dark soliton solutions of DSE have been found. Also, by using Mathematica Release 9, some graphical representations have been done to analyze the motion of these solutions.     

Classification of single travelling wave solutions to the generalized Zakharov–Kuznetsov equation

By the complete discrimination system for the polynomial method, the classification of single travelling wave solutions to the generalized Zakharov–Kuznetsov equation with p?=?2 was obtained.     

Evolution property of soliton solutions for the Whitham－Broer－Kaup equation and variant Boussinesq equation

Using the standard Painlevé analysis approach, the (1+1)-dimensional Whitham－Broer－Kaup (WBK) and variant Boussinesq equations are solved. Some significant and exact solutions are given. We investigate the behaviour of the interactions between the multi-soliton-kink-type solution for the WBK equation and the multi-solitonic solutions and find the interactions are not elastic. The fission of solutions for the WBK equation and the fusions of those for the variant Boussinesq equation may occur after their interactions.     

The investigation of soliton solutions and conservation laws to the coupled generalized Schrödinger–Boussinesq system

This paper employed the principle of undetermined coefficients and Bernoulli sub-ODE methods to acquire the topological, non-topological, periodic wave and algebraic solutions of the coupled generalized Schrödinger–Boussinesq system (CGSBs). The concept of Lie point symmetry is applied to derive the point symmetries of the CSGE. The problem on nonlinear self-adjointness of the CSGE has not been solved in previous time. In the present paper, we solve this problem and find an explicit form of the differential substitution providing the nonlinear self-adjointness. Then we use this fact to construct a set of conserved vectors using the classical symmetries admitted by the equation and the general conservation laws (Cls) theorem presented by Ibragimov. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.     

Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the Boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic function solutions, combined Jacobian elliptic function solutions, soliton solutions, triangular function solutions.     

The classification of the single travelling wave solutions to the generalized Pochhammer–Chree equation

By the complete discrimination system for the polynomial, we invest the classifications of single travelling wave solutions to the generalized Pochhammer–Chree (PC) equation with p?=?1/2 and p?=?3/2.     

Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications

The propagation of the optical solitons is usually governed by the higher order nonlinear Schrödinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.     

New doubly periodic and multiple soliton solutions of the generalized （3＋l）-dimensional KP equation with variable coefficients

A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.     

Exact solutions of the Wheeler–DeWitt equation and the Yamabe construction

Exact solutions of the Wheeler–DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff(CBS) equation. This model describes the(2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation(ESE) method is applied to the model, and a variety of novel solitarywave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration(VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme's performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.     

Soliton solutions and conservation laws of the Gilson–Pickering equation

This paper obtains the soliton solutions of the Gilson–Pickering equation. The G′/G method will be used to carry out the solutions of this equation and then the solitary wave ansatz method will be used to obtain a 1-soliton solution of this equation. Finally, the invariance and multiplier approach will be applied to recover a few of the conserved quantities of this equation.