The dynamical behavior of mixed type lump solutions on the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation

This paper studies the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq (KPB) equation via the Hirota’s bilinear form and symbolic computation. Mixed type lump solutions are presented, which include rational function, trigonometric function and hyperbolic function. The propagation and the dynamical behaviors of these mixed type of lump solutions are shown by some three-dimensional and contour plots.

Lump Solutions of a Nonlinear PDE Combining with a New Fourth-Order Term Dx2Dt2∗

A nonlinear PDE combining with a new fourth-order term Dx2Dt2 is studied. Adding three new fourth-order derivative terms and some second-order derivative terms, we formulate a combined fourth-order nonlinear partial differential equation, which possesses a Hirota’s bilinear form. The class of lump solutions is constructed explicitly through Hirota’s bilinear method. Their dynamical behaviors are analyzed through plots.

The study of complexitons and periodic solitary-wave solutions with fifth-order KdV equation in (2+1) dimensions

In this paper, the generalized fifth-order (2[Formula: see text]+[Formula: see text]1)-dimensional KdV equation is scrutinized via the extended homoclinic test technique (EHTT) and extended transformed rational function (ETRF) method. With the aid of Hirota’s bilinear form, various exact solutions comprising, periodic solitary-wave, kinky-periodic solitary-wave, periodic soliton and complexiton solutions are constructed. Moreover, the mechanical features and dynamic characteristics of the obtained solutions are presented by three-dimensional plots.

Lump solitons and interaction phenomenon to a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

This paper studies lump solutions and interaction solutions for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. With the help of symbolic computation and Hirota’s bilinear form, we obtain bright–dark lump solutions, lump-soliton solutions, and lump-kink solutions. Meanwhile, the dynamics of the obtained three classes of solutions are analyzed and exhibited mathematically and graphically. These results provide us with useful information to grasp the propagation processes of nonlinear waves.

Mixed type exact solutions to the (2+1)-dimensional Ito equation

Using a direct test function based on the Hirota’s bilinear form, two classes of mixed type exact solutions to the (2[Formula: see text]+[Formula: see text]1)-dimensional Ito equation are found through symbolic computations with Mathematica. These mixed type exact solutions contain exponential function, trigonometric function and hyperbolic function. The physical structures and characteristics for these resulting mixed type exact solutions are illustrated by some three-dimensional plots and contour plots.

Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. Many results are obtained by dynamic process of figures. We analyze the propagation direction and horizontal velocity of lump solutions to find some constraint conditions which include positiveness and localization. In the process of the travel of the periodic lump solutions, it appears that the energy distribution is not symmetrical. The interaction lump stripe solutions of non-elastic indicate that the lump solitons are dropped and swallowed by the stripe soliton.

Spatio-temporal dynamics and interaction of lump solutions for the (4+1)-D Fokas equation

The (4+1)-D Fokas equation is a new and important physical model. Its Hirota's bilinear form with a perturbation parameter is obtained by an appropriate trans-formation. A class of lump solutions and three different forms of spatio-temporal structure are obtained. Meanwhile, the theoretical analysis for the change of spatio-temporal structure is discussed by using the extreme value theory of multivariate function. Finally, the interaction between a stripe soliton and lump solution is discussed, and a new wave phenomenon that the lump solution is swallowed and drowned by the stripe soliton is investigated.

Analytical and Numerical Methods for the CMKdV-II Equation

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.