Resonant Collisions Among Two-Dimensional Localized Waves in the Mel'nikov Equation

We study the resonant collisions among different types of localized solitary waves in the Mel'nikov equation, which are described by exact solutions constructed using Hirota's direct method. The elastic collisions among different solitary waves can be transformed into resonant collisions when the phase shifts of these solitary waves tend to infinity . First, we study the resonant collision among a breather and a dark line soliton. We obtain two collision scenarios: (i) the breather is semi-localized in space and is not localized in time when it obliquely intersects with the dark line soliton, and (ii) the breather is semi-localized in time and is not localized in space when it parallelly intersects with the dark line soliton. The resonant collision of a lump and a dark line soliton, as the limit case of resonant collision of a breather and a dark line soliton, shows the fusing process of the lump into the dark line soliton. Then we investigate the resonant collision among a breather and two dark line solitons. In this evolution process we also obtain two dynamical behaviors: (iii) when the breather and the two dark line solitons obliquely intersect each other we get that the breather is completely localized in space and is not localized in time, and (iv) when the breather and the two dark line solitons are parallel to each other, we get that the breather is completely localized in time and is not localized in space. The resonant collision of a lump and two dark line solitons is obtained as the limit case of the resonant collision among a breather and two dark line solitons. In this special case the lump first detaches from a dark line soliton and then disappears into the other dark line soliton. Eventually, we also investigate the intriguing phenomenon that when a resonant collision among a breather and four dark line solitons occurs, we get the interesting situation that two of the four dark line solitons are degenerate and the corresponding solution displays the same shape as that of the resonant collision among a breather and two dark line solitons, except for the phase shifts of the solitons, which are not only dependent of the parameters controlling the waveforms of the solitons and the breather, but also dependent of some parameters irrelevant to the waveforms.

M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimensional Elliptic Toda Equation

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

MULTI-LINE SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL NONLINEAR HIROTA EQUATION

At present, the question of studying multidimensional nonlinear integrable equations in the framework of the theory of solitons is very interesting to foreign and Kazakh scientists. Many physical phenomena that occur in nature can be described by nonlinearly integrated equations. Finding specific solutions to such equations plays an important role in studying the dynamics of phenomena occurring in various scientific and engineering fields, such as solid state physics, fluid mechanics, plasma physics and nonlinear optics. There are several methods for obtaining real and soliton, soliton-like solutions of such equations: the inverse scattering method, the Hirota’s bilinear method, Darboux transformation methods, the tanh-coth and the sine-cosine methods. In our work, we studied the two-dimensional Hirota equation, which is a modified nonlinear Schrödinger equation. The nonlinear Hirota equation is one of the integrating equations and the Hirota system is used in the field of study of optical fiber systems, physics, telecommunications and other engineering fields to describe many nonlinear phenomena. To date, the first, second, and n-order Darboux transformations have been developed for the two- dimensional system of Hirota equations, and the soliton, rogue wave solutions have been determined by various methods. In this article, we consider the two-dimensional nonlinear Hirota equations. Using the Lax pair and Darboux transformation we obtained the first and the second multi-line soliton solutions for this equation and provided graphical representation.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Soliton molecules and some related interaction solutions of the (2+1)-dimensional Kadomtsev–Petviashvili hierarchy

The [Formula: see text]-soliton solutions of the (2+1)-dimensional Kadomtsev–Petviashvili hierarchy are first constructed. One soliton molecule satisfies the velocity resonance condition, the breather with the periodic solitary wave, the lump soliton localized in all directions in the space are showed successively for [Formula: see text]. Interaction of one soliton molecule and a line soliton, the soliton molecule hybrid a line soliton with the breather/lump soliton are presented for [Formula: see text]. Moreover, the elastic interaction between two-soliton molecules, the interaction between one soliton molecule, and a breather and the elastic collision between the lump soliton and one soliton molecule are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number, and the long-wave limit ideas. Figures are presented to demonstrate these dynamics features.

Lump, mixed lump-soliton, and periodic lump solutions of a (2+1)-dimensional extended higher-order Broer–Kaup System

In this paper, a (2+1)-dimensional extended higher-order Broer–Kaup system is introduced and its bilinear form is presented from the truncated Painlevé expansion. By taking the auxiliary function as the ansatzs including quadratic, exponential, and trigonometric functions, lump, mixed lump-soliton, and periodic lump solutions are derived. The mixed lump-soliton solutions are classified into two cases: the first one describes the non-elastic collision between one lump and one line soliton, which exhibits fission and fusion phenomena. The second one depicts the interaction consisting of one lump and two line soliton, which generates a rogue wave excited from two resonant line solitons.

Lie Symmetries and Low-Order Conservation Laws of a Family of Zakharov-Kuznetsov Equations in 2 + 1 Dimensions

In this work, we study a generalised (2+1) equation of the Zakharov–Kuznetsov (ZK)(m,n,k) equation involving three arbitrary functions. From the point of view of the Lie symmetry theory, we have derived all Lie symmetries of this equation depending on the arbitrary functions. Line soliton solutions have also been obtained. Moreover, we study the low-order conservation laws by applying the multiplier method. This family of equations is rich in Lie symmetries and conservation laws. Finally, when the equation is expressed in potential form, it admits a variational structure in the case when two of the arbitrary functions are linear. In addition, the corresponding Hamiltonian formulation is presented.

Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.