New Mixed Solutions Generated By Velocity Resonance In The (2+1)-Dimensional Sawada-Kotera Equation

Based on the mixed solutions of the (2+1)-dimensional Sawada-Kotera equation, the collisions among lump waves, line waves, and breather waves are studied in this paper. By introducing new constraints, the lump wave does not collide with other waves forever. Under the condition of velocity resonance, the soliton molecules consisting of a lump wave, a line wave and any number of breather waves are derived for the first time. In particular, the interaction of a line wave and a breather wave will generate two breathers under certain conditions, which is very interesting. Additionally, the method can also be extended to other (2+1)-dimensional integrable equations.

Rogue and lump waves for the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation in a liquid or lattice

Two-layer fluid models are used to depict some nonlinear phenomena in fluid mechanics, medical science and thermodynamics. In this paper, we investigate a (3[Formula: see text]1)-dimensional Yu-Toda-Sasa-Fukuyama equation for the interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice. Via the Kadomtsev-Petviashvili hierarchy reduction, we derive the rational solutions in the determinant forms and semi-rational solutions. The [Formula: see text]th-order lump waves and multi-lump waves are obtained, where [Formula: see text] is a positive integer. We observe the second-order lump waves: Two-lump waves interact with each other and separate into two new lump waves. Two-lump waves are observed: Overtaking interaction takes place between the two-lump waves; After the interaction, the two-lump waves propagate with their original velocities and amplitudes. Studying the semi-rational solutions, we show the fusion between a lump wave and a bell-type soliton and fission of a bell-type soliton. Interaction between a line rogue wave and a bell-type soliton is shown.

Hybrid solutions of a (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation in an incompressible fluid

Incompressible fluids are studied in such disciplines as ocean engineering, astrophysics and aerodynamics. Under investigation in this paper is a [Formula: see text]-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation in an incompressible fluid. Based on the known bilinear form, BLMP hybrid solutions comprising a lump wave, a periodic wave and two kink waves, and hybrid solutions comprising a breather wave and multi-kink waves are derived. We observe the interaction among a lump wave, a periodic wave and two kink waves. Fission of a kink wave is observed: A kink wave divides into a breather wave and three kink waves. On the contrary, we see the fusion among a breather wave and three kink waves: The breather wave and three kink waves merge into a kink wave. Finally, we observe the interaction among a breather wave and four kink waves.

Interaction solutions of a variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources

Under investigation is a generalized variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources. Our main job is divided into four parts: (i) lump wave solution, (ii) interaction solutions between lump and solitary wave, (iii) breather wave solution and (iv) interaction solutions between lump and periodic wave. Furthermore, the interaction phenomenon of waves is shown in some 3D- and contour plots.

Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.

Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research.