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.

Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns

In this research, we explore the dynamics of Caudrey–Dodd–Gibbon–Sawada–Kotera equations in (1 + 1)-dimension, such as N-soliton, and breather solutions. First, a logarithmic variable transform based on the Hirota bilinear method is defined, and then one, two, three and N-soliton solutions are constructed. A breather solution to the equation is also retrieved via N-soliton solutions. All the solutions that have been obtained are novel and plugged into the equation to guarantee their existence. 2-D, 3-D, contour plot and density plot are also presented.

The order-n breather and degenerate breather solutions of the (2+1)-dimensional cmKdV equations

Starting with a plane wave seed, the order-[Formula: see text] breather for the (2+1)-D complex modified Korteweg-de Vries (cmKdV) equations is obtained by the use of Darboux transformation. The dynamic evolution of order-2 and order-3 breather solutions is shown in the form of pictures. Afterward, we obtain the order-[Formula: see text] degenerate breather solution by using the Taylor expansion concerning the limits [Formula: see text] and focus on the order-2 degenerate breather solution. We show the dynamic evolution with time and discuss the degradation process from a breather solution through getting [Formula: see text] closer and closer to [Formula: see text]. Furthermore, the approximate trajectories of the order-2, order-3, order-4 degenerate breather solutions are depicted by explicit expressions, respectively.

Двухкомпонентный векторный бризер

The two-component vector breather solution of the modified Benjamin–Bona–Mahony equation is considered. By means of the generalized perturbation reduction method, the equation is reduced to the coupled nonlinear Schrodinger equations for auxiliary functions. Explicit analytical expressions for the profile and parameters of the two-component vector breather, the components of which oscillating with the sum and difference of the frequencies and wave numbers are obtained.

Ghost Interaction of Breathers

Mutual interaction of localized nonlinear waves, e.g., solitons and modulation instability patterns, is a fascinating and intensively-studied topic of nonlinear science. Here we report the observation of a novel type of breather interaction in telecommunication optical fibers, in which two identical breathers propagate with opposite group velocities. Under controlled conditions, neither amplification nor annihilation occurs at the collision point and most interestingly, the respective envelope amplitude, resulting from the interaction, almost equals another envelope maximum of either oscillating and counterpropagating breather. This ghost-like breather interaction dynamics is fully described by an N-breather solution of the nonlinear Schrödinger equation.