Comment on ‘Study of lump solutions to an extended Calogero-Bogoyavlenskii-Schiff equation involving three fourth-order terms’ (2020 Phys. Scr. 95 095207)

Of current interest, in nonlinear optics, fluid dynamics and plasma physics, the paper commented (i.e., Phys. Scr. 95, 095207, 2020) has investigated a (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff system. Hereby, we make the issue raised in that paper more complete. Using the Hirota method and symbolic computation, we construct three sets of the bilinear auto-Bäcklund transformations for that system, along with some analytic solutions. As for the amplitude of the relevant wave in nonlinear optics, fluid dynamics or plasma physics, our results depend on the coefficients in that system.

Topologically non-trivial solution in a dissipative φ4 model with Lorentz-invariance violation

In this paper a (1+1)-dimension equation of motion for φ4-theory is considered for the case of simultaneously taking into a account of the processes of dissipation and violation the Lorentz-invariance. A topological non-trivial solution of one-kink type for this equation is constructed in an analytical form. To this end, the modified direct Hirota method for solving the nonlinear partial derivatives equations was used. A modification of the method lead to special conditions on the parameters of the model and the solution.

X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

Water waves are observed in the rivers, lakes, oceans, etc. Under investigation in this paper is a (2+1)-dimensional Hirota-Satsuma-Ito system arising in the shallow water waves. Via the Hirota method and symbolic computation, we derive some X-type and resonance Y-type soliton solutions. We also work out some hybrid solutions consisting of the resonance Y-type solitons, solitons, breathers and lumps. Graphics we present reveal that the hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps describe the interactions between the resonance Y-type solitons and solitons/breathers/lumps, respectively. The obtained results rely on the water-wave coefficient in that system.

The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation

According to the homoclinic breather limit method, we obtain the homoclinic breather wave and rational wave of a nonlinear evolution differential equation. The n-soliton wave solutions are derived by utilizing the Hirota method. In addition, the graphs of these solutions are shown by selecting the appropriate parameters.

Lump, mixed lump-stripe, mixed rogue wave-stripe and breather wave solutions for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

In this paper, the investigation is made on a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. We obtain the lump, mixed lump-stripe, mixed rogue wave-stripe and breather wave solutions via the Hirota method and symbolic computation. Accordingly, we observe that the shape and amplitude of the lump keep unchanged, while the mixed lump-stripe solutions present that (1) the lump/stripe and one-stripe waves fuse into a one-stripe wave; (2) the one-stripe wave splits into a lump/stripe and a one-stripe wave. We graphically show the interaction between a rogue wave and a pair of stripe waves, and observe that the amplitude of the rogue wave reaches the maximum value at a certain position. Besides, the breather wave propagates steadily in a certain direction.

THE LOCALIZED RESONANCE EXCITATIONS ARE DESCRIBED BY EXACT SOLUTIONS OF THE MODIFIED KORTEWEG-DE VRIES EQUATION

The exact soliton solutions of modified Korteweg-de Vries equation are obtained by procedure based on Hirota method. It has shown that thesе solutions described the bound state of soliton-antisoliton pairs which are formed in result resonance interaction of two solitons. Keywords: exact solution, rational-exponential solution, Hirota method

THE EXACT SOLUTIONS OF THE “MIXED” TYPE FOR SINE-GORDON MODEL

The exact soliton solution ("mixed" type) of sine-Gordon equation is obtained by procedure based on Hirota method. It has shown that this solution is described the bound state of degenerate (resonance) soliton and kink.

Bright–dark soliton dynamics and interaction for the variable coefficient three-coupled nonlinear Schrödinger equations

Under investigation in this paper is the variable coefficient three-coupled nonlinear Schrödinger (CNLS) equations, which govern the dynamics of solitonic excitations along three-spine [Formula: see text]-helical protein with inhomogeneous effect. Via the Hirota method and symbolic computation, the exact two-bright-one-dark (TBD) and one-bright-two-dark (BTD) soliton solutions are constructed analytically. The propagation properties are discussed for TBD and BTD solitons when the variable coefficient is a hyperbolic secant function. Figures are plotted to reveal the following interactions of TBD and BTD two solitons: (1) Evolution without interactions of double-parabola-shaped solitons, of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (2) Evolution with periodic interaction of double-parabola-shaped solitons and of parabola-[Formula: see text]-shaped solitons; (3) Collision of double-[Formula: see text]-shaped solitons and of parabola-[Formula: see text]-shaped solitons.