Homoclinic Breather Research Articles

Multiwave and interaction solutions and Lie symmetry analysis to a new (2 + 1)-dimensional Sakovich equation

In this study, we construct multi wave solutions for the (2 + 1)-dimensional Sakovich equation by utilizing the logarithmic transformation of the dependent variables and symbolic computation with the ansatz function technique. We used three different method including multi waves method, double exponential form and homoclinic breather approach. 3D graphics of the solutions in different structures are drawn for some values for the parameters. Furthermore, the Lie symmetry analysis is performed for the (2 + 1)-dimensional Sakovich equation. Using the Lie symmetry groups approach, we construct the transformation groups and vector fields. We obtain the symmetry reductions and invariant solutions of the equation via these vector fields.

Multi-wave, breather wave and lump solutions of the Boiti–Leon–Manna–Pempinelli equation with variable coefficients

In this paper, a variable-coefficient Boiti–Leon–Manna–Pempinelli equation is to be investigated. We obtain abundant multi-wave, breather wave and lump solutions by using the three waves method, the homoclinic breather approach and the Hirota’s bilinear method. All solutions have been verified to be correct with the help of Mathematica software. We study the propagation characteristics of these solutions by some three-dimensional images. The obtained results are useful for understanding fluid propagating and incompressible fluid.

Benjamin-Ono equation: Rogue waves, generalized breathers, soliton bending, fission, and fusion

In this work, we construct various interesting localized wave structures of the Benjamin-Ono equation describing the dynamics of deep water waves. Particularly, we extract the rogue waves and generalized breather solutions with the aid of bilinear form and by applying two appropriate test functions. Our analysis reveals the control mechanism of the rogue waves with arbitrary parameters to obtain both bright and dark type first and second-order rogue waves. Additionally, a generalization of the homoclinic breather method, also known as the three-wave method, is used for extracting the generalized breathers along with bright, dark, anti-dark, rational solitons. Interestingly, we have observed the manipulation of breathers along with soliton interaction, bending, fission, and fusion. Our results are discussed categorically with the aid of clear graphical demonstrations.

Multi-wave, breather wave, and interaction solutions of the Hirota–Satsuma–Ito equation

Under investigation is a (2+1)-dimensional generalized Hirota–Satsuma–Ito (gHSI) equation. Multi-wave solutions are presented using the three-wave method. Breather wave solutions are derived based on the homoclinic breather approach. Interaction solutions between lump and two solitary wave solutions are constructed via Hirota direct method. Through giving some special values to the parameters, dynamical behaviors of the obtained solutions are demonstrated in some 3D and contour plots.

Multi-waves, breather wave and lump–stripe interaction solutions in a (2 $$+$$ 1)-dimensional variable-coefficient Korteweg–de Vries equation

Under investigation is a ( $$2 + 1$$ )-dimensional variable-coefficient Korteweg–de Vries equation, which is used to describe the electrostatic wave potential in plasmas or shallow-water waves in fluids. Our work is mainly divided into three aspects: (1) lump–stripe interaction solutions are obtained by the Hirota’s bilinear form with variable coefficients; (2) multi-waves solutions are presented based on three waves method with variable coefficients; (3) breather wave solutions are given by the homoclinic breather approach. Furthermore, interaction phenomenon of these obtained solutions are demonstrated by some 3D- and contour plots.

Kinky breathers, W-shaped and multi-peak solitons interaction in (2 + 1)-dimensional nonlinear Schrödinger equation with Kerr law of nonlinearity

The purpose of this paper is to construct multiwave solutions for the (2 + 1)-dimensional nonlinear Schrodinger equation by utilizing the logarithmic transformation and symbolic computation with the ansatz function method. We used three different techniques, namely, three waves method, double exponential form and homoclinic breather approach. By selecting appropriate values of the parameter, 3d plots are drawn to obtained kinky breathers, W-shaped and multi-peak solitons. Furthermore we observed three different types of very interesting interactional phenomena between multi-peak solitons and multi-kink waves.

Homoclinic breather wave, rouge wave and interaction solutions for a (3+1)-dimensional KdV-type equation

Based on symbolic computations, homoclinic breather wave, rouge wave and interaction solutions to the (3+1)-dimensional KdV-type equation are constructed. Two homoclinic breather solutions and a rouge wave solution are obtained by using the extended homoclinc test. Subsequently, we study interaction solutions between a lump solution and 1-kink soliton or 2-kink soliton of the KdV-type equation. In addition, periodic interaction solutions of the KdV-type equation are also studied. Moreover, dynamics behavior of these obtained solutions are exhibited using graphical analysis.

Rogue waves, homoclinic breather waves and soliton waves for a (3 + 1)-dimensional non-integrable KdV-type equation

PurposeThe purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.FindingsThe results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.Originality/valueThese results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Homoclinic breather waves, rogue waves and solitary waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

PurposeThe purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.Design/methodology/approachThe authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.FindingsThe results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.Originality/valueThese results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Rogue wave dynamics in barotropic relaxing media

In this work, we deal with a nonlinear wave equation, namely the Vakhnenko equation, which models the propagation of nonlinear wave in the barotropic relaxing media. Based on the homoclinic breather limit method, we seek rogue wave solution to the above equation. The results show that rogue wave or giant wave can exist in such a medium.

Homoclinic breather and rogue wave solutions to Maccari equation

In this work, a new extended homoclinic test combining with breather period limit method (BPLM), for seeking homoclinic breather and rogue wave solution of nonlinear evolution (2+1)D Maccari equation, is proposed. Two new families of homoclinic breather wave solution and rational homoclinic solution (homoclinic rogue wave) for these equations are obtained, respectively. Moreover, dynamical features of homoclinic breather and rogue wave solution are exhibited. This result shows that both homoclinic breather and rogue wave can occur in the same nonlinear system.

Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation

We consider a (2+1)-dimensional generalized breaking soliton (gBS) equation, which describes the interaction of the Riemann wave propagated along the y-axis with a long wave propagated along the x-axis. By using Bell’s polynomials, we derive a bilinear form of the gBS equation. Based on the resulting Hirota’s bilinear equation, we explicitly construct its soliton solutions. Furthermore, by using the extended homoclinic test theory, its homoclinic breather waves and rogue waves are well derived, respectively. It is hoped that our results can enrich the dynamical behavior of the gBS-type equations.

Dynamics of the breather waves, rogue waves and solitary waves in an extend Kadomtsev–Petviashvili equation

Under investigation in this work is an extend Kadomtsev–Petviashvili (eKP) equation, which appears in the study of multi-component plasmas. By considering Bell’s polynomials, an effective and straightforward way is presented to succinctly derive its bilinear form and soliton solutions. Moreover, the homoclinic breather limit method is employed to construct the breather wave and rogue wave solutions of the equation. Finally, the dynamic behaviors of breather waves, rogue waves and solitary waves are discussed with graphic analysis. It is hoped that our results can be useful for explaining and enriching the dynamic behavior of these KP-type equations.

On breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCHKP) equation is investigated, which describes the role of dispersion in the formation of patterns in liquid drops. We succinctly construct its bilinear formalism. By further using homoclinic breather limit approach, some exact solutions including breather waves, rogue waves and solitary waves of the equation are well presented. Our results show that rogue waves can come from the extreme behavior of the breather solitary waves for the (2+1)-dimensional gCHKP equation.

On the breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation

In this paper, we consider a generalized (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera- Sawada (CDGKS) equation. By using the Bell polynomial, we derive its bilinear form. Based on the homoclinic breather limit method, we construct the homoclinic breather wave and the rational rogue wave solutions of the equation. Then by using its bilinear form, some solitary wave solutions of the CDGKS equation are provided by a very natural way. Moreover, some prominent characteristics for the dynamic behaviors of these solitons are analyzed by several graphics. Our results show that the breather wave can be transformed into rogue wave under the extreme behavior.

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

Solitary waves, rogue waves and homoclinic breather waves for a (2 + 1)-dimensional generalized Kadomtsev–Petviashvili equation

We study a (2 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which characterizes the formation of patterns in liquid drops. By using Bell’s polynomials, an effective way is employed to succinctly construct the bilinear form of the gKP equation. Based on the resulting bilinear equation, we derive its solitary waves, rogue waves and homoclinic breather waves, respectively. Our results can help enrich the dynamical behavior of the KP-type equations.

Solitons and breather waves for a (2+1)-dimensional Sawada–Kotera equation

Under investigation in this letter is a (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada-Kotera equation. With the aid of the bilinear forms derived from the Bell polynomials, the Nth-order soliton solutions are obtained via the Pffafian method, and breather solutions are derived with the ansätz method. Analytic solutions obtained via the Pffafian method are the bell-type solitons. Two different kinds of the homoclinic breathers are seen, one of which is real and the other of which is complex, with two breathers interacting with each other. Homoclinic breather wave can evolve periodically along a straight line with a certain angle with the x axis and y axis, and its velocity, amplitude and width remain unchanged during the propagation. Homoclinic breather wave is not only space-periodic but also time-periodic. Interaction between the two breathers is elastic, which is similar to that of the solitons.

Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation

In this paper, the homoclinic breather limit method is employed to find the breather wave and the rational rogue wave solutions of the (2+1)-dimensional Ito equation. Moreover, based on its bilinear form, the solitary wave solutions of the equation are also presented with a detailed derivation. The dynamic behaviors of breather waves, rogue waves and solitary waves are analyzed with some graphics, respectively. The results imply that the extreme behavior of the breather solitary wave yields the rogue wave for the (2+1)-dimensional Ito equation.

Breathers, quasi-periodic and travelling waves for a generalized -dimensional Yu-Toda-Sasa-Fukayama equation in fluids

Under investigation in this paper is a generalized -dimensional Yu–Toda–Sasa–Fukayama equation for the interfacial wave in a two-layer fluid or the elastic quasi-plane wave in a liquid lattice. By virtue of the binary Bell polynomials, bilinear form of this equation is obtained. With the help of the bilinear form, N-soliton solutions are obtained via the Hirota method, and a bilinear Bäcklund transformation is derived to verify the integrability. Homoclinic breather waves are obtained according to the homoclinic test approach, which is not only the space-periodic breather but also the time-periodic breather via the graphic analysis. Via the Riemann theta function, quasi one-periodic waves are constructed, which can be viewed as a superposition of the overlapping solitary waves, placed one period apart. Finally, soliton-like, periodical triangle-type, rational-type and solitary bell-type travelling waves are obtained by means of the polynomial expansion method.