Lump Wave Research Articles

Bilinear Bäcklund, Lax pairs, breather waves, lump waves and soliton interaction of (2+1)-dimensional non-autonomous Kadomtsev–Petviashvili equation

Bilinear Bäcklund, Lax pairs, breather waves, lump waves and soliton interaction of (2+1)-dimensional non-autonomous Kadomtsev–Petviashvili equation

Multiple lump molecules and interaction solutions of the Kadomtsev–Petviashvili I equation

In this paper, a modified version of the solution in form of a Gramian formula is employed to investigate a new type of multiple lump molecule solution of the Kadomtsev–Petviashvili I equation. The high-order multiple lump molecules consisting of M N-lump molecules are constructed by means of the Mth-order determinant and the non-homogeneous polynomial in the degree of 2N. The interaction solutions describing P line solitons radiating P of the M N-lump molecules are constructed. The dynamic behaviors of some specific solutions are analyzed through numerical simulation. All the results will enrich our understanding of the multiple lump waves of the Kadomtsev–Petviashvili I equation.

Interactions of rogue and solitary wave solutions to the (2 + 1)-D generalized Camassa–Holm–KP equation

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use a probable transformation to build bilinear formulation to the model by Hirota bilinear technique. We derive a single lump waves, multi-soliton solutions to the model from this bilinear form. We present various dynamical properties of the model such as one, two, three and four solitons. The double periodic breather waves, periodic line rogue wave, interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves in a line are established. Among the results, most of the properties are unexplored in the prior research. Furthermore, with the assistance of Maple software, we put out the trajectory of the obtained solutions for visualizing the achieved dynamical properties.

Resonant collisions among localized waves in the (2+1)-dimensional Hirota–Satsuma–Ito equation

In this paper, we study the resonant collisions among different types of localized solitary waves in the (2+1)-dimensional Hirota–Satsuma–Ito equation, which are described by N-soliton solutions constructed using bilinear method. Through the asymptotic analysis and limit treatment of the phase shift of these localized waves, the elastic collisions among different localized waves can be transformed into resonant collisions. Hereby, we study the resonant collision between a breather/ lump and a bright line soliton and find two collision situations: (i) the breather is semi-localized in space and the shape of the breather is not localized during the propagation and (ii) the lump wave generates from the bright line wave. At the same time, we investigate the resonant collision between a breather/lump and two bright line solitons. In these evolution processes, we also gain two dynamical behaviors: (iii) the breather is always localized in space and the shape of the breather is not localized during the propagation, and (iv) the lump wave appears from a bright line soliton and then disappears into the other bright line soliton. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in researching the physical phenomena in shallow water waves and nonlinear optics.

Various breathers, Lumps, line solitons and their interaction solutions for the (2+1)-dimensional variable-coefficient Sawada–Kotera equation

The main concern of this paper is to investigate the various of interaction solutions to the (2+1)-dimensional variable-coefficient Sawada–Kotera equation. Analytical one-, two-, three- and four-soliton solutions of this equation are constructed based on its Hirota bilinear form. Through granting appropriate parameters and coefficients, special cases lead to resonance X-type soliton, S-type, U-type or periodic-type soliton solutions. The interaction solutions of multiple line solitons, between S-type line soliton and breather, U-type or S-type line soliton and lump wave, two U-type or S-type or periodic-type breathers, two S-type lump waves, among U-type or S-type two line solitons and one lump are obtained analytically, and some figures are provided with a better understanding of the dynamic behavior. We are confident that the results obtained in this paper are novel, which may be helpful to study other higher-dimensional nonlinear evolution equations

Degenerate lump wave solutions of the Mel’nikov equation

Degenerate lump wave solutions of the Mel’nikov equation

Trajectory equation of a lump before and after collision with other waves for (2+1)-dimensional Sawada–Kotera equation

In this paper, we employ the Hirota bilinear and long wave limit methods to obtain the hybrid solutions consisting of mth order lumps and nth order breathers, mth order lumps and n line waves for (2+1)-dimensional Sawada–Kotera (SK) equation. Then, according to the characteristics of a lump wave moving along a straight line, the trajectory equation of a lump before and after collision with a line wave, two line waves and breather waves are obtained by approximating the solution of SK equation along some parallel orbits at infinity. Furthermore, we generalize the above cases to the collision of a lump wave with n line waves and the nth order breather waves, and the corresponding trajectory equations are presented. We also give the expression of phase shift for the lump wave, and verify that when ϑi≠0, the height of the lump wave before and after collision does not change, which reveals that collisions between a lump and other waves are elastic.

Lump and interaction dynamics of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation

This paper mainly considers the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation describing propagation dynamics of nonlinear waves appearing in physics, biology and electrical networks. A simple method to derive the multiple lumps and lump molecules is proposed by means of the solutions of Grammian form and polynomial functions. The interaction dynamical behaviors of the lumps and lump molecules are analyzed with the aid of numerical simulation. The interaction solutions including lumps, lump molecules and kink solitons are obtained, where the lumps can be emitted and absorbed by solitons. The phenomenon that one lump wave exchanges between two bound solitons is presented. A special Grammian τ-function is employed to construct the lump chain. Based on an analytical method related to dominant domain, we systematically analyze the interactions between lump chain and solitons. The lump chain can be emitted by a soliton and then absorbed by other solitons when the parameter is in a certain range. A large number of new solutions obtained in this paper are helpful to study the wave phenomena existing in the theory of water wave, soliton, optical fiber communication and other relevant scientific fields.

Dynamical analysis of rational and semi‐rational solution for a new extended (3 + 1)‐dimensional Kadomtsev‐Petviashvili equation

This paper proposes a new extended (3 + 1)‐dimensional Kadomtsev‐Petviashvili equation that portrays a unique dispersion effect about . Its integrability is confirmed via the WTC‐Kruskal algorithm in Painlevé sense. ‐soliton, breather, and ‐type solitary wave are derived systematically at first. Then, the mixed solution composed of soliton and breather is obtained. In addition, the “long wave” limit is employed to construct rational and semi‐rational solution. The rational solution can be classified as rogue wave, ‐type solitary wave, and lump wave. The semi‐rational solution has the form a hybrid of two solitons, a hybrid of rogue wave and soliton, a hybrid of lump and soliton(s), and a hybrid of lump and breather. The results may help simulate complex waves and their interactions in fluid.

New type of multiple lump and rogue wave solutions of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this paper, a new version of the solution in form of Grammian for the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation is obtained with the aid of the polynomial function. A family of multiple lump solutions are constructed. All the center points of the lumps form a triangular structure as the auxiliary variable tends to the negative or positive large values. A unified scheme for constructing the high-order rogue wave solutions is proposed. This new type of rogue wave reflects the appearance and disappearance of multiple lump waves in the background of N-kink solitons. The results can well mimic complex waves and explain complex dynamics in fluids.

Nonlinear dynamics for different nonautonomous wave structure solutions

Abstract Based on the positive quadratic function method, the rich nonautonomous solutions of a generalized (2+1)-dimensional variable-coefficient breaking soliton equation with different wave structures are given. In this case, due to the influence of nonlinearity and dispersion, the characteristics, amplitude and velocity of nonautonomous wave will change with time. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions are studied. For different choices of arbitrary functions in these solutions, the corresponding dynamic properties are demonstrated.

A new extended (2+1)-dimensional Kadomtsev–Petviashvili equation with N-solitons, periodic solutions, rogue waves, breathers and lump waves

In this work, a new extended integrable (2+1)-dimensional Kadomtsev–Petviashvili equation is proposed and investigated, which models slowly varying perturbation wave in dispersion fluids. First, the WTC-Kruskal algorithm is applied to exploring the corresponding compatibility condition for this equation in Painlevé sense. Then, N-soliton, periodic, breather solution as well as the mixed form composing of breather and soliton(s) have been derived via Hirota bilinear method and symbolic computation. Moreover, we have derived the rational and semi-rational solutions in terms of “long wave” limit. The rational solutions can be classified as first order line rogue waves and lumps, while the semi-rational solutions have the forms: a hybrid of first order line rogue wave and one soliton, a hybrid of second order line rogue wave and one soliton, a hybrid of lump and soliton(s), a hybrid of lump and breather, a hybrid of lump and periodic solution.

Computation and observation of novel interaction based on the mixed solutions to a generalized Bogoyavlensky–Konopelchenko equation

A generalized Bogoyavlensky–Konopelchenko equation is introduced by using p -generalized bilinear differential operators. The lump solutions, one-lump-one-kink and one-lump-two-kink solutions are derived with symbolic computations. For the two types of mixed solutions, assuming v x and v y represent velocities of the kink waves along the x -axis and the y -axis, we find that the velocities of the lump wave and the kink waves along the vector ( v x , v y ) are equal, while the velocities of the lump wave and the kink waves along the vector ( − v y , v x ) are not equal. The results imply there is no fission and fusion between the lump wave and the kink waves. The lump wave will not be drowned by kink waves. The results might be helpful to explain the complicated real-world phenomena.

Dynamics of mixed lump-soliton solutions to the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli like equation

In this paper, a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) like equation is proposed based on the generalized bilinear operators in a prime number p = 3 . By combining an exponential function with a quadratic function, an interaction between a lump wave and one kink soliton is generated. Furthermore, the lump and the interaction solutions are illustrated by three-dimensional plots, density plots, contour plot and propagation pattern along x -axis by selecting the appropriate parameters.

Lumps with their some interactions and breathers to an integrable (2 + 1)-dimensional Boussinesq equation in shallow water

In this paper, the lumps with their interactions (lump-single and lump-double stripes), and breather wave solutions are constructed to the new integrable (2 + 1)-dimensional Boussinesq equation via the Hirota bilinear method. The propagations of the obtained lumps, lump-stripes, and breather wave solutions of the aforesaid equation are presented through some 3D and 2D graphs. All of the graphs are given to show their correct wave profiles. The graphical outputs demonstrate that lump waves propagate in all directions throughout space. However, the amplitudes and forms of lump waves remain constant during their propagation in any direction. On the other hand, the lump wave travels along with the stripes for the lump and stripes solutions. Furthermore, two homoclinic test approaches are used to produce the breather wave solutions to the aforesaid equation. It is found that breather waves propagate in a specific direction on a constant period. In addition, the effects of constant coefficients of the model are presented also graphically. It is found that the amplitudes of all obtained wave solutions are changed with the change of constant coefficients of the model..The outcomes of the study can be useful for a better understanding of wave propagation dynamics in shallow water under gravity.

Painlevé analysis, integrability property and multiwave interaction solutions for a new (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff equation

In this letter a novel (4+1)-dimensional KdV–Calogero–Bogoyavlenkskii–Schiff (KdV-CBS) equation is investigated. The Painlevé analysis shows that this higher dimensional nonlinear equation passes the integrability test. Its integrable properties, including bilinear Bäcklund transformation, Lax pair as well as N-soliton solution, are systematically derived using the binary Bell polynomial method. Furthermore, incorporating the inheritance solving technique into direct algebraic method, we construct the multiwave solutions of 1-lump and (N−2)-soliton (N>2). Multiple choices of parameters in the obtained solutions lead to rich interactions among lump waves, kink waves and breather waves. Several interesting interactions of the multiwave solutions are shown by graphs.

Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this paper, the (3+1)-dimensional nonlinear evolution equation is studied analytically. The bilinear form of given model is achieved by using the Hirota bilinear method. As a result, the lump waves and collisions between lumps and periodic waves, the collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions for the given model are constructed. Furthermore, some new traveling wave solutions are developed by applying the exp(−ϕ(ξ)) expansion method. The 3D, 2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters. As a result, a collection of bright, dark, periodic, rational function and elliptic function solutions are established. The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.

Regarding on the Results for the Fractional Clannish Random Walker’s Parabolic Equation and the Nonlinear Fractional Cahn-Allen Equation

In this research, the Ψ , Φ -expansion scheme has been implemented for the exact solutions of the fractional Clannish Random Walker’s parabolic (FCRWP) equation and the nonlinear fractional Cahn-Allen (NFCA) equation. Some new solutions of the FCRWP equation and the NFCA equation have been obtained by using this method. The diverse variety of exact outcomes such as intersection between rough wave and kinky soliton wave profiles, intersection between lump wave and kinky soliton wave profiles, soliton wave profiles, kink wave profiles, intersection between lump wave and periodic wave profiles, intersection between rough wave and periodic wave profiles, periodic wave profiles, and kink wave profiles are taken. Comparing our developed answers and that got in previously written research papers presents the novelty of our investigation. The above techniques could also be employed to get exact solutions for other fractional nonlinear models in physics, applied mathematics, and engineering.

Different Wave Structures for the (2+1)-Dimensional Korteweg-de Vries Equation

In this article, a (2+1)-dimensional Korteweg-de Vries equation is investigated. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Meanwhile, the interaction solutions between lump and periodic waves are presented. What is more, we derive the interaction solutions among lump, periodic, and solitary waves. Based on the extended homoclinic test technique, some new double periodic-soliton solutions are presented. Finally, some 3D and density plots are simulated and displayed to respond the dynamic behavior of these obtained solutions.

Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid

In this script, we consider the modified Oskolkov equation in incompressible viscoelastic Kelvin–Voigt fluid and fluid dynamics. A dominant direct algebraic method namely modified simple equation method (MSE) uses to retrieve various dynamical structural solutions of the nonlinear models. As a results, we can derive a disguise version of analytic structural solutions in exponential, hyperbolic, trigonometric and Jacobi elliptic functions with some appreciate parameters. In this work, we depict the physical explanation of cross periodic lump wave, interaction of lump and kink, lump and anti-kink shape solution, double periodic wave, kink, anti-kink shape solution, disguise version of soliton solutions with 3D contour plot and 2D plot.