Lax pair,Darboux transformation and Nth-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

A new (2+1)-dimensional Heisenberg ferromagnetic spin chain equation is investigated, which can be used to describe magnetic soliton excitations in two dimensional space fields and a time field. The Lax pair of the equation is first constructed. Based on the Lax pair, initial seed solution and Darboux transformation, the analytic first-, second- and third-order rogue wave solutions are obtained, and a general expression of the $N$th-order ($N>3$) rogue wave solutions is presented. The impacts of the system parameters on the rogue waves are demonstrated through numerical visualization method.     

Diversity of interaction solutions to the (2+1)-dimensional Ito equation

We aim to show the diversity of interaction solutions to the (2+1)-dimensional Ito equation, based on its Hirota bilinear form. The proof is given through Maple symbolic computations. An interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function. Special cases lead to lump solutions, lump-soliton solutions and lump-kink solutions. Two illustrative examples of the resulting solutions are displayed by three-dimensional plots and contour plots.     

Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation is investigated. Lump-type solutions and lump solutions are obtained with aid of symbolic computation via Hirota bilinear method and the ansatz technique. By taking the function $f$ in the Hirota bilinear form of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation as the general quadratic polynomial function, a kind of lump-type solution which contains eleven parameters with six arbitrary independent parameters and two non-zero conditions is obtained. Lump solutions are found from the lump-type solutions via taking a special set of parameters, and the motion track of the lump is also described both theoretically and graphically.     

Lump solutions to a (2+1)-dimensional extended KP equation

With the aid of a computer algebra system, we present lump solutions to a ($2+1$)-dimensional extended Kadomtsev–Petviashvili equation (eKP) and give the sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions. We plot a few solutions for some specific values of the free parameters involved and finally derive one of the lump solutions of the Kadomtsev–Petviashvili (KP) equations from the lump solutions of the eKP equation.     

N-solitons,breathers, lumps and rogue wave solutions to a (3+1)-dimensional nonlinear evolution equation

Based on Hirota bilinear method, $N$-solitons, breathers, lumps and rogue waves as exact solutions of the (3+1)-dimensional nonlinear evolution equation are obtained. The impacts of the parameters on these solutions are analyzed. The parameters can influence and control the phase shifts, propagation directions, shapes and energies for these solutions. The single-kink soliton solution and interactions of two and three-kink soliton overtaking collisions of the Hirota bilinear equation are investigated in different planes. The breathers in three dimensions possess different dynamics in different planes. Via a long wave limit of breathers with indefinitely large periods, rogue waves are obtained and localized in time. It is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again. The results can be used to illustrate the interactions of water waves in shallow water. Moreover, figures are given out to show the properties of the explicit analytic solutions.     

Analysis of Lie symmetries with conservation laws and solutions for the generalized (3+1)-dimensional time fractional Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, under investigated is a generalized $(3+1)$-dimensional Camassa–Holm–Kadomtsev–Petviashvili (gCH-KP) equation, which describes the role of dispersion in the formation of patterns in liquid drops. With the help of the semi-inverse method, the Euler–Lagrange equation and Agrawal’s method, the time fractional gCH-KP equation is derived in the sense of Riemann–Liouville fractional derivatives. Further, the symmetry of the $(3+1)$-dimensional time fractional gCH-KP equation is studied by fractional order symmetry. Meanwhile, based on the new conservation theorem, the conservation laws of $(3+1)$-dimensional time fractional gCH-KP equation are constructed. Finally, the solutions to the equation are given via a bilinear method and the radial basis functions (RBFs) meshless approach.     

Rational solutions and their interaction solutions of the (2+1)-dimensional modified dispersive water wave equation

A bilinear form for the modified dispersive water wave (mDWW) equation is presented by the truncated Painlevé series, which does not lead to lump solutions. In order to get lump solutions, a pair of quartic–linear forms for the mDWW equation is constructed by selecting a suitable seed solution of the mDWW equation in the truncated Painlevé series. Rational solutions are then computed by searching for positive quadratic function solutions. A regular nonsingular rational solution can describe a lump in this model. By combining quadratic functions with exponential functions, some novel interaction solutions are founded, including interaction solutions between a lump and a one-kink soliton, a bi-lump and a one-stripe soliton, and a bi-lump and a two-stripe soliton. Concrete lumps and their interaction solutions are illustrated by 3d-plots and contour plots.     

New solitary wave solutions of (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation

In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation involving Jumarie’s modified Riemann–Liouville derivative have been derived. As a result, with the aid of symbolic computation, many types of new analytical solutions are obtained, which include new solitary wave, periodic wave and rational wave solutions. The graphical representations show that these gained solutions have abundant structures.     

The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation

In this paper, the $N$-soliton solution is constructed for the ($2+1$)-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.