Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

A study of multi-soliton solutions, breather, lumps, and their interactions for kadomtsev-petviashvili equation with variable time coeffcient using hirota method

Hybrid localized wave solutions for a generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system in a fluid or plasma

Physics of nonlinear waves on variable backgrounds and the relevant mathematical analysis continues to be the challenging aspect of the study. In this work, we consider a (3+1)-dimensional nonlinear model describing the dynamics of water waves and construct nonlinear wave solutions on spatio-temporally controllable backgrounds for the first time by using a simple mathematical tool auto-Bäcklund transformation. Mainly, we unravel physically interesting features to control and manipulate the dynamics of nonlinear waves through the background. Adapting an exponential function and general polynomial of degree two as initial seed solutions, we construct single kink-soliton and rogue wave, respectively. We choose arbitrary periodic, localized and combined wave backgrounds by incorporating Jacobi elliptic functions and investigate the modulation of these two nonlinear waves with a clear analysis and graphical demonstrations. The solutions derived in this work give us sufficient freedom to generate exotic nonlinear coherent structures on variable backgrounds and open up an interesting direction to explore the dynamics of various other nonlinear waves propagating through inhomogeneous media. 

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Localized nonlinear waves on spatio-temporally controllable backgrounds for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq model in water waves

Water wave is one of the most common phenomena in nature, and its involves mathematical modeling, aerodynamics, computer simulation, mechanical manufacturing, marine science and so on. Hence, the water wave dynamics of a (
 $$4+1$$
 )-dimensional Boiti–Leon–Manna–Pempinelli equation in incompressible fluid is investigated based on the Hirota bilinear method and homoclinic test method. We are aimed at constructing variable coefficient solutions to the equation under consideration. An existence theorem and corollary about superposition solutions of this equation are proved. The water wave dynamic behaviors of the reported results are presented by a careful choice of the arguments for the trigonometric and hyperbolic functions as the values of the variable coefficients. With the help of Mathematica, the effect of the variable coefficients on the reported results can be clearly seen by the three-dimensional profiles. Compared with the published studied, some completely new mathematical results and physical phenomena are presented in this paper. The results are beneficial to the study of water waves in fluid dynamics, and this technique has opened a new way to explain the physical properties of nonlinear phenomena.

Interaction of multiple superposition solutions for the $$(4 + 1)$$ ( 4 + 1 ) -dimensional Boiti-LeonManna-Pempinelli equation

Variable-coefficient nonlinear evolution equations offer us with more real aspects in the inhomogeneities of media and non-uniformities of boundaries than their constant coefficients in some real-world problems. A $$(3+1)$$
 -dimensional variable coefficient Date–Jimbo–Kashiwara–Miwa (vcDJKM) equation and a $$(3+1)$$
 -dimensional variable coefficient Boiti–Leon–Manna–Pempinelli (vcBLMP) equation are studied based on the homoclinic test method and construction of abundant solutions for two kinds of equations are obtained with the help of symbolic computation. In order to study the interaction superposition solutions with different functions, an existence theorem and corollary about superposition solutions of the $$(3+1)$$
 -dimensional vcBLMP equation are proved. Via three-dimensional profiles with the help of mathematics, the propagation and the dynamical behavior of these solutions are analyzed by choosing different arbitrary variable coefficients. Compare with the published studied, some completely new results are presented in this paper.

Construction of abundant solutions for two kinds of $$\mathbf {(3\varvec{+}1)}$$-dimensional equations with time-dependent coefficients

In this paper, we introduce a new (4 + 1)-dimensional KdV-like equation. By using the Bell Polynomial method, we obtain the bilinear form, bilinear Backlund transformation, Lax pair and infinite co...

Integrability aspects and some abundant solutions for a new (4 + 1)-dimensional KdV-like equation

Water waves are common phenomena in nature, which have attracted extensive attention of researchers. In this work, the new (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation with time-dependent coefficients in incompressible fluid is investigated via the Hirota bilinear method. To this end, by considering the Hirota bilinear form of the equation and utilizing the homoclinic test method along with symbolic computations, an existence theorem and corollary about hybrid solutions of this equation are proved. Furthermore, the mixed high-order lump N-soliton solutions and the hybrid solutions are obtained analytically, and the novel evolutionary behaviors are shown by 3D figures. Additionally, a comparison has been given with some recent papers to show that our results are novel.

Dynamic analysis of hybrid solutions for the new (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation with time-dependent coefficients in incompressible fluid

In this article, the bilinear form, Backlund transformation, Lax pair and infinite conservation laws of the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients are constructed based on the Bell polynomials approach. N-soliton solutions are studied by means of introducing the complex conjugate condition technique and selecting appropriate test functions and parameters, including the hybrid solution of the a-order kink waves, b-order periodic-kink waves and c-order periodic-breather waves. The homoclinic test method is applied to investigate their dynamical interaction properties between different forms of hybrid-type solutions. Besides, a number of examples are presented by choosing different types of interactions among the hybrid-type solutions. Finally, we analyze the wave propagation direction and velocity to reflect the novel evolutionary behaviors in the three-dimensional profile of the model. These results are helpful to the study of local wave interactions in nonlinear mathematical physics.

Peng-Fei Han

Papers

Construction of abundant solutions for two kinds of $$\mathbf {(3\varvec{+}1)}$$-dimensional equations with time-dependent coefficients

Integrability aspects and some abundant solutions for a new (4 + 1)-dimensional KdV-like equation

Interaction of multiple superposition solutions for the $$(4 + 1)$$ ( 4 + 1 ) -dimensional Boiti-LeonManna-Pempinelli equation

Dynamic analysis of hybrid solutions for the new (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation with time-dependent coefficients in incompressible fluid

Novel hybrid-type solutions for the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients