Mixed type exact solutions to the (2+1)-dimensional Ito equation

Liu, Jian-Guo; Ai, Guoping; Zhu, Wen‐Hui

doi:10.1142/s0217984918503438

Cited by 8 publications

(3 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Under the premise of α = β = 0, the equation ( 9) is reduced to the (1+1)-dimensional Ito equation [32]. Based on the Hirota bilinear form of equation (7), Liu et al have obtained the mixed type exact solutions by means of a direct test function [33]. Through the parallel relationship of wave numbers, zhang et al have investigated the degeneration of lump-type localized waves [34].…”

Section: Novel Exact Solutions To the (2 + 1)-dimensional Ito Equationmentioning

confidence: 99%

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Ma,

Sudao

2023

Phys. Scr.

View full text Add to dashboard Cite

Recently, searching for exact solutions to nonlinear partial differential equations has gradually become a hot research topic. It is of great scientific research and application value to reveal the law of wave propagation, explain natural phenomena accurately and apply related technologies scientifically. In this paper, bilinear neural network method (BNNM) was employed to obtain some new exact analytical solutions to the (2 + 1)-dimensional Ito equation. Based on the Hirota form of Ito equation, we constructed activation functions f in various forms containing the exp(ξ), sin(ξ), cos(ξ), cosh(ξ) and squares of polynomial functions in multi-layer neurons such as [3-2-2] and [3-2-3] neural network models. The test function f in this work is a new expression. On the other hand, these solutions have not been studied yet. As a result, we obtained several new interaction solutions, such as periodic wave solution, breather solution and bright-dark solitons solution, etc. In addition, the corresponding 3D, density and contour plots of all the solution forms were drawn and their characters and dynamic behaviors were vividly demonstrated.

show abstract

Section: Novel Exact Solutions To the (2 + 1)-dimensional Ito Equationmentioning

confidence: 99%

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Ma,

Sudao

2023

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…In this section, we find mixed waves solutions that consist of lump and solitary wave solutions. We assume that the general form of these solutions is given by

f = f_{1}^{2} + f_{2}^{2} + e^{ξ} + a_{9}, g = g_{1}^{2} + g_{2}^{2} + e^{η} + b_{9},

where

\begin{matrix} right f_{1} & left = a_{1} x + a_{2} y + a_{3} z + a_{4} t, f_{2} = a_{5} x + a_{6} y + a_{7} z + a_{8} t, & right \\ right g_{1} & left = b_{1} x + b_{2} y + b_{3} z + b_{4} t, g_{2} = b_{5} x + b_{6} y + b_{7} z + b_{8} t, \\ right ξ & left = k_{1} x + k_{2} y + k_{3} z + k_{4} t, η = k_{5} x + k_{6} y + k_{7} z + k_{8} t, \end{matrix}

and a i , b i , k j , i =1,2,…,9, j =1,2,…,8 are real constants.…”

Section: A Study Of Different Wave Structures Of Solitons For a 3d‐blmpmentioning

confidence: 99%

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

Osman

Wazwaz

2019

Math Methods in App Sciences

124

View full text Add to dashboard Cite

In this paper, we investigate a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (3D-BMLP). By using bilinear forms under certain conditions, we obtain different wave structures for the 3D-BMLP. Among these waves, lump waves, breather waves, mixed waves, and multi-soliton wave solutions are constructed. The propagation and the dynamical behavior of the obtained solutions are discussed for different values of the free parameters. KEYWORDS (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, different wave structures, general bilinear form MSC CLASSIFICATION 35Q51; 35G99; 33F10 INTRODUCTIONDuring the last decades, solitons have been put forward in the study of nonlinear phenomena. Many researchers have extensively studied the dynamical behaviors for different nonlinear evolution equation models that are closely related to the concept of soliton by using different approaches. [1][2][3] The different types of solitons include the lump waves (usually called rogue waves in the presence of certain conditions), breather waves, and mixed waves that describe the interaction between lump wave and another type of soliton waves. 1,2,4,5 It is well known that solitons, of a diversity of physical structures, have attracted much attention in scientific fields, such as optical fiber communications, fluid dynamics, propagation of waves, marine engineering, fluid dynamics, plasma physics, incompressible fluid, ocean and rogue waves, photonics, and many other applications.As stated earlier, the Boiti-Leon-Manna-Pempinelli (BLMP) equation describes the fluid propagation and can be considered as a model for incompressible fluid. The BLMP equation describes the evolution of the horizontal velocity component of the water waves propagating in the xy-plane in an infinite narrow channel of constant depth. 6,7 It is worth noting that incompressible means that the effects of pressure on the fluid density are zero, where the density and the specific volume of the fluid do not change during the flow. Water and many liquids, at constant temperature, can be considered incompressible in most cases.The main objective of this work is to seek for lump waves, breather waves, multi-soliton waves, and mixed waves for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (3D-BLMP). 3,6 These solutions are obtained by using different choices for the arbitrary functions in bilinear forms. Lump waves, as special nonlinear wave phenomena, have been observed in many fields. 1,2 Lump waves are theoretically regarded as a limit form of soliton in some ways and propagate with higher propagating energy than general solitons. Recently, research works on the lump solutions have been growing more and more that led to promising findings. Therefore, theoretically researches on lump waves are helpful to better understand and predict possible extremes for nonlinear evolution systems.

show abstract

“…Until now, some excellent methods have been established, which are inverse scattering transformation, 2 Hirota bilinear method and generalized bilinear method, [3][4][5][6][7][8] the homogeneous method, [9][10][11][12][13] the hyperbolic function method, 14 the F -expansion method, 15 the Bäcklund transformation method, 16 the extended tanh-function method, 17 similarity transformation, 18,19 algebra-geometric approach, 20 variabledetached method, 21 Painleve analysis, 22 Darboux transformation, [23][24][25][26][27] and so forth. [28][29][30][31][32][33] Among the aforementioned methods, the Darboux transformation is one of the powerful and direct methods to investigate explicit solutions. 34 In this paper, we will study the DT and explicit solutions of the modified Volterra lattice 35 :…”

Section: Introductionmentioning

confidence: 99%

N-fold Darboux transformation and conservation laws of the modified Volterra lattice

Sun

et al. 2018

Mod. Phys. Lett. B

View full text Add to dashboard Cite

In this work, we study the modified Volterra lattice. Applying the gauge transformation of the associated [Formula: see text] matrix spectral problems, we establish the N-fold Darboux transformation (DT), and then construct a few explicit solutions in terms of determinants upon using the obtained DT. Moreover, all the results are illustrated by the graphs of the solitonic evolution profiles of the aforementioned solutions. Finally, infinitely many conservation laws for the modified Volterra lattice are proposed. The obtained results of this research might be applied to the research on nonlinear phenomena in physics or engineering areas.

show abstract

Mixed type exact solutions to the (2+1)-dimensional Ito equation

Cited by 8 publications

References 43 publications

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

A general bilinear form to generate different wave structures of solitons for a (3+1)‐dimensional Boiti‐Leon‐Manna‐Pempinelli equation

N-fold Darboux transformation and conservation laws of the modified Volterra lattice

Contact Info

Product

Resources

About