scholarly journals Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Bo Ren
Keyword(s):  
Bound States ◽  
Boussinesq Equation ◽  
Rational Solution ◽  
Quadratic Function ◽  
Special Kind ◽  
Higher Order ◽  
Lump Solution ◽  
Resonance Mechanism ◽  
Lump Wave ◽  
Nonlinear Wave Field

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.

scholarly journals Painlevé Analysis, Soliton Molecule, and Lump Solution of the Higher-Order Boussinesq Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Bo Ren
Keyword(s):  
Bilinear Form ◽  
Boussinesq Equation ◽  
Higher Order ◽  
Lump Solution ◽  
Painlevé Analysis ◽  
Resonance Mechanism ◽  
Variable Transformation ◽  
Localized Excitations ◽  
Lump Wave ◽  
Multisoliton Solutions

The Painlevé integrability of the higher-order Boussinesq equation is proved by using the standard Weiss-Tabor-Carnevale (WTC) method. The multisoliton solutions of the higher-order Boussinesq equation are obtained by introducing dependent variable transformation. The soliton molecule and asymmetric soliton of the higher-order Boussinesq equation can be constructed by the velocity resonance mechanism. Lump solution can be derived by solving the bilinear form of the higher-order Boussinesq equation. By some detailed calculations, the lump wave of the higher-order Boussinesq equation is just the bright form. These types of the localized excitations are exhibited by selecting suitable parameters.

Rational solutions and bright–dark lump solutions to the BKP equation

2018 ◽  
Vol 32 (27) ◽  
pp. 1850334 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jiang-Su Geng ◽  
Meng-Yue Zhang
Keyword(s):  
Soliton Solution ◽  
Rational Solution ◽  
Lump Solution ◽  
Collision Dynamics ◽  
Rational Solutions ◽  
Long Wave ◽  
The One ◽  
Lump Wave ◽  
Bkp Equation ◽  
Graphical Illustration

In this paper, via the limit technique of long wave, the N-order rational solution of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation is derived from the N-soliton solution. In particular, the bright–dark M-lump solution can be obtained from resulting N-order rational solution. This kind of lump wave in BKP equation exhibits the one-peak-one-valley structure which is different from that in KPI equation. In addition, graphical illustration presents the collision dynamics of multi-bright–dark lump waves.

Multiple lump solutions and dynamics of the generalized (3+1)-dimensional KP equation

2020 ◽  
Vol 34 (15) ◽  
pp. 2050167 ◽  
Author(s):  
Lingchao He ◽  
Zhonglong Zhao
Keyword(s):  
Quadratic Function ◽  
Lump Solution ◽  
Bilinear Forms ◽  
Polynomial Functions ◽  
Kp Equation ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Behaviors ◽  
Lump Wave ◽  
Bilinear Equation

In this paper, the bilinear method is employed to investigate the multiple lump solutions of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation. With the aid of the variable transformation, this equation is reduced to a dimensionally reduced equation. The reduced equation can be transformed into a bilinear equation. On the basis of the bilinear forms and a special quadratic function, the 1-lump solution is constructed, which has a positive peak and two negative peaks. Three polynomial functions are introduced to derive the 3-lump solutions. The 3-lump wave has a “triangular” structure. As the parameters tend to zero, the 3-lump wave becomes the lump wave with two adjacent cusps. The 6-lump solutions are constructed. Four kinds of lump waves appear as the parameters increase. In addition, the high-order 8-lump solutions are also obtained. All the peaks of the 6-lump and the 8-lump wave tend to the same height when the parameters are sufficiently large. The dynamical behaviors of the multiple lump solutions are discussed. All the results are useful in explaining the dynamical phenomena of the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional KP equation.

scholarly journals Higher-order Floquet topological phases with corner and bulk bound states

2019 ◽  
Vol 100 (8) ◽  
Author(s):  
Martin Rodriguez-Vega ◽  
Abhishek Kumar ◽  
Babak Seradjeh
Keyword(s):  
Bound States ◽  
Higher Order ◽  
Topological Phases

On the quasi-periodic solutions for generalized Boussinesq equation with higher order nonlinearity

2014 ◽  
Vol 94 (10) ◽  
pp. 1977-1996 ◽  
Author(s):  
Yanling Shi ◽  
Junxiang Xu ◽  
Xindong Xu ◽  
Shunjun Jiang
Keyword(s):  
Periodic Solutions ◽  
Boussinesq Equation ◽  
Higher Order ◽  
Generalized Boussinesq Equation

scholarly journals Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bo Xu ◽  
Yufeng Zhang ◽  
Sheng Zhang
Keyword(s):  
Rational Solution ◽  
Soliton Solutions ◽  
Ocean Waves ◽  
Columnar Structure ◽  
Lump Solution ◽  
Spatial Structures ◽  
Kp Equation ◽  
Spreading Process ◽  
Soliton Interactions ◽  
Special Case

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional n -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

scholarly journals Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Ömer Faruk Gözükızıl ◽  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boussinesq Equation ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Higher Order ◽  
Sobolev Type ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions

By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

scholarly journals SCHRÖDINGER EQUATION FOR HEAVY MESONS EXPANDED IN 1/mQ

1996 ◽  
Vol 11 (03) ◽  
pp. 257-266 ◽  
Author(s):  
TAKAYUKI MATSUKI
Keyword(s):  
Wave Function ◽  
Perturbation Theory ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound States ◽  
Higher Order ◽  
Wave Functions ◽  
Heavy Mesons ◽  
Higher Order Corrections

Operating just once the naive Foldy-Wouthuysen-Tani transformation on the Schrödinger equation for [Formula: see text] bound states described by a Hamiltonian, we systematically develop a perturbation theory in 1/mQ which enables one to solve the Schrödinger equation to obtain masses and wave functions of the bound states in any order of 1/mQ. There also appear negative components of the wave function in our formulation which contribute also to higher order corrections to masses.

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