Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation

2021 ◽  
Author(s):  
Lingchao He ◽  
Jianwen Zhang ◽  
Zhonglong Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Water Waves ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Before And After

Abstract In this paper, we consider a generalized (2+1)-dimensional nonlinear wave equation. Based on the bilinear, the N-soliton solutions are obtained. The resonance Y-type soliton and the interaction solutions between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions are presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.

scholarly journals New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
XiaoHua Liu ◽  
CaiXia He
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Undetermined Coefficient ◽  
Planar Dynamical Systems

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

Abundant soliton wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by bilinear analysis

2021 ◽  
Author(s):  
Guanqi Tao ◽  
Jalil Manafian ◽  
Onur Alp İlhan ◽  
Syed Maqsood Zia ◽  
Latifa Agamalieva
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Gas Bubbles ◽  
Soliton Theory ◽  
Wave Solutions ◽  
Dimensional Variable ◽  
Hirota Bilinear

In this paper, we check and scan the (3+1)-dimensional variable-coefficient nonlinear wave equation which is considered in soliton theory and generated by considering the Hirota bilinear operators. We retrieve some novel exact analytical solutions, including cross-kink soliton solutions, breather wave solutions, interaction between stripe and periodic, multi-wave solutions, periodic wave solutions and solitary wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota bilinear forms and their generalized equivalences. Lastly, the graphical simulations of the exact solutions are depicted.

scholarly journals Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Jingzhu Wu ◽  
Xiuzhi Xing ◽  
Xianguo Geng
Keyword(s):  
Shallow Water ◽  
Differential Operators ◽  
Shallow Water Equation ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Bell Polynomials ◽  
Periodic Wave Solutions ◽  
Wave Solutions

The relations betweenDp-operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms withDp-operators of nonlinear equations directly and quickly. Exact periodic wave solution of a (3+1)-dimensional generalized shallow water equation is obtained with the help of theDp-operators and a general Riemann theta function in terms of the Hirota method, which illustrate that bilinearDp-operators can provide a method for seeking exact periodic solutions of nonlinear integrable equations. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

Periodic-wave solutions and asymptotic properties for a (3+1)-dimensional generalized breaking soliton equation in fluids and plasmas

2021 ◽  
pp. 2150344
Author(s):  
Rui-Dong Chen ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Ting-Ting Jia ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Asymptotic Properties ◽  
Theta Functions ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Soliton Equation ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Riemann Theta Functions ◽  
The One

In this paper, a (3+1)-dimensional generalized breaking soliton equation is investigated. Based on the one- and two-dimensional Riemann theta functions, one- and two-periodic-wave solutions are derived. We observe that the one-periodic wave is one-dimensional and is viewed as a superposition of the overlapping waves, placed one period apart. With certain parameters, the symmetric feature appears in the two-periodic wave, and the two-periodic wave degenerates to the one-periodic wave. With the series expansions, we explore the relations between the soliton and periodic-wave solutions. According to those relations, asymptotic properties for the periodic-wave solutions to approach to the soliton solutions under certain amplitude conditions are derived.

On quasi-periodic wave solutions and asymptotic behaviors to a (2 + 1)-dimensional generalized variable-coefficient Sawada–Kotera equation

2015 ◽  
Vol 29 (19) ◽  
pp. 1550101 ◽  
Author(s):  
Jian-Min Tu ◽  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Pan-Li Ma
Keyword(s):  
Periodic Solutions ◽  
Variable Coefficient ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Bell Polynomials ◽  
Nonlinear Phenomena ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Bilinear Equation

In this paper, a [Formula: see text]-dimensional generalized variable-coefficient Sawada–Kotera (gvcSK) equation is investigated, which describes many nonlinear phenomena in fluid dynamics and plasma physics. Based on the properties of binary Bell polynomials, we present a Hirota’s bilinear equation to the gvcSK equation. By virtue of the Hirota’s bilinear equation, we obtain the N-soliton solutions and the quasi-periodic wave solutions of the gvcSK equation, which can be reduced to the ones of several integrable equations such as Sawada–Kotera, modified Caudrey–Dodd–Gibbon–Sawada–Kotera, isospectral BKP equations and etc. Furthermore, we obtain the relationship between the soliton solutions and periodic solutions by considering the asymptotic properties of the periodic solutions.

Quasi-periodic wave solutions and asymptotic properties for a fifth-order Korteweg–de Vries type equation

2016 ◽  
Vol 30 (18) ◽  
pp. 1650223 ◽  
Author(s):  
Chun-Yan Qin ◽  
Shou-Fu Tian ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Ocean Dynamics ◽  
Nonlinear Phenomena ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

Under investigation in this paper is a fifth-order Korteweg–de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its [Formula: see text]-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

2016 ◽  
Vol 8 (6) ◽  
pp. 1084-1098
Author(s):  
Wei Wang ◽  
Chunhai Li ◽  
Wenjing Zhu
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Single Peak ◽  
Dynamical System Theory ◽  
Solitary Wave Solutions ◽  
Analytical Technique ◽  
Wave Solutions

AbstractDynamical system theory is applied to the integrable nonlinear wave equation ut±(u3–u2)x+(u3)xxx=0. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation correspond to the case of wave speed c=0. In the case of c≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

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