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.

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.

scholarly journals Exact Solutions of a High-Order Nonlinear Wave Equation of Korteweg-de Vries Type under Newly Solvable Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Weiguo Rui
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Exact Solutions ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
High Order ◽  
Solitary Wave Solutions ◽  
Wave Solutions

By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions. Based on our previous research works, some exact traveling wave solutions such as broken-soliton solutions, periodic wave solutions of blow-up type, smooth solitary wave solutions, and nonsmooth peakon solutions within more extensive parameter ranges are obtained. In particular, a series of smooth solitary wave solutions and nonsmooth peakon solutions are obtained. In order to show the properties of these exact solutions visually, we plot the graphs of some representative traveling wave solutions.

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 Soliton solutions of nonlinear wave equation in finite de-formation elastic cylindrical rod by solitary wave ansatz method

2016 ◽  
Vol 4 (1) ◽  
pp. 12
Author(s):  
Salam Subhaschandra Singh
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Finite Deformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Ansatz Method ◽  
Solitary Wave Ansatz Method

<p>In this paper, we consider nonlinear wave equation in finite deformation elastic cylindrical rod and obtain soliton solutions by Solitary Wave Ansatz method. It is shown that the ansatz method provides a very effective and powerful mathematical tool for obtaining solutions for Nonlinear Evolution Equations (NLEEs) in nonlinear Science.</p><div style="mso-element: para-border-div; border: none; border-bottom: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .25pt; padding: 0cm 0cm 1.0pt 0cm;"><p class="IJOPCMKeywards" style="margin-bottom: 0.0001pt; text-align: justify; border: none; padding: 0cm;"><span style="font-size: 8.0pt; mso-fareast-language: EN-US;">Elastic Rod; Finite Deformation; Nonlinear Wave Equation; Solitary Wave Ansatz Method; Soliton.</span></p></div>

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 On the wave solutions to the TRLW equation

2018 ◽  
Vol 22 ◽  
pp. 01033
Author(s):  
Tukur Abdulkadir Sulaiman ◽  
Canan Unlu ◽  
Hasan Bulut
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Models ◽  
Soliton Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
2D And 3D ◽  
Singular Soliton

In this study, a nonlinear model is investigated, namely; the time regularized long wave equation. Various solitary wave solutions are constructed such as the non-topological, compound topological-non-topological bell-type, singular and compound singular soliton solutions. Under the choice of suitable parameters values, the 2D and 3D graphs to all the obtained solutions are plotted. The reported results in this study may be helpful in explaining the physical meanings of some important nonlinear models arising in the field of nonlinear science.

Sign in / Sign up

Export Citation Format

Share Document