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.

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 On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nattakorn Sukantamala ◽  
Supawan Nanta
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Analytic Solution ◽  
Traveling Wave Solutions ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Kawahara Equation ◽  
Ansatz Method ◽  
Wave Solutions

The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

2012 ◽  
Vol 4 (5) ◽  
pp. 559-574 ◽  
Author(s):  
Xiang Li ◽  
Weiguo Zhang ◽  
Yan Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Critical Value ◽  
Oscillatory Solution ◽  
Dissipation Effect ◽  
Wave Solutions

AbstractIn this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.

scholarly journals Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Weiguo Rui
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.

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.

TRAVELING WAVE SOLUTIONS FOR NONLINEAR WAVE EQUATION WITH DISSIPATION AND NONLINEAR TRANSPORT TERM THROUGH FACTORIZATIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 645-651 ◽  
Author(s):  
H. A. ABDUSALAM ◽  
E. S. FAHMY
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Factorization Method ◽  
Nonlinear Transport ◽  
Wave Solutions ◽  
Special Cases ◽  
Transport Term

In this work, we use the factorization method to find explicit exact particular traveling wave solutions for the nonlinear wave equation with dissipation and nonlinear transport term. The two-parameter solution is obtained by using the particular solution and the known solutions for the Newell–Whitehead equation, Kolmogorov–Petrovsky–Piscounov equation, Fitzhugh–Nagumo equation, and the Burgers equation with cubic nonlinearity obtained as special cases from the solutions of the nonlinear wave equation with dissipation and nonlinear term.

scholarly journals Traveling solitary wave solutions for the symmetric regularized long-wave equation

2015 ◽  
Vol 11 (8) ◽  
pp. 5520-5528
Author(s):  
Mostafa Khater ◽  
Mahmoud AE Abdelrahman
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Long Wave ◽  
Regularized Long Wave Equation

In this paper, we employ the extended tanh function method to nd the exact traveling wave solutions involving parameters of the symmetric regularized long- wave equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. These studies reveal that the symmetric regularized long-wave equation has a rich varietyof solutions.

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