Conservation Laws and Soliton Solutions of the (1+1)-Dimensional Modified Improved Boussinesq Equation

2015 ◽  
Vol 70 (8) ◽  
pp. 669-672 ◽  
Author(s):  
Ozkan Guner ◽  
Sait San ◽  
Ahmet Bekir ◽  
Emrullah Yaşar
Keyword(s):  
Conservation Laws ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Ansatz Method ◽  
Nonlocal Conservation Laws ◽  
Dependent Model ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
Evolution Type ◽  
Partial Lagrangian

AbstractIn this work, we consider the (1+1)-dimensional modified improved Boussinesq (IMBq) equation. As the considered equation is of evolution type, no recourse to a Lagrangian formulation is made. However, we showed that by utilising the partial Lagrangian method and multiplier method, one can construct a number of local and nonlocal conservation laws for the IMBq equation. In addition, by using a solitary wave ansatz method, we obtained exact bright soliton solutions for this equation. The parameters of the soliton envelope (amplitude, widths, velocity) were obtained as function of the dependent model coefficients. Note that, it is always useful and desirable to construct exact solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.

scholarly journals New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1001 ◽  
Author(s):  
Subhadarshan Sahoo ◽  
Santanu Saha Ray ◽  
Mohamed Aly Mohamed Abdou ◽  
Mustafa Inc ◽  
Yu-Ming Chu
Keyword(s):  
Conservation Laws ◽  
Fractional Differential Equations ◽  
Soliton Solutions ◽  
Logistic Function ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Fractional Differential ◽  
Physical Phenomena

New soliton solutions of fractional Jaulent-Miodek (JM) system are presented via symmetry analysis and fractional logistic function methods. Fractional Lie symmetry analysis is unified with symmetry analysis method. Conservation laws of the system are used to obtain new conserved vectors. Numerical simulations of the JM equations and efficiency of the methods are presented. These solutions might be imperative and significant for the explanation of some practical physical phenomena. The results show that present methods are powerful, competitive, reliable, and easy to implement for the nonlinear fractional differential equations.

A new hierarchy of Korteweg–de Vries equations

1976 ◽  
Vol 351 (1666) ◽  
pp. 407-422 ◽  
Keyword(s):  
Conservation Laws ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Bäcklund Transformations ◽  
Multiple Soliton Solutions ◽  
Oscillating Solutions ◽  
Backlund Transformations ◽  
De Vries ◽  
Rank 2

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Bäcklund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

OPTICAL SOLITON SOLUTIONS OF THE LONG-SHORT-WAVE INTERACTION SYSTEM

2013 ◽  
Vol 22 (02) ◽  
pp. 1350015 ◽  
Author(s):  
AHMET BEKIR ◽  
ESIN AKSOY ◽  
ÖZKAN GÜNER
Keyword(s):  
Soliton Solution ◽  
Function Method ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Short Wave ◽  
Optical Soliton ◽  
Physical Parameters ◽  
Ansatz Method ◽  
Topological Soliton ◽  
Dependent Model

This paper, studies the long-short-wave interaction (LS) equation. An optical soliton solution is obtained by the exp-function method and the ansatz method. Subsequently, we formally derive the dark (topological) soliton solutions for this equation. By using the exp-function method, some additional solutions will be derived. The physical parameters in the soliton solutions of ansatz method: amplitude, inverse width, and velocity are obtained as functions of the dependent model coefficients.

scholarly journals Darboux Transformation, Exact Solutions and Conservation Laws for the Reverse Space-Time Fokas-Lenells Equation

2021 ◽  
Author(s):  
Jiang-Yan Song ◽  
Yu Xiao ◽  
Chi-Ping Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Space Time ◽  
Local Equation ◽  
Seed Solution ◽  
Dynamical Behaviors ◽  
Local Case ◽  
Physical Phenomena

Abstract In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.

Dark Solitons for a Generalized Korteweg-de Vries Equation with Time-Dependent Coefficients

2011 ◽  
Vol 66 (3-4) ◽  
pp. 199-204
Author(s):  
Houria Triki ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Pulse Propagation ◽  
Kdv Equation ◽  
Critical Rayleigh Number ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Time Dependent ◽  
Ansatz Method ◽  
Wave Dynamics ◽  
De Vries ◽  
Dependent Model

We consider the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value within the framework of a perturbed Korteweg-de Vries (KdV) equation. In order to study the wave dynamics of nonlinear pulse propagation in an inhomogeneous KdV media, a generalized form of the considered model with time-dependent coefficients is presented. By means of the solitary wave ansatz method, exact dark soliton solutions are derived under certain parametric conditions. The results show that the soliton parameters (amplitude, inverse width, and velocity) are influenced by the time variation of the dependent model coefficients. The existence of such a soliton solution is the result of the exact balance among nonlinearity, third-order and fourth-order nonlinear dispersions, diffusion, dissipation, and reaction.

On the solutions and conservation laws of a two-dimensional Korteweg de Vries model: Multiple exp-function method

2018 ◽  
Vol 24 (1) ◽  
pp. 27-33 ◽  
Author(s):  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Infinite Number ◽  
Function Method ◽  
Soliton Solutions ◽  
Two Dimensional ◽  
De Vries ◽  
Nonlocal Conservation Laws ◽  
Exact Soliton Solutions

AbstractUnder investigation in this paper is a two-dimensional Korteweg de Vries model, which is a spacial extension of the Korteweg de Vries model. An infinite number of nonlocal conservation laws are given which indicate the integrability of this model. Exact soliton solutions are then respectively derived by means of the multiple exp-function method.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

scholarly journals A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
Author(s):  
N Chatterjee ◽  
U S Fjordholm
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Examples ◽  
Continuity Equations ◽  
Multiple Dimensions ◽  
Nonlocal Conservation Laws ◽  
Volume Method ◽  
Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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