Multiple soliton solutions for an integrable couplings of the Boussinesq equation

2013 ◽  
Vol 73 ◽  
pp. 38-40 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Multiple Soliton Solutions ◽  
Integrable Couplings

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.

scholarly journals A New Three Dimensional Dissipative Boussinesq Equation for Rossby Waves and Its Multiple Soliton Solutions☆

2021 ◽  
Author(s):  
Liguo Chen ◽  
Liangui Yang
Keyword(s):  
Rossby Waves ◽  
Boussinesq Equation ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Auxiliary Equation ◽  
Linear Wave ◽  
Auxiliary Equation Method ◽  
Multiple Soliton Solutions ◽  
New Equation

Abstract A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a three dimensional dissipative Boussinesq equation by the reduced perturbation method, i.e.utt +e1uxx +e2(u2)xx + e3utxy + e4uxxxx + e5uxxyy = 0. It is emphasized that the new equation is different from the existing Boussinesq equations, which describe the three dimensional nonlinear Rossby waves in the atmosphere. Moreover, we explore the dispersion relation of the linear wave through the new equation. Using the trial function and auxiliary equation method, the two kinds of soliton solutions of the equation are obtained successfully. Finally, the formation mechanism of Rossby waves is discussed by multiple soliton solutions.

A nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

2021 ◽  
Author(s):  
Xi-zhong Liu ◽  
Jun Yu
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Method ◽  
Multiple Soliton Solutions ◽  
Symmetry Method

Abstract A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Multiple Soliton Solutions of the Boussinesq Equation

2005 ◽  
Vol 71 (2) ◽  
pp. 129-131 ◽  
Author(s):  
Jun Yu ◽  
Quanping Sun ◽  
Weijun Zhang
Keyword(s):  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Multiple Soliton Solutions

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

Open Physics ◽  
2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Hirota’S Method ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Integrable Couplings ◽  
Kp Equations ◽  
Hirota's Method

AbstractIn this work, we study the nonlinear integrable couplings of the KdV and the Kadomtsev-Petviashvili (KP) equations. The simplified Hirota’s method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the KdV and the KP equations, but differ only in the coefficients of the transformation used. This difference exhibits soliton solutions for some equations and anti-soliton solutions for others.

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