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.

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

2016 ◽  
Vol 30 (03) ◽  
pp. 1650008 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Wen-Rong Sun ◽  
Yu-Feng Wang ◽  
Yun-Po Wang
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Time Dependent ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Bell Polynomial ◽  
Backlund Transformations

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

scholarly journals The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
S. M. Sayed
Keyword(s):  
Conservation Laws ◽  
Traveling Wave ◽  
Geometrical Property ◽  
Traveling Wave Solutions ◽  
Structural Equations ◽  
Bäcklund Transformations ◽  
Geometrical Method ◽  
Pseudospherical Surfaces ◽  
Backlund Transformations ◽  
De Vries

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.

N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Bäcklund Transformations ◽  
Nonlinear Superposition ◽  
Backlund Transformations ◽  
De Vries

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.

Bäcklund transformations and soliton solutions for a (2 + 1)-dimensional Korteweg–de Vries-type equation in water waves

2015 ◽  
Vol 81 (4) ◽  
pp. 1815-1821 ◽  
Author(s):  
Yun-Po Wang ◽  
Bo Tian ◽  
Ming Wang ◽  
Yu-Feng Wang ◽  
Ya Sun ◽  
...  
Keyword(s):  
Water Waves ◽  
Soliton Solutions ◽  
Type Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
De Vries

Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg–de Vries Equations

2011 ◽  
Vol 66 (10-11) ◽  
pp. 625-631
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Mkdv Equation ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
De Vries ◽  
Coupled Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

We make use of Hirota’s bilinear method with computer symbolic computation to study a variety of coupled modified Korteweg-de Vries (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are obtained for each coupled equation. The resonance phenomenon of each coupled mKdV equation is proved not to exist.

Bäcklund transformation according to Gerdzikov and Kullish for the Korteweg–de Vries equation

1982 ◽  
Vol 60 (11) ◽  
pp. 1599-1606 ◽  
Author(s):  
Henri-François Gautrin
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Scattering Parameters ◽  
Backlund Transformation ◽  
Specific Change ◽  
Backlund Transformations ◽  
De Vries

A study of solutions of the Gel'fand–Levitan equation permits one to establish new Bäcklund transformations for the Korteweg–de Vries equation. To a specific change in the scattering parameters, there corresponds a family of Bäcklund transformations. A means to construct these transformations is presented.

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