Generalized Drinfeld-Sokolov system:

Conservation laws and solutions

Cite as: AIP Conference Proceedings 2293, 220004 (2020); https://doi.org/10.1063/5.0026736
Published Online: 25 November 2020

T. M. Garrido, R. de la Rosa, E. Recio, and M. S. Bruzón

ARTICLES YOU MAY BE INTERESTED IN

On a family of (2+1)-dimensional Zakharov-Kuznetsov modified equal width equations

AIP Conference Proceedings 2293, 220005 (2020); https://doi.org/10.1063/5.0027730

A new symmetry-based method for constructing nonlocally related PDE systems from
admitted multi-parameter groups
Journal of Mathematical Physics 61, 061503 (2020); https://doi.org/10.1063/1.5122319

The general bilinear techniques for studying the propagation of mixed-type periodic and
lump-type solutions in a homogenous-dispersive medium
AIP Advances 10, 105325 (2020); https://doi.org/10.1063/5.0019219

AIP Conference Proceedings 2293, 220004 (2020); https://doi.org/10.1063/5.0026736 2293, 220004

© 2020 Author(s).
 Generalized Drinfeld-Sokolov System: Conservation Laws
and Solutions
T.M. Garrido1,a) , R. de la Rosa1,b) , E. Recio1,c) and M.S. Bruzón1,d)
1
Department of Mathematics, University of Cadiz. Spain
a)
Corresponding author: tamara.garrido@uca.es
b)
rafael.delarosa@uca.es
c)
elena.recio@uca.es
d)
m.bruzon@uca.es

One of the most important and surely one of the most studied of all nonlinear evolution equations is the Korteweg de
Vries equation (KdV)

ut + λuu x + μu xxx = 0. (1)

KdV equation arises to model shallow water waves with weak nonlinearities and it has a wide variety of applications,
such as surface gravity waves [1], propagation of nonlinear acoustic waves in liquids with small volume concentra-
tions of gas bubbles [2, 3], magma flow and conduit waves [4] and many other applications [5].

Drinfeld and Sokolov [6] obtained all the simplest generalized KdV corresponding to classical Kats-Moody algebras
of ranks 1 and 2. Among them, it can be found the following generalization

ut = u xxx + uu x − vv x ,
(2)
vt = −2v xxx − uv x .

This system of equations (2) is also known for being an example of an evolutionary system for two dependent variables
that possesses a Lax pair and, after linear changes and scale transformations, yields a Hamiltonian form with the
Hamiltonian given by
H = u2 + v2 − 6uv. (3)
In the present paper, we have considered a generalization of (2). It is called the generalized Drinfeld-Sokolov system
(gDS)
ut + α1 uu x + β1 u xxx + γvδ v x = 0,
(4)
vt + α2 uv x + β2 v xxx = 0,
where α1 , β1 , α2 , β2 , γ  0 and δ  0, −1.

There are several papers in which different kinds of generalizations of system (2) have been studied. The main problem
is that many of them have different names to call the gDS system. We summarize them and we give a few examples
of the background and results obtained for these systems:

• In [7], it was studied the so called Drinfeld-Sokolov system

ut + u xxx − 6uu x − 6v x = 0,
(5)
vt − 2v xxx + 6uv x = 0.

International Conference of Numerical Analysis and Applied Mathematics ICNAAM 2019

AIP Conf. Proc. 2293, 220004-1–220004-4; https://doi.org/10.1063/5.0026736
Published by AIP Publishing. 978-0-7354-4025-8/$30.00

220004-1
 The authors concluded that the DS system (5) admits a compatible bi-Hamiltonian structure and therefore, is an
integrable system. This system can be obtained from (4) by substituting α1 = −6, α2 = 6, β1 = 1, β2 = −2, δ = 0
and γ = −6.

Moreover, system (5) is also known as the Satsuma-Hirota system in [8]. In this paper, system (5) was
transformed to the single sixth-order Drinfeld-Sokolov-Satsuma-Hirota equation by using a Miura-type
transformation. Three different methods were applied to obtain solutions for the equation.

The same pair of equations (5) were called Bogoyavlenskii Coupled KdV Equations in [9, 10]. It can be found
soliton solutions by means of the standard and non-standard truncation Painlevè expansion and a family of
exact solutions obtained by a Bäcklund transformation respectively.

• In [11], it was considered the following system

ut + 6u xxx − 6uu x − 6v x = 0,
(6)
vt − 2v xxx + 6uv x = 0,
called the Drinfeld-Sokolov-Satsuma-Hirota. Exact solutions were obtained by using the (G’/G)-expansion
method and the generalized Tanh-coth method. System (6) can be obtained from (4) by substituting
α1 = −6, α2 = 6, β1 = 6, β2 = −2, δ = 0 and γ = −6.

• In [12], there were obtained solitary wave, kink (anti-kink) wave and periodic wave solutions by using the
theory of planar dynamical systems for the so called Generalized Drinfeld-Sokolov equations that follows
ut + u xxx − 6uu x − 6 (vα ) x = 0,
(7)
vt − 2v xxx + 6uv x = 0.
Previous system (7) can be obtained from (4) by considering α1 = −6, α2 = 6, β1 = 6, β2 = −2, δ = α − 1 and
γ = −6α.

Therefore, the generalized Drinfeld-Sokolov system presented in this paper generalizes all of the previous equations,
and unifies all the cases and results under the study of the gDS system (4).

On the other hand, in order to simplify the number of arbitrary constants involved in equations (4), we have applied a
scaling equivalence transformation. It is a change of variables which acts on independent and dependent variables
t˜ = λ1 t,
x̃ = λ2 x,
(8)
ũ = λ3 u,
ṽ = λ4 v.
β1 α1
From (8), by taking β = β2 and α = α2 , system (4) can be invertible reduced into a system involving four parameters
instead of six
ut + uu x + u xxx − γvδ v x = 0,
(9)
vt + αuv x + βv xxx = 0,
where α, β  0, γ2 = 1 and δ  0, −1. Hereafter, without loss of generality, we can restrict the study of system (4) to
the system (9).

There is a great interest in determining physical properties that do not change over time. Researchers are looking
for conservations laws of partial differential equations to show if the charge, momentum, angular momentum, or
energy among others are constant over time [13, 14, 15, 16, 17]. Hence, due to the background and the physical
characteristics of the partial differential system that is under study, one of the aims of this paper is to research the
conservation laws of the gDS system (9).

To obtain the conservation laws of the system (9), we apply the direct method of multipliers [18, 19]. This method
updates the classical method to get conservation laws in which Noether’s theorem was applied with the variational

220004-2
symmetries of the equation [20].

All low-order conservation laws of generalized Drinfeld-Sokolov system (9) arise from the multipliers consisting of:
(i) For α, β  0, γ2 = 1 and δ  0, −1.
Q1 = (1, 0). (10)
(ii) For δ = 1.
Q2 = (2αu, −2γv) . (11)
(iii) For δ = 1 and 2α = β.
 
Q3 = (6β − 6) u xx + (3β − 3) u2 − 3γv2 , −6γ (uv + 6v xx ) . (12)

(iv) For δ = 1 and α = −1.

Q4 = (2tu − 2x, 2γtv) . (13)
(v) For α = 1
2 and β = 1.
Q5a = (v, u) , (14)
 
Q5b = v2 , 2uv + 12v xx . (15)
All non-trivial low-order local conservation laws of the generalized Drinfeld-Sokolov system (9) are:
(i) For α, β  0, γ2 = 1 and δ  0, −1.
T 1 = u,
γ δ+1 (16)
X1 = u xx + 12 u2 + δ+1 v .
(ii) For δ = 1.
T 2 = αu2 − γv2 ,
(17)
X2 = −2βγvv xx + βγv2x + 2αuu xx − αu2x + 3 u .
2α 3

(iii) For δ = 1 and 2α = β.

T3 = (3 − 3β) u2x + 18γv2x + (β − 1) u3 − 3γuv2 ,

   2
X3 = − 34 γ2 − 2(β−3)
3γ u + v v −
2 2 2
(3 − 3β) 12 u2 + u xx − 36γvt v x (18)
 2
−18βγ v xx + 16 uv + 6βγvu x v x − 6βγuv2x − 3γv2 u xx + (6β − 6) ut u x .

(iv) For δ = 1 and α = −1.

T 4 = γtv2 + tu2 − 2xu,   (19)
X4 = (2tu − 2x) u xx + 2βγtvv xx + (2 − tu x ) u x − βγtv2x + 23 tu2 − xu u − γxv2 .

(v) For α = 1
2 and β = 1.
T 5a = uv,
γ δ+2 (20)
X5a = vu xx − u x v x + 12 u2 v + uv xx + δ+2 v .

T 5b = uv2 − 6v2x ,
   2 (21)
X5b = δ+33 u + γv
2 δ+1 2
v + 6 v xx + 16 uv − 2vu x v x + 2uv2x + v2 u xx + 12vt v x .
For the gDS system (9) in the previous paper [21], the authors obtained the complete classification of the Lie point
symmetries:
(i) For α, β  0, γ2 = 1 and δ  0, −1.
v1 = ∂t , (22)
v2 = ∂ x , (23)
4v
v3 = 3t∂t + x∂ x − 2u∂u − ∂v . (24)
δ+1

220004-3
(ii) For α = 1. Infinitesimal generators are v1 , v2 , v3 and

v4 = t∂t + ∂v. (25)

Through these symmetries, it was obtained the corresponding reduced ordinary differential system for the travelling
wave case and also soliton solutions by means of the sine-cosine method.

We will get new solutions by means of the double reduction method [22] elsewhere. It has been successfully
applied in several research [23, 24, 25]. The basis of the method is the use of the relation among symmetries (22)-(25)
together with conservation laws (16)-(21) in order to get solutions.

ACKNOWLEDGMENTS
The authors gratefully acknowledge the guidance and contributions received by Dr. Stephen Anco from Brock Uni-
versity. The authors are grateful to the University of Cadiz and Junta de Andalucı́a (FQM-201 group) for their support.

REFERENCES
[1] L. Debnath, Mathematics of Complexity and Dynamical Systems (Springer, New York, 2011).
[2] N. Kudryashov and D. Sinelshchikov, International Journal of Non-Linear Mechanics 63, 31–38 (2014).
[3] L. Wijngaarden, Journal of Fluid Mechanics 33(3), 465–474 (1968).
[4] N. Lowman and M. Hoefer, Journal of Fluid Mechanics 718, 524–557 (2013).
[5] D. Crighton, “Applications of KdV,” in KdV’95: Proceedings of the International Symposium held in Ams-
terdam to commemorate the centennial of the publication of the equation by and named after Korteweg and
de Vries, edited by M. Hazewinkel, H. Capel, and E. de Jager (Springer, 1995), pp. 39–67.
[6] V. G. Drinfeld and V. V. Sokolov, Journal of Soviet Mathematics 30(2), 1975–2036 (1985).
[7] M. Gurses and A. Karasu, Physics Letters A 251, 247–249 (1999).
[8] A. Wazwaz, Applied Mathematics and Computation 207(1), 248 – 255 (2009).
[9] H. Hu and S. Lou, Communications in Theoretical Physics 42(4), 485–487 (2004).
[10] B. Tian and Y. Gao, Physics Letters A 208(3), 193–196 (1995).
[11] J. Heris and M. Lakestani, Exact solutions for the integrable sixth-order drinfeld-sokolov-satsuma-hirota
system by the analytical methods, International Scholarly Research Notices ( 2014).
[12] S. C. L. Wu and C. Pang, Applied Mathematical Modelling 33(11), 4126–4130 (2009).
[13] S. Anco, E. Avdonina, A. Gainetdinova, L. Galiakberova, N. Ibragimov, and T. Wolf, Journal of Physics A:
Mathematical and Theoretical 49(10) (2017).
[14] S. Anco and C. Khalique, International Journal of Modern Physics B 30 (2016).
[15] A. Adem, Journal of Applied Analysis 24(1), 27–33 (2018).
[16] M. Przedborski and S. Anco, Journal of Mathematical Physics 58 (2017).
[17] M. S. Bruzón, A. P. Márquez, T. M. Garrido, E. Recio, and R. de la Rosa, Journal of Computational and
Applied Mathematics 354, 682–688 (2019).
[18] S. Anco and G. W. Bluman, European Journal of Applied Mathematics 13(5), 545–566 (2002).
[19] S. C. Anco and G. W. Bluman, European Journal of Applied Mathematics 13(5), 567–585 (2002).
[20] S. Anco, “Generalization of noether’s theorem in modern form to non-variational partial differential equa-
tions,” in Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational
Science (Springer, New York, 2017), pp. 119–182.
[21] T. M. Garrido and M. S. Bruzón, Journal of Computational and Theoretical Transport 45(4), 290–298 (2016).
[22] A. Sjöberg, Applied Mathematics and Computation 184(2), 608–616 (2007).
[23] M. S. Bruzón, T. M. Garrido, and R. de la Rosa, Applied Mathematics and Nonlinear Sciences 2(2), 485–494
(2017).
[24] S. San, A. Akbulut, Ö. Ünsal, and F. Taşcan, Mathematical Methods in the Applied Sciences 40(5), 1703–
1710 (2017).
[25] E. Buhe, K. Fakhar, G. A. N. G. W. E. I. Wang, and T. Chaolu, Romanian Reports in Physics 67(2), 329–339
(2015).

220004-4