Rational solutions to the Burgers' equation from particular polynomials
PIERRE GAILLARD
Institut Mathématiques de Bourgogne
Université de Bourgogne Franche Comté
9 avenue Alain Savary BP 47870 21078 Dijon Cedex,
FRANCE
Abstract: From particular polynomials, we construct rational solutions to the Burgers' equation as a quotient of a
polynomial of degree n−1in xand n−1−
n
2
in t, by a polynomial of degree nin xand
n
2
in t,|n|being the
greater integer less or equal to n. We call these solutions, solutions of order n.
We construct explicitly these solutions for orders 1until 20.
Key-Words: Burgers equation, rational solutions, determinants.
Received: July 13, 2023. Revised: October 12, 2023. Accepted: October 27, 2023. Published: November 15, 2023.
1 Introduction
We consider the Burgers' equation which can be writ-
ten as
ut+uxx +uux= 0 (1)
where the subscripts xand tdenote partial deriva-
tives.
In 1915, [1] introduced this equation (1). This
equation appears in different contexts in physics as
in gas dynamics, [2], acoustics, [3], heat conduction,
[4], in soil water, [5], in hydrodynamics turbulence,
[6], [7], [8], in shock waves, [9],...
The first solutions has been constructed, [1] in 1915.
Other types of methods have been used to solve this
equation. We can quote the exp-function method,
[13], the tanh-coth method, [14], the groups actions
on coset bundles, [15], the Cole-Hopf method,
[16, 17, 18], [17], [18], the homotopy perturbation
method, [19],...
We can quote some recent results in connection with
this study as, [10], [11], [12].
Rational solutions to the Burgers' equation are con-
structed in this paper. We give solutions as a quotient
of a polynomial of degree n−1in xand n−1−n
2
in tby a polynomial of degree nin xand |[fracn2]
in t,|p|]being the greater integer less or equal to p.
We explicitly build these solutions for orders
between 1and 20.
2 Rational solutions to the Burger's
equation
We consider the following polynomials defined by
pn(x, t) = n
k=0 xk
k!
(−t)
n−k
2
n−k
2!
1−n−k−2n−k
2,
for n≥0,
pn(x, t) = 0 for n < 0.
(2)
With the choice of these polynomials, we have the fol-
lowing statement
Theorem
The function vndefined by
vn(x, t) = 2pn−1(x, t)
pn(x, t),(3)
where pnare defined by previous relations (2), is a
solution to the Burgers' equation (1)
ut+uxx +uux= 0.
Remark
In the following, we will call the solution vn, the
solution of order nof the Burgers' equation, (1).
Remark
More explicitly, the previous polynomials can be
written as
p2k(x, t) = n
l=0 x2l
(2l)!
(−t)k−l
k−l!,
for k≥0,
p2k+1(x, t) = n
l=0 x2l+1
(2l+1)!
(−t)k−l
k−l!,
for k≥0,
pn(x, t) = 0 for n < 0.
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Proof
The proof is elementary. It is sufficient to evaluate
the expression
A=vt+v2x+vvx.
Taking into account that (pn)x=pn−1and
(pn)t=−pn−2, we can write
A=2pn−1
pnt+2pn−1
pn2x+2pn−1
pn2pn−1
pnx
=−2pn−3pn+2pn−1pn−2
p2
n
+2pn−2pn−2p2
n−1
p2
nx
+ 2pn−1
pn2pn−2pn−2p2
n−1
p2
n
=1
p3
n−2pn−3p2
n+ 2pn−2pn−1pn
+pn(2pn−3pn+ 2pn−2pn−1
−4pn−1pn−2)
−2pn−1(2pn−2pn−2p2
n−1)
+4pn−2pn−1pn−4p3
n−1)
The simplifications then give A= 0 and the result.
3 Explicit first order solutions
All these rational solutions are singular. At each or-
der, we see the appearence of curves of singularities.
The patterns of singularities are lines, as in figures [1],
[3],[5],[7],[9],[11],[13],[15],[17],[19], or horseshoe,
as in figures [2],[4],[6],[8],[10],[12],[14],[16],[18],
[20], type depending on the order of the solution, .
3.1 First order solutions
Proposition
The function vdefined by
v1(x, t) = 2
x(4)
is a solution to the Burgers' equation (1).
Figure 1. Solution of order 1to (1).
3.2 Solutions of order two
Proposition
The function v2defined by
v2(x, t) = −4x
−x2+ 2 t,(5)
is a solution to the Burgers' equation (1).
Figure 2. Solution of order 2to (1).
3.3 Solutions of order three
Proposition
The function v3defined by
v3(x, t) = 6 −x2+ 2 t
x(−x2+ 6 t),(6)
is a solution to the Burgers' equation (1).
Figure 3. Solution of order 3to (1).
3.4 Solutions of order four
Proposition
The function v4defined by
v4(x, t) = −8x−x2+ 6 t
x4−12 x2t+ 12 t2,(7)
is a solution to the Burgers' equation (1).
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Figure 4. Solution of order 4to (1).
3.5 Solutions of order five
Proposition
The function v5defined by
v5(x, t) = 10 x4−12 x2t+ 12 t2
x(x4−20 x2t+ 60 t2),(8)
is a solution to the Burgers' equation (1).
Figure 5. Solution of order 5to (1).
3.6 Solutions of order six
Proposition
The function v6defined by
v6(x, t) = −12 xx4−20 x2t+ 60 t2
−x6+ 30 x4t−180 x2t2+ 120 t3,(9)
is a solution to the Burgers' equation (1).
Figure 6. Solution of order 6to (1).
3.7 Solutions of order seven
Proposition
The function v7defined by v7(x, t) = 14n(x,t)
d(x,t),
n(x, t) = (−x6+ 30 x4t−180 x2t2+ 120 t3),
d(x, t) = x(−x6+ 42 x4t−420 x2t2+ 840 t3)
is a solution to the Burgers' equation (1).
Figure 7. Solution of order 7to (1).
3.8 Solutions of order eight
Proposition
The function v8defined by v8(x, t) = −16n(x,t)
d(x,t),
n(x, t) = x(−x6+ 42 x4t−420 x2t2+ 840 t3),
d(x, t) = x8−56 x6t+ 840 x4t2−3360 x2t3+
1680 t4,
is a solution to the Burgers' equation (1).
Figure 8. Solution of order 8to (1).
3.9 Solutions of order nine
Proposition
The function v9defined by v9(x, t) = 18n(x,t)
d(x,t),
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n(x, t) = x8−56 x6t+ 840 x4t2−3360 x2t3+
1680 t4,
d(x, t) = x(x8−72 x6t+ 1512 x4t2−10080 x2t3+
15120 t4),
is a solution to the Burgers' equation (1).
Figure 9. Solution of order 9to (1).
3.10 Solutions of order ten
Proposition
The function v10 defined by v10(x, t) = 20 n(x,t)
d(x,t),
n(x, t) = x(x8−72 x6t+ 1512 x4t2−10080 x2t3+
15120 t4),
d(x, t) = −x10 + 90 x8t−2520 x6t2+ 25200 x4t3−
75600 x2t4+ 30240 t5,
is a solution to the Burgers' equation (1).
Figure 10. Solution of order 10 to (1).
3.11 Solutions of order eleven
Proposition
The function v11 defined by v11(x, t) = 22 n(x,t)
d(x,t),
n(x, t) = −x10 + 90 x8t−2520 x6t2+ 25200 x4t3−
75600 x2t4+ 30240 t5,
d(x, t) = x(−x10 + 110 x8t−3960 x6t2+
55440 x4t3−277200 x2t4+ 332640 t5),
is a solution to the Burgers' equation (1).
Figure 11. Solution of order 11 to (1).
3.12 Solutions of order twelve
Proposition
The function v12 defined by v12(x, t) = n(x,t)
d(x,t),
n(x, t) = −24x(−x10 + 110 x8t−3960 x6t2+
55440 x4t3−277200 x2t4+ 332640 t5),
d(x, t) = x12 −132 tx10 +5940 t2x8−110880 t3x6+
831600 t4x4−1995840 t5x2+ 665280 t6
is a solution to the Burgers' equation (1).
Figure 12. Solution of order 12 to (1).
3.13 Solutions of order thirteen
Proposition
The function v13 defined by v13(x, t) = n(x,t)
d(x,t),
n(x, t) = 26(x12 −132 tx10 + 5940 t2x8−
110880 t3x6+ 831600 t4x4−1995840 t5x2+
665280 t6),
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d(x, t) = x(x12 −156 tx10 + 8580 t2x8−
205920 t3x6+ 2162160 t4x4−8648640 t5x2+
8648640 t6)
is a solution to the Burgers' equation (1).
Figure 13. Solution of order 13 to (1).
3.14 Solutions of order fourteen
Proposition
The function v14 defined by v14(x, t) = n(x,t)
d(x,t),
n(x, t) = −28x(x12 −156 tx10 + 8580 t2x8−
205920 t3x6+ 2162160 t4x4−8648640 t5x2+
8648640 t6),
d(x, t) = −x14 + 182 tx12 −12012 t2x10 +
360360 t3x8−5045040 t4x6+ 30270240 t5x4−
60540480 t6x2+ 17297280 t7
is a solution to the Burgers' equation (1).
Figure 14. Solution of order 14 to (1).
3.15 Solutions of order fifthteen
Proposition
The function v15 defined by v15(x, t) = n(x,t)
d(x,t),
n(x, t) = 30(−x14 + 182 tx12 −12012 t2x10 +
360360 t3x8−5045040 t4x6+ 30270240 t5x4−
60540480 t6x2+ 17297280 t7),
d(x, t) = x(−x14 + 210 tx12 −16380 t2x10 +
600600 t3x8−10810800 t4x6+ 90810720 t5x4−
302702400 t6x2+ 259459200 t7)
is a solution to the Burgers' equation (1).
Figure 15. Solution of order 15 to (1).
3.16 Solutions of order sixteen
Proposition
The function v16 defined by v16(x, t) = n(x,t)
d(x,t),
n(x, t) = −32x(−x14 + 210 tx12 −16380 t2x10 +
600600 t3x8−10810800 t4x6+ 90810720 t5x4−
302702400 t6x2+ 259459200 t7),
d(x, t) = x16 −240 tx14 + 21840 t2x12 −
960960 t3x10 + 21621600 t4x8−242161920 t5x6+
1210809600 t6x4−2075673600 t7x2+518918400 t8
is a solution to the Burgers' equation (1).
Figure 16. Solution of order 15 to (1).
3.17 Solutions of order seventeen
Proposition
The function v17 defined by v17(x, t) = n(x,t)
d(x,t),
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n(x, t) = 34(x16 −240 tx14 + 21840 t2x12 −
960960 t3x10 + 21621600 t4x8−242161920 t5x6+
1210809600 t6x4−2075673600 t7x2+
518918400 t8),
d(x, t) = x(x16 −272 tx14 + 28560 t2x12 −
1485120 t3x10 + 40840800 t4x8−588107520 t5x6+
4116752640 t6x4−11762150400 t7x2+
8821612800 t8)
is a solution to the Burgers' equation (1).
Figure 17. Solution of order 17 to (1).
3.18 Solutions of order eighteen
Proposition
The function v18 defined by v18(x, t) = n(x,t)
d(x,t),
n(x, t) = −36x(x16 −272 tx14 + 28560 t2x12 −
1485120 t3x10 + 40840800 t4x8−588107520 t5x6+
4116752640 t6x4−11762150400 t7x2+
8821612800 t8),
d(x, t) = −x18 + 306 tx16 −36720 t2x14 +
2227680 t3x12 −73513440 t4x10 +
1323241920 t5x8−12350257920 t6x6+
52929676800 t7x4−79394515200 t8x2+
17643225600 t9
is a solution to the Burgers' equation (1).
Figure 18. Solution of order 18 to (1).
3.19 Solutions of order nineteen
Proposition
The function v19 defined by v19(x, t) = n(x,t)
d(x,t),
n(x, t) = 38(−x18 + 306 tx16 −
36720 t2x14 + 2227680 t3x12 −73513440 t4x10 +
1323241920 t5x8−12350257920 t6x6+
52929676800 t7x4−79394515200 t8x2+
17643225600 t9),
d(x, t) = x(−x18 + 342 tx16 −46512 t2x14 +
3255840 t3x12 −126977760 t4x10 +
2793510720 t5x8−33522128640 t6x6+
201132771840 t7x4−502831929600 t8x2+
335221286400 t9)
is a solution to the Burgers' equation (1).
Figure 19. Solution of order 19 to (1).
3.20 Solutions of order twenty
Proposition
The function v20 defined by v20(x, t) = n(x,t)
d(x,t),
n(x, t) = −40x(−x18 + 342 tx16 −
46512 t2x14 + 3255840 t3x12 −126977760 t4x10 +
2793510720 t5x8−33522128640 t6x6+
201132771840 t7x4−502831929600 t8x2+
335221286400 t9),
d(x, t) = x20 −380 tx18 + 58140 t2x16 −
4651200 t3x14 + 211629600 t4x12 −
5587021440 t5x10 + 83805321600 t6x8−
670442572800 t7x6+ 2514159648000 t8x4−
3352212864000 t9x2+ 670442572800 t10
is a solution to the Burgers' equation (1).
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Figure 20. Solution of order 20 to (1).
4 Conclusion
We have given an expression of rational solutions to
the Burgers' equation involving particular polynomi-
als.
In particular, we have constructed explicit solutions
to the Burgers' equation for the orders n= 1 until
n= 20.
All these solutions are singular. We can classify them
by the pattern of their singulatities.
The singularities of these solutions depend on the or-
ders of the solutions. When we consider odd order
solutions we have always the line x= 0 of singular-
ities. In the case of even order solutions n= 2p, the
singularities form horseshoe patterns with pbranches.
It will be interesting to construct solutions of this
equation depending on some real parameters.
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Contribution of Individual Authors to the
Creation of a Scientific Article
Pierre Gaillard is the only one who directed the re-
search, established the results and wrote the arti-
cle.
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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