Rogue waves of the Hirota equation in terms of quasi-rational
solutions depending on multi-parameters
PIERRE GAILLARD,
Institut de math´ematiques de Bourgogne,
Universit´e de Bourgogne Franche Comt´e,
9 avenue Alain Savary BP 47870
21078 Dijon Cedex, France
FRANCE
Abstract:Quasi-rational solutions to the Hirota equation are given. We con- struct explicit expressions
of these solutions for the first orders. As a byproduct, we get quasi-rational solutions to the focusing
NLS equation and also rational solutions to the mKdV equation. We study the patterns of these
configurations in the (x, t) plane.
Key-Words: Hirota equation - quasi-rational solutions.
Received: July 15, 2022. Revised: January 14, 2023. Accepted: February 11, 2023. Published: March 7, 2023.
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We consider the Hirota (H) equation in
the following normalization
iut+α(uxx + 2|u|2u)−iβ(uxxx
+6|u|2ux)=0,(1)
with as usual the subscript meaning
the partial derivatives and α,βreal
numbers.
Hirota proposed this equation in 1973
in a slightly different formulation
iut+ρ1uxx +δ1|u|2u+iσ1uxxx
+3iα1|u|2ux= 0,(2)
with α1,ρ1,σ1,δ1real numbers satis-
fying α1ρ1=σ1δ1and built a kind of
soliton in [1].
This last equation can be rewritten as
(1) by choosing α1=−2β,δ1= 2α,
ρ1=α,σ1=−β.
Maccari introduced in 1998 [2]a two
dimensional extension of this equation.
This equation is used to identify many
kinds of nonlinear phenomena in the
fields of physics, in particular in opti-
cal fibers. It describes the evolution of
the slowly varying amplitude of a non-
linear train in weakly nonlinear sys-
tems. Hirota equation is an integrable
equation which has a number of phys-
ical applications, such as the propaga-
tion of optical pulses [3] or in plasma
physics, [4].
A lot of methods to solve this equation
have been presented such as general
projective Riccati equation method [5],
Darboux transformation [6, 7] or the
trace method [8].
Here, we try to construct other types
of solutions to the Hirota equation, be-
longing to the AKNS hierarchy, in or-
der to get as particular case some solu-
tions to the NLS and mKdV equations.
We construct quasi-rational solutions
for the first orders.
Theorem 2.1 The function v(x, t)de-
fined by
v(x, t) = 1−41+4iαt
1 + (2 x+ 12 βt)2+ 16 α2t2!e2iat (3)
is a solution to the Hirota equation (1)
iut+α(uxx + 2|u|2u)−iβ(uxxx + 6|u|2ux) = 0.
Proof: Just replace the expression of
the solution given by (3) and check that
(1) is satisfied.
We give another type of solution with
one real parameter a1.
Theorem 2.2 The function v(x, t)de-
fined by
v(x, t) = 1−41 + i(4 αt −24 a1)
1 + (2 x+ 12 βt)2+ (4αt −24 a1)2!
×ei(2 αt−6a1)(4)
is a solution to the Hirota equation (1)
iut+α(uxx + 2|u|2u)−iβ(uxxx + 6|u|2ux) = 0,
with a1arbitrary real parameter.
Proof: It is sufficient to replace the
expression of the solution given by (4)
and check that (1) is satisfied.
Remark 2.1 It is possible to add other
real parameters, but this does not rad-
ically change the structure of the solu-
tion as in the case of a parameter a1.
Therefore, we do not mention these pa-
rameters in the following.
1. Introduction 2. Quasi-rational solutions of
order 1 to the Hirota equation
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Figure 1. Solution of order 1 to (1),
a1= 0; with α=β= 1.
Figure 2. Solution of order 1 to (1),
a1= 0.2; with α=β= 1.
Figure 3. Solution of order 1 to (1),
a1= 10; with α=β= 1.
Remark 2.2 If we choose, α= 1,β=
0and the other parameters equal to 0,
we recover the classical Peregrine
breather, solution to the NLS equation
iut+uxx + 2|u|2u= 0,(5)
v(x, t) = 1−41 + 4 it
1+4x2+ 16 t2e2it.(6)
This solution was first found by Pere-
grine [9].
It was rediscovered by the present au-
thor by using degeneracy of Riemann
theta functions in [10].
Remark 2.3 If we choose, α= 0,β=
1and the parameters equal to 0, we re-
cover a non singular rational solution
to the mKdV equation
ut=uxxx + 6|u|2ux(7)
v(x, t) = 1−4
1 + (2x+ 12t)2.(8)
This solution is different from those con-
structed by the author in [11].
Theorem 3.1 The function v(x, t)de-
fined by
v(x, t) = 1−12 n(x, t)
d(x, t)e2iα t (9)
with
n(x, t) = (2 x+ 12 β t)4+ 6 (16 α2t2+
1)(2 x+12 β t)2+192 β t(2 x+12 β t)+
1280 α4t4+ 288 α2t2−3 + i(4 α t(2 x+
3. Quasi-rational solutions of
order 2 to the Hirota equation
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12 β t)4+ 2 (64 α3t3−12 α t)(2 x
+ 12 β t)2+ 768 α t2β(2 x+ 12 β t)
+ 1024 α5t5+ 128 α3t3−60 α t)
and
d(x, t) = ((2 x+ 12 β t)2+ 16 α2t2+
1)3−192 β t(2 x+12 β t)3−24 (16 α2t2−
1)(2 x+12 β t)2+576 (16 α2t2+1)β t(2 x+
12 β t) + 6144 α4t4+ 8 + 1536 α2t2+
9216 β2t2
is a solution to the Hirota equation (1).
Proof: By replacing the expression of
the given solution with (9), we check
that the relation (1) is satisfied.
We can give also the solution depend-
ing on 2 real parameters.
The function v(x, t) defined by
Theorem 3.2
v(x, t) = 1−12 n(x, t)
d(x, t)ei(2 α t−6a1)(10 )
with
n(x, t) = (2 x+12 β t+60 b1)4+6 ((4 α t−
24 a1)2+1)(2 x+12 β t+60 b1)2−24 (−8β t −
80 b1)(2 x+ 12 β t + 60 b1) + 5 (4 α t −
24 a1)4+18 (4 α t−24 a1)2−384 a1(4 α t−
24 a1)−3+i((4 α t−24 a1)(2 x+12 β t+
60 b1)4+ 2 ((4 α t −24 a1)3−12 α t +
168 a1)(2 x+12 β t+60 b1)2−24 (4 α t−
24 a1)(−8β t−80 b1)(2 x+12 β t+60 b1)
+ (4 α t −24 a1)5+ 2 (4 α t −24 a1)3−
192 a1(4 α t −24 a1)2−60 α t + 552 a1)
and
d(x, t) = ((2 x+12 β t+60 b1)2+(4 α t−
24 a1)2+ 1)3+ 24 (−8β t −80 b1)(2 x+
12 β t + 60 b1)3−24 ((4 α t −24 a1)2−
48 a1(4 α t −24 a1)−1)(2 x+ 12 β t +
60 b1)2−72 ((4 α t−24 a1)2+1)(−8β t−
80 b1)(2 x+ 12 β t + 60 b1) + 24 (4 α t −
24 a1)4−384 a1(4 α t−24 a1)3+96 (4 α t−
24 a1)2−3456 a1(4 α t−24 a1)+36864 a12+
144 (−8β t −80 b1)2+ 8
is a solution to the Hirota equation (1).
Proof: By replacing the expression of
the solution given by (10), we check
that the relation (1) is satisfied.
Figure 4. Solution of order 1 to (1);
a1= 0, b1= 0 with α=β= 1.
Figure 5. Solution of order 1 to (1);
a1= 1, b1= 0 with α=β= 1.
Figure 6. Solution of order 1 to (1),
a1= 10, b1= 0 with α=β= 1.
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We observe a very fast evolution of the
structure of the solution according to
the value of th parameter a1toward
three peaks.
Figure 7. Solution of order 1 to (1),
a1= 0, b1= 0.01 with α=β= 8.
Figure 8. Solution of order 1 to (1),
a1= 0, b1= 0.05 with α=β= 8.
Figure 9. Solution of order 1 to (1),
a1= 0, b1= 1 with α=β= 8.
The evolution previously observed is
the same as in the case of the parame-
ter a1.
Remark 3.1 If we choose, α= 1,β=
0and the parameters equal to 0, we re-
cover solutions to the NLS equation
iut+uxx + 2|u|2u= 0,(11)
uk(x, t) = (1 −12 n(x, t)
d(x, t))e2it (12)
with
n(x, t) = 16 x4+ 24 (16 t2+ 1)x2−3 +
1280 t4+ 288 t2+i(64 tx4+ 8 (64 t3−
12 t)x2+ 1024 t5+ 128 t3−60 t)
and
d(x, t) = (4 x2+16 t2+1)3+8−96 (16 t2−
1)x2+ 6144 t4+ 1536 t2.
We recover the solutions given by the
author in [10].
Remark 3.2 If we choose, α= 0,β=
1and the parameters equal to 0, we re-
cover rational solutions to the mKdV
equation
ut=uxxx + 6|u|2ux(13)
u(x, t) = (1 −12 n(x, t)
d(x, t)) (14)
with
n(x, t) = (2 x+ 12 t)4+ 6 )2 x+ 12 t)2+
192 t(2 x+ 12 t)−3
and v(x, t) = ((2 x+ 12 t)2+ 1)3−192
, t(2 x+ 12 t)3+ 24 (2 x+ 12 t)2
+ 576 t(2 x+ 12 t) + 8 + 9216 t2.
These solutions are different from these
presented in [12].
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Theorem 4.1 The function v(x, t)de-
fined by
v(x, t) = 1−24 n(x, t)
d(x, t)e2iα t (15)
with
n(x, t) = (2 x+12 β t)10+15 (1+16 α2t2)
(2 x+ 12 β t)8+ (210 −960 α2t2
+ 12800 α4t4)(2 x+ 12 β t)6
+(92160 α2t3β+5760 β t)(2 x+12 β t)5
+(7200 α2t2−38400 α4t4+286720 α6t6
−450 + 345600 β2t2)(2 x+ 12 β t)4
+ (4915200 α4t5β−57600 β t
+ 1843200 α2t3β)(2 x+ 12 β t)3
+ (1728000 α4t4+ 1720320 α6t6
+2949120 t8α8−43200 α2t2(5+256 β2t2)
−675 −691200 β2t2)(2 x+ 12 β t)2
+ (55050240 α6t7β+ 1382400 α2t3β
−86400 β t −22118400 β3t3
−7372800 α4t5β)(2 x+ 12 β t)
+ 8970240 α6t6+ 32440320 t8α8
+ 11534336 t10α10 + 675 + 115200 α4t4
(−17 + 1792 β2t2) + 10800 α2t2(−3
+1024 β2t2)+1728000 β2t2+i((−60 α t
+ 320 α3t3)(2 x+ 12 β t)8+ (−600 α t
−8960 α3t3+10240 α5t5)(2 x+ 12 β t)6
+ (122880 α3t4β−23040 α t2β)
(2 x+ 12 β t)5+ (−28800 α3t3
−215040 α5t5+ 163840 α7t7+ 1800 α t
(−3 + 768 β2t2))(2 x+ 12 β t)4
−14254080 α7t7+ (3932160 α5t6β
−230400 α t2β+ 2457600 α3t4β)(2 x
+12 β t)3+(1751040 α5t5−983040 α7t7
+1310720 t9α9−57600 α3t3(7+256 β2t2)
+2700 α t(7+1024 β2t2))(2 x+12 β t)2
+ 6553600 t9α9+ (31457280 α7t8β
+ 43200 α t(−24 β t −2048 β3t3)
−5529600 α3t4β−41287680 α5t6β)(2 x
+ 12 β t) + 4194304 t11α11 + 4 α t(2 x
+12 β t)10+92160 α5t5(−107+1792 β2t2)−
14400 α3t3(11+5120 β2t2)−2700 α t(−7+
3584 β2t2))
and
d(x, t) = 2024+(−1920 α2t2+120)(2 x+
12 β t)8+(7680 α2t2−61440 α4t4+2320+
138240 β2t2)(2 x+12 β t)6+(1474560 α4t5β−
552960 α2t3β+51840 β t)(2 x+12 β t)5+
(−368640 α4t4+ 3360 + 3840 α2t2(56 +
8640 β2t2)+2073600 β2t2)(2 x+12 β t)4+
(31457280 α6t7β+16588800 α2t3β+88473600 α4t5β−
345600 β t+44236800 β3t3)(2 x+12 β t)3+
(55050240 α6t6+15728640 t8α8−61440 α4t4(−326+
2880 β2t2)+7680 α2t2(−76+8640 β2t2)+
12144 −6220800 β2t2)(2 x+ 12 β t)2+
(188743680 α8t9β+ 47185920 α6t7β−
66355200 α4t5β+1036800 α2t2(40 β t−
2048 β3t3)+648000 β t−132710400 β3t3)(2 x+
12 β t)+22809600 β2t2+243793920 t8α8+
125829120 t10α10 + 2123366400 β4t4−
960 β t(2 x+12 β t)9+327680 α6t6(191+
4032 β2t2)+61440 α4t4(599+8640 β2t2)+
384 α2t2(3881+777600 β2t2)+(1+(2 x+
12 β t)2+ 16 α2t2)6
is a solution to the Hirota equation (1).
Proof: Check that the relation (1) is
satisfied when we replace the expres-
sion of the solution defined by (15).
We can give also the solutions to the
Hirota equation depending on 4 real
parameters. Because of the length of
the expression, we only give it in the
appendix.
We give patterns of the modulus of the
solutions in function of the parameters.
4. Quasi-rational solutions of
order 3 to the Hirota equation
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Figure 10. Solution of order 3 to (1),
a1= 0, b1= 0, a2= 0, b2= 0 with
α=β= 1.
Figure 11. Solution of order 3 to (1),
a1= 1, b1= 0, a2= 0, b2= 0 with
α=β= 1.
Figure 12. Solution of order 3 to (1),
to the right a1= 2, b1= 0, a2= 10,
b2= 0 with α=β= 1.
Figure 13. Solution of order 3 to (1),
a1= 0, b1= 0, a2= 0.1, b2= 0 with
α=β= 1.
Figure 14. Solution of order 3 to (1),
a1= 0, b1= 10, a2= 1, b2= 0 with
α=β= 1.
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Figure 15. Solution of order 3 to (1),
a1= 10, b1= 0, a2= 8, b2= 0 with
α=β= 1.
Figure 16. Solution of order 3 to (1),
a1= 0, b1= 0.1, a2= 0, b2= 0 with
α=β= 1.
Figure 17. Solution of order 3 to (1),
a1= 0, b1= 1, a2= 0, b2= 0 with
α=β= 1.
Figure 18. Solution of order 3 to (1),
a1= 2, b1= 10, a2= 0, b2= 0 with
α=β= 1.
Figure 19. Solution of order 3 to (1),
a1= 0, b1= 0, a2= 0, b2= 0.01 with
α=β= 1.
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Figure 20. Solution of order 3 to (1),
a1= 0, b1= 10, a2= 0, b2= 0.1 with
α=β= 1.
Figure 21. Solution of order 3 to (1),
a1= 10, b1= 0, a2= 0, b2= 1 with
α=β= 1.
In these examples, we note the appear-
ance of triangles with six peaks.
Remark 4.1 If we choose, α= 1,β=
0and the parameters equal to 0, we re-
cover solutions to the NLS equation
iut+uxx + 2|u|2u= 0,(16)
uk(x, t) = (1 −24 n(x, t)
d(x, t))e2iα t (17)
with
n(x, t) = 16 x4+ 24 ((4 t−24000)2+
1)x2+ 9215999997 + 5 (4 t−24000)4+
18 (4 t−24000)2−1536000 t+i(16 (4 t−
24000)x4+ 8 ((4 t−24000)3−12 t+
168000)x2+ (4 t−24000)5+ 2 (4 t−
24000)3−192000 (4 t−24000)2−60 t+
552000)
and
d(x, t) = (4 x2+ (4 t−24000)2+ 1)3+
119808000008 −96 ((4 t−24000)2
−192000 t+ 1151999999)x2+ 24 (4 t−
24000)4−384000 (4 t−24000)3+96 (4 t−
24000)2−13824000 t.
We recover the solutions constructed in
[?].
Remark 4.2 If we choose, α= 0,β=
1and the parameters equal to 0, we re-
cover rational solutions to the mKdV
equation
ut=uxxx + 6|u|2ux(18)
uk(x, t) = (1 −24 n(x, t)
d(x, t))e2iα t (19)
with
n(x, t) = (2 x+ 12 t)4+ 6 (2 x+ 12 t)2+
192 t(2 x+ 12 t)−3
and d(x, t) = ((2 x+12 t)2+1)3−192 t(2 x+
12 t)3+ 24 (2 x+ 12 t)2+ 576 t(2 x+
12 t) + 8 + 9216 t2.
These solutions are different from these
given in [11, 12].
Quasi-rational solutions to the Hirota
equation have been given for the first
orders. As a byproduct, we have re-
cover quasi-rational solutions to the NLS
equation and rational solutions to the
mKdV equation.
We can mention some other recent works
about this equation. For example, in
[13], by using the inverse scattering trans-
form an explicit soliton solution for-
mula for the Hirota equation has been
5. Conclusion
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constructed. This formula allows to
get, as a particular case, the N-soliton
solution, the breather solution and, most
relevantly, a new class of solutions called
multi-pole soliton solutions.
By using the Hirota direct method, lump-
soliton to the HIrota equation were con-
structed in [14].
In [15], the mixed localized wave solu-
tions of the Hirota equation have been
constructed through the modified Dar-
boux transformation. One of them is
the mixed 1-breather and 1-rogue wave
solution and the other two are the mixed
1-breather and 2-rogue wave solution,
and the mixed 2-breather and 1-rogue
wave solution. These mixed localized
wave solutions are presented graphically
by choosing proper parameters and their
dynamic behavior is briefly studied.
It will relevant to describe a more gen-
eral formulation of the the solutions to
the Hirota equation in terms of wron-
skians or Fredholm determinants as in
the works [16, 17, 18] for NLS, KP or
KdV equations.
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Soliton perturbation theory for
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770778, 2008
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Math. J. Chin. Univ., V. 23, N. 1,
5764, 2008
[5] Q. Wang, Y. Chen, B. Li,
H. Zhang, New exact travel-
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Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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Appendix
Solution of order 3 to the Hirota equation depending on 4 real parameters :
The function v(x, t) defined by
v(x, t) = 1−24 n(x, t)
d(x, t)e2iα t (20)
with
n(x, t) = 675 + 15 (1 + (4 α t −24 a1+ 120 a2)2)(2 x+ 12 β t + 60 b1−280 b2)8+
(210−60 (4 α t−24 a1+120 a2)2+50 (4 α t−24 a1+120 a2)4+480 (4 α t−24 a1+
120 a2)(16 a1−160 a2))(2 x+ 12 β t + 60 b1−280 b2)6+ (−720 (4 α t −24 a1+
120 a2)2(−8β t −80 b1+ 560 b2) + 5760 β t −11520 b1+ 564480 b2)(2 x+ 12 β t +
60 b1−280 b2)5+ (450 (4 α t −24 a1+ 120 a2)2−150 (4 α t −24 a1+ 120 a2)4+
70 (4 α t −24 a1+ 120 a2)6+ 1200 (4 α t −24 a1+ 120 a2)3(16 a1−160 a2)−450 +
5400 (−8β t −80 b1+ 560 b2)2−1800 (16 a1−160 a2)2+ 3600 (4 α t −24 a1+
120 a2)(16 a1−224 a2))(2 x+ 12 β t + 60 b1−280 b2)4+ (−2400 (4 α t −24 a1+
120 a2)4(−8β t −80 b1+ 560 b2) + 28800 (4 α t −24 a1+ 120 a2)(−8β t −80 b1+
560 b2)(16 a1−160 a2)−57600 β t −806400 b1+ 7257600 b2−7200 (4 α t −24 a1+
120 a2)2(−16 β t −128 b1+ 672 b2))(2 x+ 12 β t + 60 b1−280 b2)3+ (6750 (4 α t −
24 a1+ 120 a2)4+ 420 (4 α t −24 a1+ 120 a2)6+ 45 (4 α t −24 a1+ 120 a2)8−
2700 (4 α t−24 a1+120 a2)2(5+4 (−8β t−80 b1+560 b2)2−12 (16 a1−160 a2)2)−
675 −10800 (−8β t −80 b1+ 560 b2)2−10800 (16 a1−160 a2)2+ 21600 (4 α t −
24 a1+120 a2)(32 a1−384 a2)−7200 (4 α t−24 a1+120 a2)3(32 a1−128 a2))(2 x+
12 β t+60 b1−280 b2)2+(−1680 (4 α t−24 a1+120 a2)6(−8β t−80 b1+560 b2)−
28800 (4 α t−24 a1+120 a2)3(−8β t−80 b1+560 b2)(16 a1−160 a2)−10800 (4 α t−
24 a1+120 a2)2(−8β t−272 b1+3248 b2)−86400 β t−1209600 b1+10886400 b2+
43200 (−8β t−80 b1+560 b2)3+ 43200 (−8β t−80 b1+560 b2)(16 a1−160 a2)2+
3600 (4 α t −24 a1+ 120 a2)4(−8β t + 80 b1−1680 b2)−86400 (4 α t −24 a1+
120 a2)((−8β t−80 b1+560 b2)(16 a1−160 a2)+(16 a1−160 a2)(32 b1−448 b2)−
64 (−8β t −80 b1+ 560 b2)a2))(2 x+ 12 β t + 60 b1−280 b2)−720 (4 α t −24 a1+
120 a2)7(16 a1−160 a2) + 450 (4 α t −24 a1+ 120 a2)4(−17 + 28 (−8β t −80 b1+
560 b2)2+ 12 (16 a1−160 a2)2)−3600 (4 α t −24 a1+ 120 a2)3(48 a1−1376 a2)−
720 (4 α t−24 a1+120 a2)5(272 a1−3168 a2)+10800 (4 α t−24 a1+120 a2)(−16 a1+
224 a2+ 4 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2) + 4 (16 a1−160 a2)3) +
675 (4 α t −24 a1+ 120 a2)2(−3 + 16 (−8β t −80 b1+ 560 b2)2+ 16 (16 a1−
160 a2)2−128 (−8β t−80 b1+ 560 b2)(32 b1−448 b2)−8192 (16 a1−160 a2)a2)−
86400 (−8β t −80 b1+ 560 b2)(32 b1−448 b2)−11059200 (16 a1−160 a2)a2+
(2 x+ 12 β t + 60 b1−280 b2)10 + 2190 (4 α t −24 a1+ 120 a2)6+ 27000 (−8β t −
80 b1+560 b2)2+91800 (16 a1−160 a2)2+495 (4 α t−24 a1+120 a2)8+11 (4 α t−
24 a1+ 120 a2)10 + 86400 (32 b1−448 b2)2+ 353894400 a22+i(−21600 (−8β t −
80 b1+ 560 b2)2(16 a1−160 a2) + (4 α t −24 a1+ 120 a2)(2 x+ 12 β t + 60 b1−
280 b2)10 + (−60 α t + 840 a1−6600 a2+ 5 (4 α t −24 a1+ 120 a2)3)(2 x+ 12 β t +
60 b1−280 b2)8+ (−600 α t −240 a1+ 58800 a2−140 (4 α t −24 a1+ 120 a2)3+
10 (4 α t −24 a1+ 120 a2)5+ 240 (4 α t −24 a1+ 120 a2)2(16 a1−160 a2))(2 x+
12 β t + 60 b1−280 b2)6+ (−240 (4 α t −24 a1+ 120 a2)3(−8β t −80 b1+ 560 b2)−
1440 (−8β t−80 b1+560 b2)(16 a1−160 a2)+ 720 (4 α t−24 a1+120 a2)(−8β t −
176 b1+ 1904 b2))(2 x+ 12 β t + 60 b1−280 b2)5+ (−450 (4 α t −24 a1+ 120 a2)3−
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Volume 22, 2023
210 (4 α t −24 a1+120 a2)5+ 10 (4 α t −24 a1+ 120 a2)7+ 300 (4 α t −24 a1+
120 a2)4(16 a1−160 a2) + 450 (4 α t −24 a1+ 120 a2)(−3 + 12 (−8β t −80 b1+
560 b2)2−4 (16 a1−160 a2)2)−14400 a1+ 259200 a2+ 1800 (4 α t −24 a1+
120 a2)2(16 a1−224 a2))(2 x+ 12 β t + 60 b1−280 b2)4+ (−480 (4 α t −24 a1+
120 a2)5(−8β t −80 b1+ 560 b2) + 14400 (4 α t −24 a1+ 120 a2)2(−8β t −80 b1+
560 b2)(16 a1−160 a2) + 7200 (4 α t −24 a1+ 120 a2)(−8β t −48 b1+ 112 b2)−
2400 (4 α t −24 a1+ 120 a2)3(−16 β t −128 b1+ 672 b2)−14400 (−8β t −80 b1+
560 b2)(16 a1−160 a2)−14400 (16 a1−160 a2)(32 b1−448 b2) + 921600 (−8β t −
80 b1+ 560 b2)a2)(2 x+ 12 β t + 60 b1−280 b2)3+ (1710 (4 α t −24 a1+ 120 a2)5−
60 (4 α t −24 a1+ 120 a2)7+ 5 (4 α t −24 a1+ 120 a2)9−900 (4 α t −24 a1+
120 a2)3(7+4 (−8β t−80 b1+560 b2)2−12 (16 a1−160 a2)2)+675 (4 α t−24 a1+
120 a2)(7 + 16 (−8β t −80 b1+ 560 b2)2+ 16 (16 a1−160 a2)2)−345600 a1+
4492800 a2−21600 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2)−21600 (16 a1−
160 a2)3+691200 (4 α t−24 a1+120 a2)2a2−1800 (4 α t−24 a1+120 a2)4(64 a1−
448 a2))(2 x+ 12 β t + 60 b1−280 b2)2+ (−240 (4 α t −24 a1+ 120 a2)7(−8β t −
80 b1+ 560 b2)−7200 (4 α t −24 a1+ 120 a2)4(−8β t −80 b1+ 560 b2)(16 a1−
160 a2) + 10800 (4 α t −24 a1+ 120 a2)(−24 β t −400 b1+ 3920 b2+ 4 (−8β t −
80 b1+ 560 b2)3+ 4 (−8β t −80 b1+ 560 b2)(16 a1−160 a2)2) + 3600 (4 α t −
24 a1+120 a2)3(−24 β t −176 b1+ 784 b2) +720 (4 α t −24 a1+ 120 a2)5(−56 β t−
400 b1+ 1680 b2) + 21600 (−8β t −80 b1+ 560 b2)(16 a1−160 a2) + 43200 (16 a1−
160 a2)(32 b1−448 b2)−2764800 (−8β t −80 b1+ 560 b2)a2−43200 (4 α t −
24 a1+120 a2)2((−8β t −80 b1+560 b2)(16 a1−160 a2) + (16 a1−160 a2)(32 b1−
448 b2)−64 (−8β t −80 b1+ 560 b2)a2))(2 x+ 12 β t + 60 b1−280 b2)−90 (4 α t −
24 a1+ 120 a2)8(16 a1−160 a2) + 90 (4 α t −24 a1+ 120 a2)5(−107 + 28 (−8β t −
80 b1+ 560 b2)2+ 12 (16 a1−160 a2)2) + 5400 (4 α t −24 a1+ 120 a2)2(176 a1−
2464 a2+ 4 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2) + 4 (16 a1−160 a2)3)−
120 (4 α t−24 a1+120 a2)6(80 a1−1248 a2)+900 (4 α t−24 a1+120 a2)4(464 a1−
4000 a2)−225 (4 α t−24 a1+120 a2)3(11+80 (−8β t−80 b1+560 b2)2+80 (16 a1−
160 a2)2+128 (−8β t −80 b1+ 560 b2)(32 b1−448 b2) +8192 (16 a1−160 a2)a2)−
675 (4 α t−24 a1+120 a2)(−7+56 (−8β t−80 b1+560 b2)2+88 (16 a1−160 a2)2−
128 (−8β t −80 b1+ 560 b2)(32 b1−448 b2)−128 (32 b1−448 b2)2−524288 a22) +
5529600 (−8β t−80 b1+560 b2)2a2−5529600 (16 a1−160 a2)2a2−172800 (−8β t−
80 b1+560 b2)(16 a1−160 a2)(32 b1−448 b2)+64800 (16 a1−160 a2)3−870 (4 α t−
24 a1+120 a2)7+25 (4 α t−24 a1+120 a2)9+(4 α t−24 a1+120 a2)11 −151200 a1+
1857600 a2)
and
d(x, t) = 2024−777600 (−8β t−80 b1+560 b2)(32 b1−448 b2)−82944000 (16 a1−
160 a2)a2+ (−120 (4 α t −24 a1+ 120 a2)2+ 360 (4 α t −24 a1+ 120 a2)(16 a1−
160 a2) + 120)(2 x+ 12 β t + 60 b1−280 b2)8+ (480 (4 α t −24 a1+ 120 a2)2−
240 (4 α t−24 a1+ 120 a2)4+ 960 (4 α t −24 a1+120 a2)3(16 a1−160 a2) + 2320+
2160 (−8β t −80 b1+ 560 b2)2+ 5040 (16 a1−160 a2)2−1440 (4 α t −24 a1+
120 a2)(64 a1−960 a2))(2 x+ 12 β t + 60 b1−280 b2)6+ (−720 (4 α t −24 a1+
120 a2)4(−8β t −80 b1+ 560 b2)−17280 (4 α t −24 a1+ 120 a2)(−8β t −80 b1+
560 b2)(16 a1−160 a2)+ 4320 (4 α t −24 a1+ 120 a2)2(−8β t −176 b1+ 1904 b2)+
51840 β t+103680 b1+2177280 b2)(2 x+12 β t + 60 b1−280 b2)5+(−1440 (4 α t −
24 a1+120 a2)4+720 (4 α t −24 a1+120 a2)5(16 a1−160 a2)+240 (4 α t−24 a1+
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DOI: 10.37394/23206.2023.22.24
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Volume 22, 2023
120 a2)2(56 + 135 (−8β t −80 b1+560 b2)2−45 (16 a1−160 a2)2) + 32400 (4 α t −
24 a1+ 120 a2)(16 a1−288 a2) + 7200 (4 α t −24 a1+ 120 a2)3(48 a1−544 a2) +
3360 + 32400 (−8β t −80 b1+ 560 b2)2−54000 (16 a1−160 a2)2+ 86400 (−8β t −
80 b1+ 560 b2)(32 b1−448 b2) + 5529600 (16 a1−160 a2)a2)(2 x+ 12 β t + 60 b1−
280 b2)4+ (−960 (4 α t −24 a1+ 120 a2)6(−8β t −80 b1+ 560 b2) + 57600 (4 α t −
24 a1+ 120 a2)3(−8β t −80 b1+ 560 b2)(16 a1−160 a2)−43200 (4 α t −24 a1+
120 a2)2(−24 β t −272 b1+ 2128 b2)−7200 (4 α t −24 a1+ 120 a2)4(−48 β t −
448 b1+2912 b2)−345600 β t −5529600 b1+ 53222400 b2−86400 (−8β t −80 b1+
560 b2)3−86400 (−8β t−80 b1+ 560 b2)(16 a1−160 a2)2+ 172800 (4 α t −24 a1+
120 a2)((−8β t−80 b1+560 b2)(16 a1−160 a2)−(16 a1−160 a2)(32 b1−448 b2)+
64 (−8β t −80 b1+ 560 b2)a2))(2 x+ 12 β t + 60 b1−280 b2)3+ (13440 (4 α t −
24 a1+120 a2)6+240 (4 α t−24 a1+120 a2)8−240 (4 α t−24 a1+120 a2)4(−326+
45 (−8β t−80 b1+560 b2)2−135 (16 a1−160 a2)2)+480 (4 α t−24 a1+120 a2)2(−76+
135 (−8β t −80 b1+ 560 b2)2+ 1215 (16 a1−160 a2)2)−129600 (4 α t −24 a1+
120 a2)3(32 a1−256 a2)−12960 (4 α t−24 a1+120 a2)5(32 a1−256 a2)−64800 (4 α t−
24 a1+ 120 a2)(−96 a1+ 1280 a2+ 4 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2) +
4 (16 a1−160 a2)3) + 12144 −97200 (−8β t −80 b1+ 560 b2)2+ 32400 (16 a1−
160 a2)2+518400 (32 b1−448 b2)2−33177600 (16 a1−160 a2)a2+2123366400 a22)(2 x+
12 β t + 60 b1−280 b2)2+ (−360 (4 α t −24 a1+ 120 a2)8(−8β t −80 b1+ 560 b2)−
17280 (4 α t−24 a1+120 a2)5(−8β t−80 b1+560 b2)(16 a1−160 a2)−1440 (4 α t−
24 a1+120 a2)6(−8β t−240 b1+2800 b2)+32400 (4 α t−24 a1+120 a2)4(−8β t+
112 b1−2128 b2) + 64800 (4 α t −24 a1+ 120 a2)2(40 β t + 752 b1−7728 b2+
4 (−8β t−80 b1+560 b2)3+4 (−8β t−80 b1+560 b2)(16 a1−160 a2)2)−777600 (4 α t−
24 a1+120 a2)((−8β t−80 b1+560 b2)(16 a1−160 a2)+2 (16 a1−160 a2)(32 b1−
448 b2)−128 (−8β t−80 b1+560 b2)a2)−172800 (4 α t−24 a1+120 a2)3(3 (−8β t−
80 b1+ 560 b2)(16 a1−160 a2) + (16 a1−160 a2)(32 b1−448 b2)−64 (−8β t −
80 b1+560 b2)a2)+648000 β t+8553600 b1−74390400 b2+259200 (−8β t−80 b1+
560 b2)3+ 1296000 (−8β t −80 b1+ 560 b2)(16 a1−160 a2)2−1036800 (−8β t −
80 b1+560 b2)2(32 b1−448 b2)+1036800 (16 a1−160 a2)2(32 b1−448 b2)−132710400 (−8β t−
80 b1+ 560 b2)(16 a1−160 a2)a2)(2 x+ 12 β t + 60 b1−280 b2) + 120 (−8β t −
80 b1+ 560 b2)(2 x+ 12 β t + 60 b1−280 b2)9+ 1440 (32 b1−448 b2)(2 x+ 12 β t +
60 b1−280 b2)7−120 (4 α t −24 a1+ 120 a2)9(16 a1−160 a2) + 80 (4 α t −24 a1+
120 a2)6(191 + 63 (−8β t −80 b1+ 560 b2)2+ 27 (16 a1−160 a2)2)−2160 (4 α t −
24 a1+ 120 a2)5(240 a1−4576 a2) + 21600 (4 α t −24 a1+ 120 a2)3(−368 a1+
3488 a2+ 4 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2) + 4 (16 a1−160 a2)3)−
1440 (4 α t −24 a1+ 120 a2)7(80 a1−864 a2) + 240 (4 α t −24 a1+ 120 a2)4(599 +
135 (−8β t −80 b1+ 560 b2)2−225 (16 a1−160 a2)2−360 (−8β t −80 b1+
560 b2)(32 b1−448 b2)−23040 (16 a1−160 a2)a2)−16200 (4 α t−24 a1+120 a2)(496 a1−
6240 a2+ 80 (−8β t −80 b1+ 560 b2)2(16 a1−160 a2) + 16 (16 a1−160 a2)3+
128 (−8β t−80 b1+560 b2)(16 a1−160 a2)(32 b1−448 b2)−4096 (−8β t−80 b1+
560 b2)2a2+4096 (16 a1−160 a2)2a2)+24 (4 α t−24 a1+120 a2)2(3881+12150 (−8β t−
80 b1+ 560 b2)2+ 28350 (16 a1−160 a2)2+ 21600 (−8β t −80 b1+ 560 b2)(32 b1−
448 b2) + 21600 (32 b1−448 b2)2+ 88473600 a22) + 1036800 (−8β t −80 b1+
560 b2)2(16 a1−160 a2)2+ 356400 (−8β t −80 b1+ 560 b2)2+ 874800 (16 a1−
160 a2)2+3720 (4 α t−24 a1+120 a2)8+120 (4 α t−24 a1+120 a2)10+518400 (32 b1−
448 b2)2+ 2123366400 a22+ (1 + (2 x+ 12 β t + 60 b1−280 b2)2+ (4 α t −24 a1+
120 a2)2)6+ 518400 (−8β t −80 b1+ 560 b2)4+ 518400 (16 a1−160 a2)4
is a solution to the Hirota equation (1).
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DOI: 10.37394/23206.2023.22.24
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Volume 22, 2023
