Citation: Fokas, A.S.; Cao, Y.; He, J.
Multi-Solitons, Multi-Breathers and
Multi-Rational Solutions of
Integrable Extensions of the
Kadomtsev–Petviashvili Equation in
Three Dimensions. Fractal Fract. 2022,
6, 425. https://doi.org/10.3390/
fractalfract6080425
Academic Editors: Burcu Gürbüz and
Arran Fernandez
Received: 24 May 2022
Accepted: 22 July 2022
Published: 31 July 2022
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fractal and fractional
Article
Multi-Solitons, Multi-Breathers and Multi-Rational Solutions
of Integrable Extensions of the Kadomtsev–Petviashvili
Equation in Three Dimensions
Athanassios S. Fokas 1,2,3, Yulei Cao 4 and Jingsong He 5,*
1 DAMTP, University of Cambridge, Cambridge CB3 0WA, UK; tf227@cam.ac.uk
2 Viterbi School of Engineering, USC, Los Angeles, CA 90089, USA
3 Centre of Mathematics, Academy of Athens, 11527 Athens, Greece
4 School of Mathematical Sciences, USTC, Hefei 230026, China; caoyulei@mail.ustc.edu.cn
5 Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
* Correspondence: hejingsong@szu.edu.cn or jshe@ustc.edu.cn
Abstract: The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are proto-
typical examples of integrable evolution equations in one and two spatial dimensions, respectively.
The question of constructing integrable evolution equations in three-spatial dimensions has been one
of the most important open problems in the history of integrability. Here, we study an integrable
extension of the KP equation in three-spatial dimensions, which can be derived using a specific reduc-
tion of the integrable generalization of the KP equation in four-spatial and two-temporal dimensions
derived in (Phys. Rev. Lett. 96, (2006) 190201). For this new integrable extension of the KP equation,
we construct smooth multi-solitons, high-order breathers, and high-order rational solutions, by using
Hirota’s bilinear method.
Keywords: Lax pair; bilinear method; high-order localized wave
MSC: 35C08; 35Q51; 37K10; 37K35
1. Introduction
There exists a distinctive class of nonlinear evolution equations in one and two spatial
dimensions, called integrable [1,2]. Such prototypical equations in one spatial dimension are
the celebrated Korteweg–de Vries (KdV) [3,4] and the nonlinear Schrodinger (NLS) [5,6]
equations. Every integrable nonlinear evolution equation in one spatial dimension has
several integrable versions in two spatial dimensions. Two such physically significant
generalizations of the KdV are the Kadomtsev–Petviashvili I (KPI) and II (KPII) equations.
Similarly, two analogous generalizations of the NLS are the Davey–Stewartson I (DS I)
and II (DSII) equations. It is important to emphasize that integrable nonlinear evolution
equations arise in a variety of physical applications. For example, in the context of fluid
mechanics, the KP equations arise in the weakly nonlinear, weakly dispersive, weakly
two-dimensional limit of inviscid, irrotational, water waves, and in the case of KPI when
the surface tension is dominant [7]. Other significant applications of the KP and DS
equations are discussed, for example, in weakly dispersive media [2,8–10] in optics and
hydrodynamics [2,10–13].
One of the most important open questions in the field of integrability has been the
question of the existence of integrable evolution equations in three spatial dimensions.
Substantial progress regarding this question was presented in [14], where 4 + 2 gener-
alizations of the KP and of the DS equations were presented, namely equations in four
spatial and two temporal dimensions. The solution of the Cauchy problem of the 4 + 2
generalization of the KP is presented in [15]. The solution of the Cauchy problem of the
Fractal Fract. 2022, 6, 425. https://doi.org/10.3390/fractalfract6080425 https://www.mdpi.com/journal/fractalfract
Fractal Fract. 2022, 6, 425 2 of 13
4+ 2 generalization of the DS equation is sketched in [14] and presented in detail in [16].
Incidentally, another approach toward integrability in multidimensions is presented in [17],
where it is shown that there exist integrable nonlinear evolution equations in any number
of dimensions; however, these equations have the major disadvantage that they involve a
nonlocal commutator.
The question of reducing integrable equations from 4+ 2 to equations in three spatial
dimensions, although discussed in many papers, including [14,15], as well as [18,19],
remained open. In particular, regarding 3+ 1 reductions of the KP equation, it should be
noted that, although the real reduction discussed in [14] involves two equations, many
authors have erroneously analyzed only one of these two equations, ignoring the other one.
Here, we discuss a constructive approach for obtaining integrable equations in three spatial
dimensions, and by implementing this approach to KP, we derive the results presented
below. Our paper is a companion of [20], where a highly novel nonlinear transform pair in
three spatial dimensions, capable of solving the Cauchy problem of these equations, is also
derived. Specifically, regarding Equation (1) below, we note that the analysis of its Cauchy
problem performed in [20] reveals that the time dependence of the associated spectral
function (the nonlinear Fourier transform of the solution) is of the form exp(iTt − Rt),
where T and R are real functions of the real spectral parameters k1, k2, and k3 (in the linear
limit, these three variables in the Fourier space correspond to the three real variables x, y, z
in the physical space). Because of the occurrence of the term Rt, it is not clear that this initial
value problem is well posed. However, in the recent breakthrough presented in [20], it is
explicitly shown that the Cauchy problem of Equation (1) is indeed well posed. Remarkably,
the novel nonlinear Fourier transform introduced in [20], which is capable of solving (1),
gives rise to a completely new transform for solving the linear limit of Equation (1), namely,
the equation obtained from Equation (1) after eliminating the nonlinear term uux.
The following equation for the complex-valued function u(x, y, z, t), where x, y, z, t, are
real, independent variables, provides an integrable generalization of the KP equation in
three-spatial dimensions,
uxt + αuxxxx + β(uux)x +
γ
4
uyy − γ4 uzz + i
γ
2
uyz = 0, u ∈ C, x, y, z, t ∈ R, (1)
where α, β,γ are arbitrary complex constant parameters.
As is well known, the most important property of integrable equations is that they
admit a Lax pair formulation. The derivation of Equation (1), as well as its Lax pair, is
presented in Section 2. Two-dimensional integrable equations, like KP and DS equations,
possess a variety of particular solutions, which can be obtained using different techniques,
including the inverse scattering method, the d-bar method, the Darboux transformation,
the Hirota bilinear method and the KP hierarchy reduction method [21–30]. In this connec-
tion, smooth multi-soliton and high-order breather solutions, as well as high-order rational
solutions of Equation (1), are constructed in Sections 3 and 4, respectively, via Hirota’s
bilinear method. These results are discussed further in Section 5.
2. Integrable Extensions of the KP in Three Dimensions and Their Lax Pairs
Starting with the KP equation and complexifying x, y, t, namely, introducing the
complex variables
x = x1 + i x2, y = y1 + i y2, t = t1 + i t2, x1, x2, y1, y2, t1, t2 ∈ R, (2)
it is shown in [14] that the equation
qt¯ =
1
4
qx¯x¯x¯ − 32qqx¯ +
3
4
∂−1x¯ qy¯y¯, q ∈ C, (3)
Fractal Fract. 2022, 6, 425 3 of 13
is an integrable generalization of the KP equation in the 4-spatial dimensions x1, x2, y1, y2
and the two temporal dimensions t1, t2. In Equation (3), ∂−1 means integral, bar denotes
complex conjugation, and Equation (2) implies the following identities:
∂t¯ =
1
2 (∂t1 + i∂t2), ∂x¯ =
1
2 (∂x1 + i∂x2), ∂y¯ =
1
2 (∂y1 + i∂y2), (∂
−1
x¯ f )(x1, x2) =
1
pi
∫∫
R2
f (x′1,x′2)
x−x′ dx
′
1dx
′
2, (4)
where we have assumed that f vanishes as |x1|, |x2| → ∞. Assuming q is real and sup-
pressing the t2 dependence, the (4 + 2)-dimensional KP (3) yields two (4 + 1)-dimensional
equations; see Equation (10) or equivalent Equation (11) in [14]. These equations have
provoked extensive studies [18,19]. In addition, similar considerations are valid for the
(4 + 2)-dimensional Davey–Stewartson equation [16].
Replacing in Equation (3) q and y by Qq and y → yγ¯1 , where Q and γ1 are complex
constants, we find ∂y¯ → γ1∂y¯ and
qt¯ =
1
4
qx¯x¯x¯ − 32Qqqx¯ +
3
4
γ21∂
−1
x¯ qy¯y¯. (5)
In order to obtain three-dimensional reductions of this equation, which is the first
purpose of this paper, we let ξ = ax1 + bx2, where a and b are all real constants. Hence,
∂x¯ = A∂ξ , where A = 12 (a+ i b). Replacing in Equation (5) ∂x¯ by A∂ξ , we find a (3 + 2)-
dimensional generalization of the KP
qt¯ =
A3
4
∂3ξq−
3A
2
Qqqξ +
3γ21
4A
∂−1ξ ∂
2
y¯q. (6)
Letting τ = a˜t1 + b˜t2, where a˜, b˜ are real constants, and seeing ∂t¯ = A˜∂τ , where
A˜ = 12 (a˜+ i b˜), Equation (6) reduces to a three-dimensional KP equation for the complex-
valued function q,
qτ + αqξξξ + βqqξ + γ∂−1ξ ∂
2
yq = 0, (7)
where
α = − A
3
4A˜
, β =
3AQ
2A˜
, γ = − 3γ
2
1
4AA˜
= γ˜γ21. (8)
Renaming in Equation (7) q, τ, ξ, y1, y2 by u, t, x, y, z, Equation (7) becomes Equation (1).
Remark 1. If we set A˜ = − A34 ,Q = −A2,γ1 = δA2 (or equivalently γ˜ = 3A4 ) in Equation (7),
then
qτ + qξξξ + 6qqξ + 3δ2∂−1ξ ∂
2
yq = 0, (9)
which indeed looks like a three-dimensional extension of the usual KP equation. In order to obtain a
smooth solution, which will be shown in Section 3, δ2 is a complex-valued constant.
The second purpose of the paper is to show the integrability of the three-dimensional
KP Equation (1) by providing its Lax pair and different kinds of solutions. The Lax pair of
Equation (3) [14] is given by
Dyµ = Mµ, M = D2x − q, (10)
Dtµ = Nµ, N = D3x −
3
2
qDx − N0, N0 = 34 (qx¯ + (∂
−1
x¯ qy¯)), (11)
where µ is an appropriate eigenfunction, and the operators Dx, Dy, Dt are defined by
Dx = ∂x¯ + k, Dy = ∂y¯ + k2, Dt = ∂t¯ + k
3.
Fractal Fract. 2022, 6, 425 4 of 13
The Lax pairs (10) and (11) can be declassified in the form
∂y¯µ = M˜µ, M˜ = ∂2x¯ + 2k∂x¯ − q, (12)
∂t¯µ = N˜µ, N˜ = ∂
3
x¯ + 3k∂
2
x¯ + (3k
2 − 3
2
q)∂x¯ + (−32qk− N0). (13)
Using in these equations the scaling transformations q→ Qq and y→ yγ¯1 , we find the
following Lax pair for Equation (5):
∂y¯µ = [
1
γ1
(∂2x¯ + 2k∂x¯ −Qq)] · µ, (14)
∂t¯µ = [∂
3
x¯ + 3k∂
2
x¯ + (3k
2 − 3
2
Qq)∂x¯ + (−32qQk)−
3Q
4
(qx¯ + γ1(∂−1x¯ qy¯))] · µ. (15)
The crucial reduction ∂x¯ = A∂ξ , ∂t¯ = A˜∂τ imply the following Lax pair for (7):
µy¯ = [
1
γ1
(A2∂2ξ + 2kA∂ξ −Qq)] · µ = M1 · µ, (16)
µt¯ =
1
A˜
[A3∂3ξ + 3kA
2∂2ξ + (3k
2 − 3
2
Qq)A∂ξ + (−32qQk)−
3Q
4
(Aqξ +
γ1
A
(∂−1ξ qy¯))] · µ = N1 · µ, (17)
where A, A˜,Q are expressed by virtue of α, β,γ via equations
A = i(
3α
γ˜
)
1
4 , A˜ =
i
4α
(
3α
γ˜
)
3
4 ,Q =
β√
12αγ˜
. (18)
3. Soliton and Breather of the Three-Dimensional KP Equation
In this section, we derive soliton and breather solutions of the three-dimensional KP
equation (1) by using the Hirota bilinear method and certain perturbation technique [31].
Equation (1) can be transformed into the bilinear form(
DxDt + αD4x +
γ
4
D2y −
γ
4
D2z +
γ
2
i DyDz
)
f · f = 0, (19)
through the variable transformation
u = 12
α
β
(ln f )xx, (20)
where f is a complex function, and D is the Hirota’s bilinear differential operator [31]
defined by
Dmx D
n
t f (x, t) · g(x, t) = (
∂
∂x
− ∂
∂x′
)m(
∂
∂t
− ∂
∂t′
)n f (x, t)g(x
′
, t
′
)
∣∣∣∣∣
x′=x,t′=t
.
Based on this bilinear form, the three-dimensional KP Equation (1) admits the follow-
ing N-soliton solutions:
u = 12
α
β
(ln f )xx, f = ∑
µ=0,1
exp
(
(N)
∑
j<k
µjµkAjk +
N
∑
j=1
µjηj
)
, (21)
with
ηj = k j
(
x+ pjy+ qjz−
[
1
4
(pj + i qj)2 + αk2j
]
t
)
+ η0j ,
eAjk =
(
pj − pk + i qj − i qk
)2
γ− 12α(k j − kk)2(
pj − pk + i qj − i qk
)2
γ− 12α(k j + kk)2 ,
(22)
Fractal Fract. 2022, 6, 425 5 of 13
where the above parameters must satisfy the condition
γIq2j − 2γRpjqj − 4αIk2j − γI p2j = 0, (23)
which assures that these solutions are smooth. Here, k j, pj, qj, η0j are arbitrary real param-
eters, and the subscript j denotes an integer. The subscripts R and I denote the real and
imaginary parts of the corresponding parameters, respectively.
Taking in Equation (21) N = 1, we obtain the one-soliton solution of the three-
dimensional KP equation:
u =
3βk2
α cosh2( ξˆ2 )
=
3βk2
α
sech2(
ξˆ
2
). (24)
The modulus of this solution is given by
|u| = 6|β|k
2
|α|(coshA + cosB) , (25)
with
ξˆ = A + iB, α = αR + i αI , γ = γR + iγI ,
A = k
[
x+ p1y+ q1z− k14
(
4αRk21 + γRp
2
1 − γRq21 − 2γI p1q1
)
t
]
,
B =
k1
4
[
γIq21 − γI p21 − 2γRp1q1 − 4αIk21
]
t.
It can be seen from the above expression that the one-soliton is nonsingular, provided
that cos(B) ≥ 0, which means −pi2 < B = k14
[
γIq21 − γI p21 − 2γRp1q1 − 4αIk21
]
t < pi2 .
For simplicity, we set
[
γIq21 − γI p21 − 2γRp1q1 − 4αIk21
]
= 0, as noted in Equation (23),
to guarantee the smoothness of the one-soliton solution (24).
Taking in (21) N = 2, α = 1+ i and γ = 1− i, imposing the constraints
−q21 − 2p1q1 − 4k21 + p21 = 0, −q22 − 2p2q2 − 4k22 + p22 = 0,
and choosing k1 = 1, k2 = 1, p1 = 32 , p2 = −3, q1 =
√
2− 3
2
, q2 = 3+
√
14, η01 = η
0
2 = 0, we
obtain a smooth two-soliton solution of the three-dimensional KP equation:
u =12
α
β
(ln f )xx, f = 1+ eη1 + eη2 + A12eη1+η2 ,
η1 = x+
3
2
y+
(
2
√
2− 3
2
)
z+
1− 3√2
4
t,
η2 = x− 3y+
(
3+
√
14
)
z+
(
7+ 3
√
14
)
t,
A12 =
(
55+ 18
√
14− 9√2− 2√7
)
+ i
(
26+ 2
√
7
)
(
103+ 18
√
14− 9√2− 2√7
)
+ i
(
14+ 2
√
7
) .
It is straightforward, using Equation (21), to derive multi-soliton solutions of the
three-dimensional KP equation. Such solutions are depicted in Figure 1. The effect of
changing the variables z and t is simply to move the profile of the soliton in the (x, y)-plane,
without any deformations.
Fractal Fract. 2022, 6, 425 6 of 13
Figure 1. Plots of |u| of soliton solutions for the three-dimensional KP Equation (1) generated by
Equation (21) with parameters α = 1 + i, β = 1, η0j (j = 1, 2, 3, ) = 0 and variables t = 0, z = 0. (a):
One-soliton solution with parameters N = 1,γ = 1 + 203 i, k1 = 1, p1 =
1
2 , q1 = 1; (b): Two-soliton
solution with parameters N = 2,γ = 1− i, k1 = 1, k2 = 1, p1 = 32 , p2 = −3, q1 =
√
2− 3
2
, q2 =
3 +
√
14; (c): Three-soliton solution with parameters N = 3,γ = 1− i, k1 = k2 = k3 = 1, p1 =
1
2 , p2 = −3, p3 = −2, q1 =
√
2− 3
2
, q2 = 3+
√
14, q3 = 4; (d): Four-soliton solution with parameters
N = 4,γ = 1− i, k1 = k2 = k3 = k4 = 1, p1 = 12 , p2 = −3, p3 = −2, p4 = −2, q1 =
√
2− 3
2
, q2 =
3+
√
14, q3 = 4, q4 = 0.
In order to derive breather solutions of the three-dimensional KP equation, we use the
constraints,
N = 2n, k∗j = kn+j, p
∗
j = pn+j, q
∗
j = qn+j, η
0∗
j = η
0
n+j. (26)
For the particular case of Equation (26),
N = 2, k∗1 = k2, p
∗
1 = p2, q
∗
1 = q2, η
0∗
1 = η
0
2 ,
we obtain a first-order breather solution of Equation (1) u = 12 αβ (ln f )xx with f given
below:
f = 1+ [cosh(l1) + sinh(l1)] · [cosh(B1t) + sinh(B1t)] · [cos(A2t) + i sin(A2t)] · [cos(l2) + i sin(l2)]
+ [cosh(l1) + sinh(l1)] · [cosh(B1t)− sinh(B1t)] · [cos(A2t) + i sin(A2t)] · [cos(l2)− i sin(l2)]
+ A˜12[cosh(2l1) + sinh(2l1)] · [cos(2A2t) + i sin(2A2t)],
A˜12 =
[
(p21I − q21I)γR − 12αRk21I − 2γI p1Iq1I
]
+ i
[
(p21I − q21I)γI − 12αIk21I + 2γRp1Iq1I
][
(p21I − q21I)γR + 12αRk21R − 2γI p1Iq1I
]
+ i
[
(p21I − q21I)γI + 12αIk21R + 2γRp1Iq1I
] ,
l1 =k1Rx+ [k1Rp1R − k1I p1I ]y+ [k1Rq1R − k1Iq1I ]z+ A1t,
l2 =k1Ix+ [k1I p1R + k1Rp1I ]y+ [k1Iq1R + k1Rq1I ]z+ B2t,
A1 =− αRk31R +
1
4
[−C1γR + 12αRk21I + 2C2γI ]k1R +
1
2
D1k1I , C1 = p21R − p21I − q21R + q21I ,
B1 =− αIk31I +
1
4
[C1γI + 12αIk21R + 2C2γR]k1I +
1
2
D2k1R, C2 = p1Rq1R − p1Iq1I ,
A2 =− αIk31R +
1
4
[−C1γI + 12αIk21I − 2C2γR]k1R +
1
2
D2k1I , D1 = (p1Rp1I − q1Rq1I)γR − (p1Rq1I − p1Iq1R)γI ,
B2 =αRk31I +
1
4
[−C1γR − 12αRk21R + 2C2γI ]k1I −
1
2
D1k1R, D2 = (p1Rp1I − q1Rq1I)γI − (p1Rq1I + p1Iq1R)γR,
(27)
where the parameters satisfy
− 2αI(p1Rq1I + p1Iq1R) + 2γI(q1Rq1I − p1Rp1I)− 8γRk1Rk1I = 0,
− C1 − 2αI(p1Rq1R + p1Iq1I) + 4γR(k21I − k21R) = 0.
(28)
Without loss of generality, we study the dynamics of the first-order breather in the
(x, y)-plane. The velocities of the first-order breathers along the x and y directions are
given by
Vx =
k1R[p1IA1 − p1RB2] + k1I [p1RA1 − p1IB2]
p1I
[
k21R + k
2
1I
] , Vy = − k1IA1 − k1RB2p1I[k21R + k21I] . (29)
Fractal Fract. 2022, 6, 425 7 of 13
The period of the first-order breather is given by T =
√
Tx + Ty, where Tx and Ty are
given by
Tx = −2pi[k1Rp1R − k1I p1I ]p1I
[
k21R + k
2
1I
] , Ty = 2pik1Rp1I[k21R + k21I] . (30)
Note that changing z and t only results in moving the profile of |u| in a parallel manner
in the (x, y)-plane, and it shows that there exist four types of breathers:
• The line breather. If k1R = 0, k1Rp1R − k1I p1I = 0, A1 6= 0, k1I 6= 0, k1I p1R +
k1Rp1I 6= 0, B2 6= 0, the first-order breather (27) is a linear breather in the (x, y)
plane. The four pictures (a–d) (Figure 2) describe a series of periodic line waves
that suddenly appear from the constant background plane and then annihilate in the
original background.
• The oblique breather. If k1R 6= 0, k1Rp1R − k1I p1I 6= 0, A1 6= 0, k1I 6= 0, k1I p1R +
k1Rp1I 6= 0, B2 6= 0, the first-order breather (27) is an oblique breather in the (x, y)
plane, which has periodicity in both x and y directions; see Figure 2e.
• The x-periodic breather. If k1R = 0, k1Rp1R − k1I p1I 6= 0, A1 6= 0, k1I 6= 0, k1I p1R +
k1Rp1I = 0, B2 6= 0, the first-order breather (27) has periodicity only in the x-direction
in the (x, y) plane, and is localized in the y-direction; see Figure 2f.
• The y-periodic breather. If k1R 6= 0, k1Rp1R − k1I p1I = 0, A1 6= 0, k1I = 0, k1I p1R +
k1Rp1I 6= 0, B2 = 0, the first-order breather (27) has periodicity only in the y-direction
in the (x, y) plane, and is localized in the x-direction; see Figure 2g. Since the coefficient
of t in l1 is not zero, the effect of t is to move the breather along the x-axis. Similarly,
since the coefficient of z in l2 is not zero, the effect of z is to move the breather along
the y-axis.
Figure 2. (a–d): Plots of |u| for the first-orderline breather (27) of the three-dimensional KP Equation
(1) with parameters N = 2, α = 1+ i, β = 1,γ = 1+ i, k1 = i, p1 = 13 , q1 =
1+
√
34
3
i, η01 = 0, z = 0;
(e): An oblique breather with parameters N = 2, α = 1− i, β = 1,γ = 1 + 203 i, k1 = 1 + i, p1 =
1 + 2i, q1 = 1 + (
√
6 + 2)i, η01 = 0, z = 0, t = 0; (f): An x-periodic breather with parameters N =
2, α = 1− i, β = 1,γ = 1+ i, k1 = i, p1 = i, q1 =
√
2+ i, η01 = η
0
2 = 0, z = 0, t = 0; (g): A y-periodic
breather with parameters α = 1+ 2i, β = 20,γ = 1+ i, k1 = 14 , p1 =
i
2 , q1 =
i
2 , η
0
1 = 0, z = 0, t = 0.
Taking in (26)
N = 4, k∗1 = k2, k
∗
3 = k4, p
∗
1 = p2, p
∗
3 = p4, q
∗
1 = q2, q
∗
3 = q4, η
0∗
1 = η
0
2 , η
0∗
3 = η
0
4 ,
a second-order breather of three-dimensional KP Equation (1) can be derived. The dynamic
of this solution is shown in Figure 3.
Fractal Fract. 2022, 6, 425 8 of 13
Figure 3. Second-order breathers in the (x, y)-plane of the three-dimensional KP Equation (1) with
parameters N = 4, α = 1− i, β = 1,γ = 1 + i and variables z = 0, t = 0. (a): k1 = 1 + i, k3 =
1 + i, p1 = 1 + 2i, p3 = 1 + 3i, q1 = 1 + (2 +
√
6)i, q3 =
2+
√
8+ 2
√
17√
8+ 2
√
17
+ (3 +
√
8+ 2
√
17)i, η01 =
0, η03 = 2pi; (b): k1 = i, k3 =
i
2 , p1 = 2i, p3 = 2i, q1 = 4i, q3 = (2 +
√
7)i, η01 = −pi, η03 = pi; (c):
k1 = 14 , k3 =
1
4 , p1 =
i
2 , p3 =
i
3 , q1 =
1+
√
3
2
i, q3 =
2+
√
17
6
i, η01 = 0, η
0
3 = pi.
In addition, mixed solutions composed of solitons and breathers, are also given for
N > 2. Taking in (21)
N = 3, k∗1 = k2, p
∗
1 = p2, q
∗
1 = q2, η
0∗
1 = η
0
2 , k3 = 1, p3 = −2, q3 = 4, η03 = 0,
a hybrid solution consisting of a first-order breather and one soliton is constructed, see
Figure 4a. For larger N, we obtain mixed solutions consisting of several first-order breathers
and solitons. For example, taking in (21),
N = 4, k∗1 = k2, p
∗
1 = p2, q
∗
1 = q2, η
0∗
1 = η
0
2 , k3 = 1, k4 = 1,
p3 =
3
2
, p4 = −3, q3 =
√
2− 3
2
, q4 = 3+
√
14, η03 = η
0
4 = 0,
we derive the hybrid solution consisting of one first-order breather and two solitons, see
Figure 4b.
Figure 4. Hybrid solutions of the three-dimensional KP Equation (1) consisting of a breather and
solitons with parameters α = 1+ i, β = 1,γ = 1− i, η0j (j = 1, 2, 3, 4) = 0 and variables z = 0, t = 0.
(a): N = 3, k1 = k∗2 = −i, p1 = p∗2 = i, q1 = q∗2 =
√
2− i, k3 = 1, p3 = −2, q3 = 4; (b): N = 4, k1 =
k∗2 = i, p1 = p∗2 = −i, q1 = q∗2 =
√
2+ i, k3 = 1, k4 = 1, p3 =
3
2
, p4 = −3, q3 =
√
2− 3
2
, q4 = 3+
√
14.
4. Rational Solution of the Three-Dimensional KP Equation
In order to derive rational solutions of the three-dimensional KP Equation (1), we take
in Equation (21)
N = 2n, η0j = ipi (1 ≤ j ≤ N), (31)
Fractal Fract. 2022, 6, 425 9 of 13
and compute a suitable limit as k j → 0. In this way, the function f , defined in (21), becomes
a polynomial function. This construction yields an n-th rational solution, u = 12 αβ (ln f )xx,
of Equation (1), in which f is written as follows:
f = f [n] =
N
∏
k=1
θk +
1
2
(N)
∑
k,j
αkj
N
∏
l 6=k,j
θl + · · ·+ 1M!2M
(N)
∑
i,j,...,m,n
M︷ ︸︸ ︷
αkjαkl · · · αmn
N
∏
p 6=k,j,...m,n
θp + · · · , (32)
with
θj = x+ pjy+ qjz− γ4
(
pj + i qj
)2t, αjk = 24α1
2
[
(pj − pk)2 − (qj − qk)2
]
+ i (pj − pk)(qj − qk)
. (33)
To illustrate this formula of rational solutions, we provide concrete examples by taking
in Equation (32)
N = 2, p1 = p∗2 = pR + i pI , q1 = q∗2 = qR + i qI , α = αR + i αI , γ = γR + iγI , (34)
where we impose the constraints
pR =
γ2R + γ
2
I − γR
√
γ2R + γ
2
I
γI
√
γ2R + γ
2
I
qR, pI =
√
γ2R + γ
2
I − γR
γI
qI , (35)
In this way, we obtain a first-order rational solution of the three-dimensional KP
Equation (1):
u = 12
α
β
(ln f )xx, f = L 21 +L
2
2 + α12, α12 =
6γ2I (αR + i αI)(√
γ2R + γ
2
I − γR
)(
γ2R + γ
2
I
)
q2I
,
L1 = x+
γ2R + γ
2
I − γR
√
γ2R + γ
2
I
γI
√
γ2R + γ
2
I
qRy+ qRz+
(
q2R − q2I
)(
γ2R + γ
2
I − γR
√
γ2R + γ
2
I
)√
γ2R + γ
2
I
2γ2I
t,
L2 =
γ2R + γ
2
I − γR
√
γ2R + γ
2
I
γI
√
γ2R + γ
2
I
qIy+ qIz+
(
γ2R + γ
2
I − γR
√
γ2R + γ
2
I
)√
γ2R + γ
2
I
γ2I
qRqI t.
(36)
If
αR
(√
γ2R + γ
2
I − γR
)
> 0 or αI 6= 0, (37)
this first-order rational solution is nonsingular. Without loss of generality, taking in (36)
αR = 1, αI = 1,γR = 1,γI = 1, qR = 1, qI = 1, that is, α = 1 + i,γ = 1 + i, q = 1 + i,
we obtain the following analytical expression of first-order rational solution of the three-
dimensional KP Equation (1):
u = 8
(√
2− 1
)
(1+ i)
1−√2
3
[
x+ (
√
2− 1)y+ z
]2
+ 68+52
√
2
3
(
1
2y+
√
2+1
2 z+ t
)2
+ 1+ i
√
2−1
3
[
x+ (
√
2− 1)y+ z
]2
+ 68+52
√
2
3
(
1
2y+
√
2+1
2 z+ t
)2
+ 1+ i
. (38)
The first-order rational solution given in (38) is a lump in the (x, y)-plane, which reaches
the maximum of 8(2−√2) at
(
x = (2
√
2− 2) t, y = −(√2+ 1) z− 2t
)
; see Figure 5.
Fractal Fract. 2022, 6, 425 10 of 13
Figure 5. Plots of |u| of the first-order lump solution (38) of the three-dimensional KP equation with
parameters α = 1 + i, β = 1,γ = 1 + i, q1 = q∗2 = 1 + i, p1 = p∗2 = (
√
2− 1) + (√2− 1) i; (a) a
three-dimensional plot at t = 0, z = 0; (b) the temporal evolution in the (x, y)-plane of the contour
of a first-order lump at z = 0, where the blue line is x+ (
√
2− 1) y+ z = 0; (c) the evolution in the
(x, y)-plane of the contour of a first-order lump at t = 0, where the red line is x = 0.
Taking in Equation (32),
N = 2, p1 = p2 =
1
3
, q1 = q∗2 = −i, α = 1, β = −1, γ = 2, (39)
we obtain the rational solution
u = 12(ln f )xx, f =
(
x− 5
9
t+
1
3
y
)2
+ z2 − 1
9
t2 + 6− 2
3
i t z. (40)
This rational solution is nonsingular when z 6= 0. As described in Figure 6, this
solution describes the interaction of two rational parallel solitons in the (x, y)-plane. When
y = −3x, the two rational solitons fuse together and reach the minimum value 24z2+6 at
t = 0. In general, the interaction of two solitons produces a solution with larger amplitude.
Surprisingly, in this case, the amplitude of a line soliton fused by two rational solitons is
significantly smaller than the solitons eventually after separation. As shown in Figure 6e–h,
although the two solitons are completely separated as t tends to infinity, as time evolves,
they interact, producing a downward peak, which yields a decrease of the amplitude.
Finally, the two rational solitons fuse into a W-type rational soliton at t = 0.
Figure 6. Plots of |u| of the temporal evolution of the rational solution (40) for the three-dimensional
KP equation with parameters α = 1, β = −1,γ = 2, p1 = p2 = 13 , q1 = q∗2 = −i, z = 1.
Fractal Fract. 2022, 6, 425 11 of 13
Following the extensive studies of the line rogue waves for (2 + 1)-dimensional
integrable models [32–35], it is interesting to find such solutions of the three-dimensional
KP equation. The line rogue wave has infinite length and suddenly appears and disappears
in the background plane. Compared with the normal rogue waves, which are localized in
time and space, line rogue waves are localized in time, but are non-local in space, and also
only appear in high-dimensional systems. Taking in Equation (32),
N = 2, p1 = p2 =
1
3
, q1 = q∗2 = 1+ i, α = 1, β = −1, γ = 2, (41)
we obtain
u = 12(ln f )xx, f =
(
x− 1
18
t+
1
3
y+ z
)2
+ (z+ t)2 − 2
9
t2 + 6− 1
21
i t(17t+ 18x+ 6y+ 36z). (42)
This rational solution is nonsingular when z = 0. This rational solution is analogous
to a typical line rogue wave solution. Its visible line profile is preserved for the short period
of time during the evolution process; see Figure 7.
Figure 7. Plots of |u| of the temporal evolution of the line rogue wave (42) for the three-dimensional
KP equation with parameters α = 1, β = −1,γ = 2, p1 = p2 = 13 , q1 = q∗2 = 1+ i, z = 0.
5. Discussion and Conclusions
The main achievement of this work is the explicit construction of nonsingular multi-
solitons, and high-order breather and high-order rational solutions of the three-dimensional
KP Equation (1) using a Hirota bilinear method and the perturbation expansion technique.
Moreover, different patterns of first-order breathers for the three-dimensional KP equa-
tion are discussed; see Figure 2. Furthermore, mixed solutions composed of breather and
solitons are also given; see Figure 4. Additionally, high-order rational solutions of the
three-dimensional KP equation are constructed by taking the long-wave limit of N-soliton
solution. The first-order rational solutions are nonsingular, providing that the constraints
of (35) and (37) are valid. The first-order rational solution generates a lump wave in the
(x, y)-plane; its local characteristics and evolutional process are demonstrated in Figure 5.
In addition to the lump wave, the first-order rational solution (40) describes the inter-
action of two rational parallel solitons under the constraint (39). As shown in Figure 6,
the interaction of two rational solitons surprisingly makes the amplitude of this solution
significantly smaller in comparison to the amplitude of the separated line solitons. In sum-
mary, under the constraint (41), the first-order rational solution (40) in the (x, y)-plane is a
line rogue wave; see Figure 7. The dynamics of the exact solutions possessed by this new
three-dimensional model are rich and highly non-trivial. Hopefully, such solutions may be
important in realistic applications.
Author Contributions: Conceptualization, A.S.F. and J.H.; methodology, A.S.F.; software, A.S.F. and
Y.C.; formal analysis, Y.C. and J.H.; investigation, A.S.F.; resources, A.S.F.; data curation, Y.C.; writ-
ing—original draft preparation, A.S.F., Y.C. and J.H.; writing—review and editing, J.H.; supervision,
J.H.; project administration, J.H.; funding acquisition, J.H. All authors have read and agreed to the
published version of the manuscript.
Fractal Fract. 2022, 6, 425 12 of 13
Funding: This research was funded by the National Natural Science Foundation of China (Grant
No.12071304 and No.11871446), and the Guangdong Basic and Applied Basic Research Foundation
(Grant 2022A1515012554).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: The work was supported by the National Natural Science Foundation of China
(Grant No.12071304 and No.11871446), and the Guangdong Basic and Applied Basic Research
Foundation (Grant 2022A1515012554).
Conflicts of Interest: The authors declare no conflict of interest.
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