American Journal of Computational Mathematics, 2014, 4, 254-270
Published Online June 2014 in SciRes. http://www.scirp.org/journal/ajcm
http://dx.doi.org/10.4236/ajcm.2014.43022
How to cite this paper: Miroshnikov, V.A. (2014) Interaction of Two Pulsatory Waves of the Korteweg-de Vries Equation in
a Zigzag Hyperbolic Structure. American Journal of Computational Mathematics, 4, 254-270.
http://dx.doi.org/10.4236/ajcm.2014.43022
Interaction of Two Pulsatory Waves of
the Korteweg-de Vries Equation in
a Zigzag Hyperbolic Structure
Victor A. Miroshnikov
Department of Mathematics, College of Mount Saint Vincent, New York, USA
Email: victor.miroshnikov@mou ntsaintvincent .edu
Received 5 May 2014; revised 2 June 2014; accepted 8 June 2014
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
A new exact solution for nonlinear interaction of two pulsatory waves of the Korteweg-de Vries
(KdV) equation is computed by decomposition in an invariant zigzag hyperbolic tangent (ZHT)
structure. A computational algorithm is developed by experimental programming with lists of eq-
uations and expressions. The structural solution is proved by theoretical programming with sym-
bolic general terms. Convergence, tolerance, and summation of the ZHT structural approximation
are discussed. When a reference level vanishes, the two-wave solution is reduced to the two-soli-
ton solution of the KdV equation.
Keywords
KdV Equation, Two Pulsatory Waves, ZHT Structure, Experimental, Theoretical Computation
1. Introduction
Since the discovery of the N-soliton solution of the Korteweg-de Vries (KdV) equation by experimental compu-
tation [1], various analytical methods, like the inverse scattering transform, the Bäcklund transform, generalized
functions, etc. [2]-[5], were developed and implemented by theoretical computation [6]. Further developments
of the theory of solitons comprise effects of vorticity [7]-[9] and viscosity [10], while computational methods
evolved from asymptotic methods [11]-[13], to series approximations [14]-[16], and structural decompositions
in invariant structures [17] [18]. In the current paper, an invariant zigzag hyperbolic-tangent (ZHT) structure is
developed to treat nonlinear interaction of two pulsatory waves of the KdV equation. A zigzagging pattern is a
ubiquitous phenomenon in fluid-dynamic, biological, and chemical systems [19]-[21].
The structure of this paper is as follows. In section 2, experimental computation with lists of equations and
V. A. Miroshnikov
255
expression is used to develop a computational algorithm in Maple™ and explore the convergence of the ZHT
structural approx imation. Theoretical computatio n with symbolic general ter ms is utilized in Section 3 to devel-
op differentiation and multiplication formulas for the ZHT structure, show its invariance, and consider summa-
tion, convergence and tolerance of the structural approximation. The two-wave solution is visualized and com-
pared wit h t he two-soliton solution in Section 4, which is followed by a summary of main results in Section 5.
2. Experiment on Interaction of Two Pulsatory Waves in the ZHT Structure
2.1. Formulation of an Experimental Problem in the ZHT Structure
Explore experimentally a structural solution of the canonical form of the KdV equa tion
3
3
60
tx
x
φ φφ
φ
∂ ∂∂
+ +=
∂∂
(1)
for nonlin ear interaction of two pulsatory waves. Construct the stru ctural solution in the ZHT structure of alge-
braic order
3
M=
, which is used to illustrate a computational algorithm,
()
() ()()
222324 235 3
0,0 2,01,13,12,24,23,35,3
2,,
xthpZZ taZtaZta TbZ taZ taTbZtaZta Tb
φµ
=++ +++++

+

(2)
where
h
is a reference level,
and
,
mm
Z
are structural coefficients,
( )
tanhta X
µ
=
and
()
tanh
Tb qtbY
ν
= =
are structural functions,
16
2Uh
µ
= −
and
16
2Vh
ν
= −
are wave numbers such
that
0
νµ
≤<
,
Xx Uta=−−
and
x tb
YV
=−−
are propagation variables,
U
and
V
are celerities,
a
and
b
are initial locations of first and second pulsatory waves,
q
ν
µ
=
and
2
1pq= −
are interaction and
complementary parameters such that
01q≤<
and
01p<≤
, respectively. W hen
( )
1 0,pq= =
0,0
1,Z=
and
2
0,2
,Zp= −
the two-wave solution is reduced to a one-wave solution
( )
( )
( )( )
22
2
, 21
11
6 sec6,
22
x thta
h UhhUhxUta
φµ
=+−

=+ −−−−


(3)
which becomes the one-soliton solution
()( )
2
1
, sec
22
x thUxUta
U
φ

= −−


[3] for
0h=
. Consider an in-
stance of (2) with
1 8,3,1,1 3,223,34
h UVqp
µ
======
, and
14
ν
=
. The ZHT structure then is
( )
( )
() ()
23
0,0 2,01,13,1
2423 53
2,24,23,3 5,3
1
,8
.
x tZZtaZtaZtaTb
ZtaZtaTbZ taZ taTb
φ
=+++
+++
+
+
(4)
2.2. Experimental Differentiation of the ZHT Structure
Primarily, symbolic computation of a spatial derivative of differential order
1N=
of two-wave solution (4)
yields the ZHT structure of algebraic order
4MN+=
()
( )
() ()
3 24
1,03,00,12,14,1
3 52
1,2 3,25,2
24 633 54
2,34,3 6,33,4 5,4
,
A taAtaAA taA taTb
x
A taAtaAtaTb
A taAtaAtaTbA taA taTb
φ
= ++++
++ +
+++ ++
(5)
with a general term
,n
nm m
ta TbA
, where structural coefficients
,nm
A
are
V. A. Miroshnikov
256
1,02,01,1 3,02,03,1
0,11,1 2,11,13,12,2
4,13,14,2 1,21,12,2
3,23,12,24,23,35,24,25,3
2,32,2 3,3
31 31
, ,
2 122 12
3 391
, ,
4 446
91 33
, ,
4642
33 11
3, 3,
42 44
39
,
24
AZZA ZZ
AZAZZZ
AZZAZZ
AZZZZA ZZ
AZ ZA
=+ =−+
= =−++
=−+ =−+
=−−+ +=−+
=−+
4,34,2 3,35,3
6,35,3 3,43,3 5,45,3
3915 ,
244
15 99
, , .
4 44
ZZZ
AZA ZA Z
=−−+
=− =−=−
(6)
Secondly, computation of a temporal derivative of differential order
1N=
also returns the ZHT structure of
algebraic order
5MN+=
( )
()
()
()
3 24
1,03,00,1 2,14,1
3 52
1,2 3,25,2
2 4 63
2,3 4,3 6,3
3 54
3,4 5,4
T taTtaTT taTtaTb
t
T taTtaTtaTb
T taTtaTtaTb
T taTtaTb
φ
= ++++
+++
+ ++
++
(7)
with a general term
,m
nm n
Tta Tb
and the following structural coeff icients
,nm
T
:
1,02,01,1 3,02,03,1
0,11,1 2,11,13,12,2
4,13,14,2 1,21,12,2
3,23,1 2,24,2 3,35,24,2 5,3
2,32,23,3 4,
91 91
, ,
2 122 12
99 271
, ,
4 446
2713 9
, ,
4 642
39 11
9, 9,
42 44
327 ,
24
T ZZTZZ
TZTZZ Z
TZZTZ Z
TZZZZTZZ
T ZZT
=−− =−
=− =−−
=−=−
= +−−=−
= −
3 4,23,35,3
6,35,3 3,43,3 5,45,3
32745,
244
45 , 4.
99
,
44
ZZZ
TZT ZT Z
+−
==
=
=
(8)
Finally, a spatial derivativ e of order
3N=
again pr o duc e s the Z HT structure of order
6MN+=
( )
( )
( )
( )
( )
335246
1,03,05,00,1 2,14,16,1
3
357
1,2 3,25,27,2
246 83
0,3 2,34,36,38,3
3 574
1,4 3,45,47,4
2 4 65
2,5 4,5 6,5
6
2
3,
B taBtaBtaBBtaB taB taTb
x
B taBtaB taBtaTb
BB taBtaBtaBtaTb
B taBtaB taBtaTb
B taBtaBtaTb
B
φ
=++++++
++ + +
+++++
++ + +
+ ++
+
( )
3 56
5,6
taB taTb+
(9)
with a general term
,m
nm n
Bta Tb
, where structural coefficien ts
,nm
B
become
V. A. Miroshnikov
257
1,02,01,13,1 2,2
3,02,01,13,12,2 4,23,3
5,02,03,1 4,25,3
0,11,13,1 2,2
2,11,13,1 2,24
277 27 1
424 32 16
135961111,
83224 168288
812711,
8168288
9819 ,
8 3216
117 837727
32
,
23 38
BZZZZ
BZZZ ZZZ
BZZZZ
B ZZZ
BZZZ Z
=− −++
= +−−+
=− + −+
= −
−+
+
= −
+
+
,2 3,3
4,11,13,12,24,23,35,3
6,13,14,25,3 1,21,13,12,24,23,3
3,21,1 3,1 2,24,2
9,
32
8178327109915 ,
3216161232 32
405451593 2438181
, 9,
168323232832
81741153549253
32 328832
Z
BZ ZZZZZ
BZZZBZZ ZZZ
BZZZ Z
+
=−++−− +
=−+−=−−++
=−+ +−−
3,3 5,3
5,23,1 2,24,23,35,3
7,24,25,3 0,31,12,23,3
2,31,1 3,1 2,24,23,35,3
4,3
135 ,
16
243 8187381685
16881632
40540581 8181
,,
83232 1632
812431772431053405 ,
32 32883216
243
32
,
ZZ
BZZZZZ
BZZBZ ZZ
BZZZ ZZZ
BZ
+
=− −++−
=− +=−+
=−+ +−−+
= −
3,12,24,23,35,3
6,34,23,35,3 8,35,3
1,41,12,23,3 3,43,12,24,23,35,3
5,4 4
2436638912295
16816 16
405405 66152835
,,
8 16 3232
81 2437298124324311791215
,
32 1632321681616
243
8
,
,
ZZZ Z
BZ ZZBZ
BZZZB ZZZZZ
BZ
−++−
=−−+ =−
=−+−=−−++−
= −
,23,35,3 7,45,3
2,52,2 3,3
4,54,23,35,3 6,55,3
3,63,3 5,65,3
729 31233645
,
16 1632
81729 ,
8 16
8172912151215 ,
8 161616
405 405
, .
1
,
16 6
,
Z ZBZ
BZZ
BZZZBZ
B ZB Z
−+ =−
=−+
=−− +=−
=−=−
(10)
2.3. Experimental Multiplication of the ZHT Str uctur es
Symbolic computation of a product of two-wave solution (4) and first spatial derivative (5) once more produces
the ZHT structure of algebraic order
27MN+=
( )
() ()
() ()
35246
1,03,05,00,12,1 4,1 6,1
357224 6 83
1,23,2 5,27,22,34,3 6,3 8,3
3579 446810 5
3,45,4 7,4 9,44,56,58,510,5
P taPtaPtaPP taP taP taTb
x
PtaP taP taPtaTbP taP taP taPtaTb
P taP taPtaPtaTbP taP taPtaPtaTb
φ
φ
=++++++
++++++++
+ +++++++
() ()
57911668107
5,67,69,6 11,66,78,7 10,7
P taPtaP taPtaTbPtaP taPtaTb+ ++++++
(11)
V. A. Miroshnikov
258
with a general term
,m
nm n
Pta Tb
and the following structural coeff icients
,
nm
P
:
1,00,01,03,0 0,03,02,01,0
5,02,0 3,00,10,00,1
2,10,02,12,00,11,1 1,0
4,10,04,12,02,13,1 1,01,13,0
6,13,1 3,
11
, ,
88
1
, ,
8
1,
8
1,
8
PZAPZAZA
PZA PZA
PZAZAZA
P ZAZAZAZA
P ZA
 
=+ =++
 
 

== +



=++ +



=++++


=02,04,11,20,01,21,1 0,1
3,20,03,22,0 1,23,10,11,1 2,12,2 1,0
5,20,05,22,03,23,1 2,11,1 4,14,2 1,02,23,0
7,24,23,03,1 4,12,0
1
, ,
8
1,
8
1,
8
Z APZAZA
P ZAZAZAZAZA
PZAZAZA ZAZA ZA
PZAZA ZA

+=+



=++ ++



=++++ +
 +

= ++
+
+
5,2
2,30,02,31,1 1,22,20,1
4,30,04,32,0 2,33,11,21,13,24,2 0,12,2 2,13,3 1,0
6,30,06,32,04,33,13,21,1 5,24,22,12,24,15,3 1,0
,
,
1,
8
1
1
8
8
PZAZAZ A
P ZAZAZAZAZAZAZA
PZAZAZAZAZA ZA ZA

=+++



=++ ++++



=+++ +++

+
+3,3 3,0
8,35,33,0 4,24,13,15,2 2,06,3
3,40,03,41,12,32,21,2 3,30,1
5,40,05,42,0 3,43,1 2,31,1 4,34,21,22,2 3,25,3 0,13,3 2,1
7,45,3 2,
,
1,
8
1,
8
ZA
PZAZA ZAZA
P ZAZAZAZA
P ZAZAZAZAZAZAZAZA
P ZA
+
=+++

=+++



=++
+
+++ ++

+
=14,2 3,23,34,13,14,32,2 5,22,0 5,41,16,3
9,45,3 4,14,25,23,1 6,3
4,53,3 1,22,22,31,13,4
6,55,3 1,24,22,33,3 3,23,1 3,42,24,31,1 5,4
8,55,33,24,2
,
,
,
,
ZA ZAZAZA ZAZA
PZA ZAZA
PZAZ AZA
PZAZAZAZAZAZA
P ZA Z
++ ++
=++
= +
=+ ++
= +
++
+
++
4,33,3 5,23,1 5,42,26,3
10,55,3 5,24,26,35,63,3 2,32,23,4
7,65,3 2,34,2 3,43,34,32,2 5,4
9,65,3 4,34,25,43,3 6,311,65,3 6,36,73,3 3,4
8,75,3 3,43,
,
, ,
,
,,,
AZAZAZA
PZAZA PZAZA
PZAZAZAZA
PZAZAZA PZA PZA
PZA Z
++
=+=+
=+++
=++= =
= +
+
3 5,410,75,3 5,4
, .A PZA=
(12)
2.4. Experimental Solution of an Algebraic KdV Pr o b lem
Consider an experimental solution for algebraic order
4,M=
which is a smallest order required to avoid de-
generation of the subsequent computational algorithm. Substitution of temporal derivative (7), product (11) of
the two-wave solution and the first spatial derivative, third spatial derivative (9), and collection of structural
coefficients reduce differential KdV Equation (1) to an experimental algebraic KdV equation
V. A. Miroshnikov
259
()()()()
(
() ()
)
() ()
(
() ()
)
35 2
1,01,0 1,03,03,03,05,05,00,10,1 0,12,12,1 2,1
46 3
4,14,14,16,16,11,21,2 1,23,23,23,2
5 72
5,25,2 5,27,27,20,3
6666 6
6 666
66
BP TtaBPTtaBPtaBP TBP Tta
BPT taBPtaTbBPT taBPTta
BPTtaBPta TbBB
++ ++++++++ + +
+++ +++
+
+
++ +++
+++ +++
()()
(
() ()
)
()()
(
() ()
)
() ()
(
24
2,32,3 2,34,34,3 4,3
6 8335
6,36,36,38,38,31,43,43,43,45,45,45,4
794246
7,47,4 7,49,49,42,54,54,5 4,56,56,5 6,5
66
66 66
66 66
PT taBPT ta
BPTtaBPtaTbBtaBPTtaBPTta
BPTta BPtaTb Bta BPTta BPTta
++ +++
+++ ++++++++
++++++++ +++ +
+
+
( )
)
()()
( )
( )
( )
(
)
81053579116
8,5 8,510,53,65,65,67,67,69,611,6
46 8101277 911
4,76,7 6,78,710,712,77,89,811,8
13 88
13,88,9 10,
6 66666
666 6666
6 66
BPtataPTbBtaBPtaBPtaP taPtaTb
BtaBPtaPtaPtaPtaTbPtaPtaPta
PtaTbP taP
+++++++ +
++++++
++
+++
+
+
( )
10129
9 12,9
6 0.taP a Tbt=+
(13)
To vanish structural coefficients of
nm
ta Tb
, construct binomial systems by vanishing structural coefficients
of
km m
ta Tb
of odd binomial orders
1, 3,, 21kM=−
with
0,1,,.mk=
The binomial systems have two
equations, four equa tions, and five equations for
1, 3,kk= =
and
5k
, respectively.
The degenerate d bi nomial sy stems for
1k=
and
3k=
are, respectively,
1,01,0 1,00,10,10,1
60, 60,T PBTPB
+ +=+ +=
(14)
3,03,0 3,02,12,1 2,11,21,2 1,20,3
60, 60, 60, 0.TPBT PBT PBB
++=+ +=+ +==
(15)
In agreement with one-wave solution (3), structural coefficients
,nm
Z
of the two-wave solution are initia-
lized by
2
0,0 2,0
1, .
Z Zp== −
(16)
For the experimental solution,
0,0 2,0
1, 89.ZZ
== −
(17)
Solving Equations (14) and (15) with respect to
1,1 3,1 2,24,2
,,,ZZZ Z
, and
3,3
Z
yields
1,1 3,12,24,23,3
2,16 9,2,8 3,2.
ZZZ ZZ==− ==−=
(18)
For
5, 7,, 23,kM= −
the binomial system has five equations
5,0 5,04,14,14,13,23,2 3,22,32,3 2,31,4
60,60, 60, 60,0.
PBTPBT PBT PBB=++=+ +=+ +==+
(19)
Solving first and fifth equations of (19) with respect to
() ()
52,1 2kk
Z++
and
() ( )
32, 32kk
Z++
, respectively, gives
5,3 4,4
32 9,2.ZZ=−=
(20)
Substitution of (20) in second, th ird , and fourth equations of system (19) reduc es th em to identities.
For
21kM= −
, the binomial system also has five equati ons
6,16,15,25,2 5,24,34,3 4,33,43,4 3,42,5
60,6 0,6 0,6 0,0.PBT PBT PBT PBB=+ +=+ +=++==+
(21)
Solving first equation of system (21) respect to
() ()
52,1 2kk
Z++
returns
6,4
40 9.Z= −
(22)
Substitution of computed structural coefficients (17), (18), (20), and (22) in (4) yields an experimental two-
wave solution in the ZHT structure of algebraic order
4M=
( )
23242353464
9 81683240
,2 222
89 9399
xttata taTbta taTbtataTbtataTb
φ
  
=−+− +−+−+−
  
  
(23)
Substitution of the computed structural coefficients in the left-hand-side part of (13) returns an experimental
remainder of the ZHT structural approximation
V. A. Miroshnikov
260
( )
5 724683
3 5794
246810 5
3 57
5525 1055910
,722787281
2251765 13405 14230
8 83681
4055805430719081 17536
8 82927
1215 1611 319
82
e
rx ttataTbtatataTb
tatatataTb
tatatatataTb
tata ta
 
=−++− +−
 
 

+− +−+



+− +−+−

−+
+

911 6
4681012 7
791113 8
12108 9
3416 61760
3 81
405 60160
189 190904
4 81
3920 1600
54 3293
3200 320 72
9.
tata Tb
tatatatata Tb
tatatata Tb
tatataTb
+−

+−+− +−



+− ++−



+−+ −


(2 4)
Rate of convergence of the ZHT structural approximation is examined in Table 1, where a tolerance
( )
() ()
, ,
max ,
e
tx
r xt
ε
∈−∞∞ −∞∞
=
(25)
and a CPU time
τ
are given versus algebraic order
M
for various reference levels
h
. Table 1 was co m-
puted on a workstatio n in Maple 17.02 by using a six-core A MD-6300 processor with frequency 3.50 GH z and
RAM 12.0 GB. The CPU time depends mainly on order of approximation
M
. Tolerance significantly depends
upon
M
and
h
through interaction para mete r
q
. For propagatio n ce lerities
3
U=
and
1,1 3,33Vq= =
,
105 150.333,0.577,0.683=
for
1 8,0,180.125,0,0.125,h= −=−
respectively. Surface
plots of
()
,
c
R xt
show a uniform convergence of the two-wave solution in the ZHT structure.
The experimental solutions of Section 2 were computed by experimental programming with lists of equations
and expressions in the virtual environment of a global variable
Eqe
with 4 procedur es of 185 code lines in to-
tal.
3. Theory of Interaction of Two Pulsatory Waves in the ZHT Structure
3.1. Formulation of the Theoretical Problem in the ZHT Structure
Compute theoretically a stru ctural solu tion of (1) for nonlinear interac tion of two pulsatory w aves. Constru ct the
structural solution in the ZHT structure of algebraic order
M
and diffe rential or de r
N
( )
2
2,
0
1
0
,,
MN mmN k
m Nkm
m
N
K
sxthSTbCta
+−+
−+
+
==
= +∑∑
(26)
where
S
is a scale,
2,
,
m Nkm
CM
−+
→∞
is a structural coefficient,
M
and
N
are symbolic limits of sum-
Table 1. Convergence of the experimental solutions in the ZHT structure.
M
10 20 40 80 160
0.125h=
ε
4
1.8 10
×
9
6.3 10
×
18
3.7 10
×
37
6.2 10
×
75
8.3 10
×
( )
s
τ
0.156 0.407 1.782 11.672 99.343
0.000h=
ε
0.125 0.00112
8
4.0 10
×
17
2.4 10
×
36
3.8 10
×
( )
s
τ
0.312 0.609 2.250 13.312 111.718
0.125h= −
ε
1.03 0.056
5
5.7 10
×
11
2.8 10
×
24
3.4 10
×
( )
s
τ
0.250 0.578 2.328 14.282 116.812
V. A. Miroshnikov
261
mation. Other variables and parameters are the same as in (2), but instead of experimental instances of section 2
they receive symbolic values to compute theoretically a general term of the structural solution.
In agreement with (26) and (2), a two-wave solution is constructed with algebra ic order
M
and differential
order
0N=
using a generalized Einstein notation for summation, which is extended for exponents, as follows:
( )
( )
22 2
, 2,
,2 ,
mm m
mmmm
x thptaZtaZTb
φµ
+
+
=++
(27)
where
0,1, , .mM=
When
()
0 ,1 0,Tb qtbpq == ==
and initialization condition (16) is invoked, the
two- wave solution is reduced to one-wave s o lution (3).
3.2. Theoretical Differentiation of the Invariant ZHT Structure
Primarily, computation a fir s t spa tial derivative of a binomial term
nm
ta Tb
yields
( )
1 111
, 11,1,, 1
,
nm nmnn mnm
nmn mn mnm
ta Tbdata TbdataataTbdata Tb
xd
µ
− −++
− −++
+

=++

(28)
( )( )
( )( )
2
, 1, 11,1,
1,1,, 1, 1
,,
,
,,
,,,
nmnmn mnm
n mn mnmnm
da aqda an
d
dnm mdnm
d nmaandamna dm
−− −−
++++
= =
=
=
=−=−
=
=
(29)
where structural coefficients (29) of binomial derivative (28), called differential binomial coefficients, are ma-
trix functions
( )
,,
,
nm nm
da ad ij=
. Here, indices
n
and
m
, which are equal to powers of
ta
and
Tb
in the
binomial derivative, d efine names of the matrix fun ctions. Indices
i
and
j
which are equal to powe rs of
ta
and
Tb
in the binomial term
, determine definition s of the matrix functions.
Similarly, computation of a firs t temporal derivative of the binomial term
nm
ta Tb
gives
( )
1 111
, 11,1,, 1
,
nm nmnn mnm
nmn mn mnm
ta TbdttaTbdttattaTbdtta Tb
td
µ
− −++
− −++
+

=++

(30)
( )
( )
( )
( )
( )
( )
( )
( )
2 222
, 1, 11,1,
2 22
1,1,, 1, 1
22,,
,,
3,223 ,
223 ,223.
nmnmn mn m
n mn mnmnm
dtdtqqh mdtth nnm d
dtth ndt
nm
d nmdtq hmnm
µµ
µµ
−− −−
++ ++
=
= =
==−+=−+
=+=+
(31)
Thus, spatial and temporal derivatives (28) and (30) have the same structure but vary in differential binomial
coefficients (29) and (31), respectively.
Increase in order of differentiation produces a two-dimensional (2d) differentiation cascade, which is shown
in Figure 1. The one-dimensional (1d) diff erentiation cascade o f invariant hyp erbolic-secant structures is asym-
metrical as the cascade spreads only towards higher powers [18]. To the contrary of the 1d differentiation cas-
cade, the 2d differentiation cascade spreads in symmetrical square waves, which resemble circular waves on the
water surface generated by a point source. Similar to the 1d differentiatio n of the invariant trigonometric, hyper-
bolic, and elliptic stru ctures [18], the 2d differentiation of an even order preserves structure of binomial deriva-
tives and the 2d differentiation of an odd order converts structure of binomial derivatives to complementary
ones.
Finally, computation a third spatial deriv ative of the binomial term
nm
ta Tb
returns a binomial derivative of
the following structure:
() (
) ()
3331122
,31,21,22,1, 1
3
2131 13
2, 13,1,1,3,
2
2, 1,12,
nm nmnn mnn
nmn mnmnmnm
nmnnnnm
n mnmnmnmnm
nn
n mnmn
ta TbdbtaTbdbtadbta Tbdbtadbta
x
dbtaTbdb tadbtadbtadbta Tb
dbtadab tdb
µ
−− +−−
−−−+−− −−
+−−−++
+−−− + +
−++ +
= ++++
++ +++
+++
( )
( )
21
1
1 12
1, 21,2
3
,3
nm
m
n nm
nm nm
nm
nm
ta Tb
dbtadbta Tb
dbta Tb
++
+
− ++
−+ ++
+
+
+ +
+
(32)
with differential binomial coefficients
V. A. Miroshnikov
262
(a) (b)
(c) (d)
Figure 1. Differentiation cascade of the binomial structure: (a) N = 1; (b) N = 3; (c) N = 2; (d) N = 4.
( )()()( )()
( )()( )()
( )()()
64
, 3,31, 21, 2
42
1,21,22, 12, 1
2 2222
,1 ,12,12,1
12, 31,
31, 31,
2 16
,,
,,
,3, 3,
nmnmn mn m
nmnmn mn m
nmnmn mn m
dbdbqm mmdbdbqmmn
dbbq mmndbdbq mnn
db
nm nm
d nmnm
d nmmnmbq mqqmndbdbq
−−−− −−
+−+−− −−−
−−+− +−
==−−=−
=−−=
=
= =

= −=−+ == +
( )
( )()()( )()
( )()( )()()
( )()( )
22
3,3,1, 1,
22
1, 1,3,3,
2
2,12,1,1,1
1,
12, 2631,
2631 ,
,
12,
31,
,
2
,,
,,
nm nmnm nm
nm nmnm nm
n mn mnmnm
dnmd nm
d nm
mn n
dbbn nndbbqmn nn
dbbqmn nndbbn nn
dbbmnn
d nm
db dbdnmnmmq
−− −−
++ ++
−+ −+++
+

= −−=−++−


=+++=
= =
== −+ +

−==−= =
( )
( )()( )()
( )()()()()
22
2, 12, 11, 21,2
1, 21, 2, 3, 3
,,
316 ,
31,
,
3 1,
,31, 12.
n mn mnmnm
n mn mnmnm
d nmnm
d
qmm n
dbbmn ndbdbmmn
dbb nm m mndbdbmmnmm
+ ++ +−+−+
++ ++++

++

=−+ =+
=− +=−+
+
=
= +
=
=
(33)
Construct now a sequence of terms of the zigzag structure
( )
2
, 2,
mkmkmk
mkmmkm
Z taZtaTb
+++ +
+ ++
+
with
,1, ,kNNN=− −+
, derivatives of which make a contribution to the general term of a zigzag derivative
through binomial derivatives (28), (30), and (32). Substitute then the binomial derivatives of
2
,
mk mkmkmk
ta TbtaTb
+ +++ +
and collect like terms of a structural coefficient of the general term proportional to
m
Tb
. In agreement with (5), a first spatial derivative of the two-wave solution is the invariant ZHT structure of
algebraic order
MN+
and diffe rential or de r
1N=
( )
32 113
1, 1,3,
2,
m mmm
mm mm mm
p AtaAtaAtaTb
x
φµ
−++
−++
+
= +
(34)
V. A. Miroshnikov
263
where
0,1, ,m MN= +
and structural coefficients
,nm
A
are following:
()
1,, 11, 11,,
1,, 11, 11,,1,2,, 11,1
3,1,2,, 13, 1
,, ,,
,
,
,.
,
mmnmmmnmmm
mmnmmmnmmmnmmm nmmm
mmnmmmnm mm
nmi jnmi j
ADa ZDa Z
ADaZDa ZDa ZDa Z
ADa ZDa Z
d ijDa ZaZ
−+ −−−
+++−+−+− ++
+++− ++
= +
=++ +
=+
=
(3 5)
Equations (34)-(35) are complemented by truncation conditions of the two-wave solution
,
0 for 0,,
nm
Zm mM=<>
(36)
which are set by (27). Conditions (36) result in truncation conditions for the first spatial derivative:
,3,1, 1,
0 for 0,1,3; 0 for ; 0,0 for 1.
nmmmmmmm
AnnmnmAmMAAmM
+ +−
=<<−>+= >== >+
(37)
Similarly, a first temporal derivative of the two wave solution again yields the invariant ZHT structure of al-
gebraic order
MN+
and diffe rential or de r
1N=
( )
32 113
1, 1,3,
2 ,
m mmm
mm mm mm
p TtaTtaTtaTb
t
φµ
−++
−++
= ++
(38)
where structural coefficients
,
nm
T
are
( )
1,, 11,11,,1,, 11,11,,1,2,
, 11, 13,1,2,, 13, 1,,,,
,
,,,,
mmnmmmnm mmmmnmmmnm mmnm mm
nmmmmmnmmmnmmmnmi jnmi j
TDt ZDt ZTDt ZDt ZDt Z
DtZTDtZDtZDt ZtZd ij
−+−− −+++− +−+
−++++ +−++
=+=+
+=
+
+=
(39)
and truncation conditions are given by (36).Truncation conditions for the first temporal derivative become
,3,1, 1,
0 for 0,1,3; 0 for ; 0,0 for 1.
nmmmm mm m
TnnmnmTmMTTmM
+ +−
=<<−>+= >== >+
(40)
Finally, computation of a third spatial derivative of the two-wave solution produces the invariant ZHT struc-
ture of algebraic order
MN+
and differential order
3N=
( )
352 31135
3,1,1,3,5,
3
,2
mm mmmm
mmmmmm mmmm
p BtaBtaBtaBtaBtaTb
x
φµ
−−++ +
− −++ +
=+++ +
(4 1)
where structural coefficients
,nm
B
are
3,,33, 31,22,22, 11,13,,
1,,31, 31,22, 21, 2,2,11,1
2,11, 11,,3,2,
,
mmnmmmnmmmnm mmnmmm
mmnmmmnmm mnmmmnmmm
nmm mnmmmnmm
BDbZDb ZDb ZDbZ
BDbZDbZDbZDb Z
DbZDb ZDbZ
−+− −−+− −− +−−−
−+−−++− −−+−+−−
− ++−−−+
=+++
= +
+
++
++
2, 11, 1
1,1,2,22,11, 1,11, 11,,
1,2,, 11, 12, 13, 11,22,2
3,2,11, 13
,
,
mn mmm
m mn mmmnmmmnmm mn mmm
nmm mnmmmn mmmnmmm
mmnmmmn
Db Z
BDbZDbZDb ZDb Z
Db ZDb ZDbZDbZ
BDb ZDb
− −++
+++−++−−+ +−+
−+−++−−++−− ++
++++−+
++
+
= +
++
=
++
+
,,1,2,2, 11,1
,13,11, 22,21,24, 2,33, 3
5,3,2,2, 13, 11,24,2, 35,3
,
,
,
mmm nmmm nmmm
nmmmnmmmnmm mnmmm
mmnmmmnm mmnmmmnmmm
nm i
ZDb ZDbZ
DbZDbZ DbZDbZ
BDb ZDbZDbZDb Z
Db Z
+++ −++
−+++− ++−− ++−++
+++ +−+++−++−++
+
++ +
= +
+
++
+
( )
,, ,
,,
jnmi j
i Zjdb=
(42)
and truncation conditions are set by (36). Truncation conditions for the third spatial derivative are
,5, 3,
1,1,3,
0 for 0,3,5; 0 for ;0 for 1;
0 for 2; 0,0 for 3.
nmm mmm
mmmm mm
Bnnm nmBmMBmM
BmM BBmM
++
+ −−
=<<−>+=>=> +
= >+==>+
(4 3)
Equations (34), (38), and (41) show that the ZHT structure is invariant with respect to differentiation of orders
1N=
and
3N=
, which only modify algebraic orders and structural coefficients. The zigzag structure of two-
wave solution (27) and its derivatives (34) and (38) together with product of (27) and (34) are shown in Figure
2 in a virtual space of computational indices
n
and
m
of structural coefficients
,nm
A
,
,nm
T
,
,nm
B
V. A. Miroshnikov
264
(a)
(b)
(c)
(d)
Figure 2. The invariant ZHT structure in the virtual space (n, m): (a)—(27),
(b)—(34), (c)—(38), (d)—(53).
and
,
nm
P
, which coincides with a virtual space of computational powers
n
and
m
of
nm
ta Tb
. Differentia-
tion increases the width of the invariant ZHT structure and the effect of multip lication is similar to that of diffe-
rentiation.
3.3. Theoretical Multiplication of the Invariant ZHT Structures
Continuation of spatial derivatives (34) and (38) in the invariant ZHT structure to any differential order
N
in
the generalized Einstein notatio n gives
22 2
2,
,2
NmNkm
m Nkm
N
N
pta Tb
xR
φµ
+ −+
−+
=
(44)
V. A. Miroshnikov
265
where
0,1, ,1kN=+
and
0,1, ,.m MN= +
The spatial derivative of order
1
N
then becomes
12 2
1
1
21
1
,
2 .
NmNkm
mN k m
N
N
pta Tb
xQ
µ
φ
+ −+
+
−+ +
=
(45)
Differentiation of (45) with respect to
x
and reduction of all terms to a general term by substitutions
1mm=+
and
1kk=−
in a term proportional to
1m
Tb
,
1kk=−
in a term proportional to
22
,
mN km
ta Tb
−+ +
1mm=−
in a term proportional to
1m
Tb +
yields
()( )
()()
22 21,2 ,1
2
2 1,1
22,
211
2
2
11 ,
NmN k mmN km
mN km
mN k mmN km
N
N
mN kQmQ
mN kQm
x
tTq aQ
p
b
µ
φ
+
−+ −−+−
−+
−+ +−++
− −+−−−
+ −+ −+ +
=
(46)
where struc tural coefficients of (44) a re c on ne c ted with st ructura l coeffici e nt s of (46) by a recurrent re lation
()()
()( )
2,2 1,2,1
22, 1
2
1,
21 1
211 .
mN kmmN k mmN km
mN k mmNkm
RmN kQmQ
m NkQmqQ
−+−+ −−+−
−+ +−++
− −+ −− −
+−+ +++
=
(47)
Thus, the invariance of the ZHT structure with respect to spatial differentiation of any order is proved by in-
duction.
Set up two spatial derivatives of the two-wave solution in the invariant ZHT structures of algebraic order
M
and differential orders
1
N
and
2
N
with structural coefficients
1
2,mNkm
Q
−+
and
2
2,
m Nkm
R
−+
as
12
1
112 2
12
2
222 2
22
2, 2,
2 ,2,
NmN kNmNk
mm
mN kmmN
NN
NN
km
pta TbptaTb
x
QR
x
φφ
µµ
+−++− +
−+− +
∂∂
= =
∂∂
(48-49)
where
1
0,1, ,1kN=+
and
1
0,1, ,m MN= +
for the first structure,
2
0,1, ,1kN= +
and
2
0,1, ,m MN= +
for the second structure. Product of (48) with a binomial substitution
,m nkl==
and (49)
with a binomial substitution
,mmnkkl=−=−
returns
12
1
12 12
12
2
42
2,2 2,
4
,4
N NmN Nkm
nNln mnNkl
NN
mn
NN
Qpta Tb
xx R
φφ µ
+ +−− +
− +−−+ −−
∂∂ =
∂∂
(50)
where
0,1, ,lk=
,
12
0,1, 2k NN= ++
,
0,1, ,nm=
and
12
0,1,,2
mMNN= ++
.
Thus, the ZHT struc ture is also invariant with respect to multiplication since a general term of the product is
the ZHT structure of algebraic or der
12
2MN N
++
and differentiation order
12
NN+
12 12
1
2
2
1
12
4
42
2
4,
N NmNNkm
m
NN
NN NN k
ptaTbP
xx
φµ
φ
+ +−− +
−−+
∂∂ =
∂∂
(51)
where the structural coefficients are
12 12
22,22,
.
mNNknNlnm n Nklm n
P QR
−− +−+−− +−−
=
(52)
Summation of a general term of the product of two-wave solution (27) and first spatial derivative (34) by
(51)-(52) with
12
0,1,0,1,,3NNk===
yields
( )
511 35
1, 1,3
4, 5,
,4
m mmmm
mm mm mmmm
p PtaPtaPtaPtaTb
x
φ
φµ
−++ +
−++ +
=++ +
(53)
where structural coefficients
12,mkm
P
−+
are obtained for
and
by constructing a list of sums of
general terms for
0,1, ,lk=
with truncation conditions
4, 6,
0, 0,
nn nn
ZZ
++
= =
which follows from (27),
1,1,,1,1,,1,2,
3,3,,1,2,5,5,,3, 2,
, ,
, .
mmmnmn nnmmmnmn nnmnmn nn
mmmnmnnnmnmn nnmmmnmn nnmnmn nn
PAZPAZAZ
PAZAZ PAZAZ
−−− −+−+−−− −+
+−+−−+−++−+−−+− +
==+
=+=+
(54)
Structure of the two-wave solution is reduced to that of a spatial der ivative b y subs titution
22
0,0 0,02.ZZh p
µ
= +
(55)
Truncation conditions for product of the two-wave solution and the first spatial derivative become
V. A. Miroshnikov
266
, 5,
3, 1, 1,3,
0 for 0,1,5; 0 for 2;
0,0,0,0 for 21.
nmm m
mmmmmmmm
Pnnm nmPm M
PP PPmM
+
++−−
=<<− >+=>
==== >+
(56)
3.4. Theoretical Solution of the Algebraic KdV Problem
Substitution of temporal derivative (38), product (53) of the two-wave solution and the first order spatial deriva-
tive, third order spatial d erivative ( 41), and collection of structural coefficients reduce differential KdV Equation
(1) to a theoretical algebr aic KdV equatio n
( )
() ()
( )
523 3254521
3,1,1, 1,
3254521 3254523
1,1, 1,3,3,3,
54 525
5, 5,
22 24 2
2 2422242
24 20,
mm
mmmmmmmm
mm
mmmm mmmmmm mm
mm
mm mm
pB tapTpPpBta
pTpPpB tapTpPpBta
pPpBta Tb
µµµµ
µ µµµµµ
µµ
−−
−− −−
++
+ +++++
+
++
++ +
++ ++++
++=
(57)
which is complemented by truncation conditions (40), (56), and (43).
For this equation to be satisfied exactly for all para meters and variables, all structural coefficients of
nm
ta Tb
should vanish. Therefore, five structural coefficients of (57), which are supposed to be vanished, are
222 222 2
3, 1,1,1, 1,1,1,
22 222 2
3,3, 3,5,5,
0, 120, 120,
12 0,12 0.
mmmmmmmmmmmm mm
mmmm mmmm mm
BTpPB TpPB
TpP BpP B
µµµ µµ
µµ µµ
−−− −++ +
+ ++++
=+ +=+ +=
+ +=+=
(58)
These equations constitute a polynomial system of equations with respect to leading structural coefficients
,nm
Z
of increasing orders. To combine equations with respect to
,
nm
Z
of same binomial orders
k
, construct
binomial systems as structural coe ff icie nts of
km m
ta Tb
, where
1, 3,21kM=−
for
0,1,mk=
.
The binomial syst ems of orde rs
1k=
and
3k=
have two and four equations, respectively,
22 222 2
1,01,0 1,00,10,1 0,1
120, 120,TpP BTpP B
µµ µµ
+ +=+ +=
(59)
22222 222 22
3,03,03,02,12,12,1 1,21,21,20,3
120, 120,120,0.TpPBTpP B TpPBB
µµµµµµµ
++=++=++==
(60)
Substituting structural coefficients
,, ,
,,
nmnmnm
TPB
through
,nm
Z
by (39)-(40), (54)-(56), (42)-(43), using
initialization (16), and solving (59)-(60) with respect to
1,1 3,1 2,24,2
,,,ZZZZ
and
3,3
Z
gives
22
1,1 3,12,24,23,3
2,,2,2 ,2.ZZpZZpZ==−==− =
(61)
For binomial orders
5,7, 23,kM= −
where
25kl= +
for
0,1, 4,lM= −
the binomial system has
five equations
5,
22 2222
5,4, 14, 14, 1
222
3, 23, 23,2
22 22
2, 32,32,31,4
120, 120,
12 0,
120, 0.
ll
llllllll
llll ll
llll llll
pPBTpPB
TpP B
TpP BB
µµµ µ
µµ
µµ µ
+
+++++ ++
++++ ++
++++ ++++
+= ++=
+ +=
++= =
(62)
Solving first and fifth equations of (62) with respect to
() ()
52,1 2kk
Z++
and
( )()
32, 32kk
Z++
respectively, yields
22 2
5,34,46,45,51, 1,
4 ,2,5 ,2,,2,
M MMM
ZpZZpZZMpZ
+−
=−==−==− =
(6 3)
where last two terms are obtained by induction. In agreement with the experimental solution of section 2.4,
second, third, and fourth equations of system (62) are satisf ied identically.
For binomial orders
7,9, 21kM= −
, where
25kl= +
for
3lM= −
, the binomial system also has five
equations
22222 2
2, 32, 31, 21, 21, 2
22 222 22
, 1, 1, 11,1,1,2, 1
120, 120,
12 0,12 0,0.
M MMMMMMMMM
MMMMMMM MM MMMMM
pPB TpPB
TpP BTpP BB
µµµµ
µµ µµµ
+ −+ −+−+−+−
− −−− −−−+
+=+ +=
+ +=+ +==
(64)
Solving first equation of (64) with respect to
() ()
52,1 2kk
Z++
returns
V. A. Miroshnikov
267
( )
22 2
6,4 7,52,
5,6,1 ,
MM
ZpZp ZMp
+
=− =−=−+
(6 5)
where a last term is also generated by induction, the proof of which will be completed in Section 4.
Finally, a general solution fo r th e ZHT structural coefficients may be written as
( )
2
, 2,
2,1 ,
mmmm
Z Zmp
+
= =−+
(66)
and general term (27) of the two-wave solution in the ZHT structure becomes
( )()
222 2
, 221
m mm
x thptamptaTb
φµ
+

=+−+

(67)
for
1, 2,mM=
.Theoretical formula for the ZHT structural sum of the two-wave solution is
( )()
222222
1
, 2121.
Mm mm
m
xthpptatamp taTb
φµ
+
=


=+− +−+



(68)
Substitution of initialized (16) and computed (66) stru ctural coefficients in the left -hand-side part of theoreti-
cal algebraic KdV Equation (57) yields a theoretical remaind er of the invariant ZHT structure
( )
() (
) ()
( )
3213 22
1,23,2,1 2,1
411135
4, 11,1,3,5,
311 35
3,1,1, 3,5,
,2
MMMMM
tMMMMMMMM
MMM MMMM
M MMMMMM MMM
mmmm m
mmmm mmmmmm
rxtpCtaCtaTbCtaC ta
Cta TbCtaCtaCtaCtaTb
CtaC taCtaCtaCtaT
µ
+ +−+
+−+−−+ −
+−− +++
+−−+ ++
−−+++
− −++ +
=+++
++++ +
++++ +
21
1
,
Mm
mM
b
+
= +
(69)
where a structural coefficient
22 2
,,, ,
12
nm nmnmnm
C TpPB
µµ
=++
(70)
and truncation conditions (40), (56), and (43) are invoked.
The theoretical solutions of Section 3 in the invariant ZHT structure were computed using theoretical pro-
gramming methods with symbolic general terms by the generalized Einstein notation in the virtual environment
of a global variable
Eqt
with 16 procedures of 923 code lines in total. The theore tical formulas for two-wave
solution (68), first spatial derivative (34)-(36), first temporal derivative (38)-(40), third spatial derivative
(41)-(43), product of the two-wave solution and the first spatial derivative (53)-(56), and structural remainder
(69)-(70) were justified by the correspondent experimental solutions for algebraic order
40M=
.
4. Discussion and Visualization
Through an expansion variable
z taTbqtatb= =
, where
1z<
as
11,11,
ta tb
−≤≤ −≤≤
and
01q≤<
,
summation of the ZHT structural approximation yields
( )
( )
( )
222222
2
21
,,
1
pptata Tb
xt htaTb
µ
φ
−−
= +
(71)
since a partial sum of the Taylor series expansion of (71) in
z
of order
M
returns the same expression as the
ZHT structural sum (68) with the same general term as general term (67) of the invariant ZHT struc tur e.
A functional form of (71) expressed through two regular hyperbolic functions
( )
tanh X
µ
and
( )
tanh vY
,
five parameters
,,,,hqp
µν
, and two propagation variables
Xx Uta=−−
and
x tb
Y V=−−
becomes
( )()()()
()( )
2 2222 22
2
tanhanh tanh
t
21t
a
,.
1 nhtanh
XXp Y
XY
pq
xt hq
µ
φµ µν
µν

−−

= +


(7 2)
Differentiation and substitution of Equation (72) into differential KdV Equation (1) completes the proof by
induction of (66)-(68) and returns another verification of (71) as Equation (1) is satisf ied identically.
Conversion of (72) through two singular hyperbolic functions
( )
coth X
µ
and
( )
csch X
µ
, two regular
hyperbolic functions
( )
tanh Y
ν
and
( )
sech Y
ν
, three parameters
and two variables
X
and
Y
gives
V. A. Miroshnikov
268
( )
( )
()()
()( )
222 2
2
22
csch sech
coth
2
,tanh .xt hXY
XY
µν ν
µ
µν
µν
µ
φν


= +

+
(73)
When
0h=
, the two-wave solution reduces to the conventional form [6] of the two-soliton solution in two
singular hyperbol i c f u nc t ions
()
coth 2UX
and
( )
csch 2UX
, two regul ar hyperbolic funct ions
( )
tanh 2VY
and
( )
sech 2VY
, two parameters
U
and
V
, and two propagation variables
X
and
Y
() ( )
()( )
( )( )
22
2
csch sec
22
co
h
,th2tan2 .
2 h
UVU
xt UX VVY
UUX VVY
φ


=

+
(74)
A remainder of the Taylor series approximation of two-wave solution (71)
( )()
2222
1
,22 1mm
smM
rxtpmptataTb
µ
= +

= −+

(75)
converges slower than the ZHT structural remainder (69) because of the infinite limit of summation and trunca-
tion condition s (40), (56), and (43). So, the method of decomposition in the invariant ZHT structure is more ro-
bust than the method of expansion in the Taylor series.
Interaction of two pulsatory waves is visualized by spatiotemporal plots in Figure 3 for positive and negative
reference levels
h
. Negative values of
h
considerably increase amplitudes and decrease dispersions of pul-
satory waves compared with those of solitons with
0h=
, because pulsatory waves propagate on a more shal-
low water in this case. The effect of positive values of
h
is opposite and results in decrease of amplitudes and
increase of dispersions. Similar to interaction of two solitons, interaction of two pulsatory waves is also con-
servative and preserves one-wave solutions before and after a nonlinear interaction at the moment of merging.
Animations of the two-wave solution show that the merging process may be considered as a flow of a faster flu-
id of the first pulsatory wave into the second pulsatory wave with a slower fluid.
5. Conclusions
The analytical methods of solving PDEs by undetermined coefficients and series expansions are generalized by
the computational method of solving nonlinear PDEs by decomposition in the invariant ZHT structure. The
computational algorithm is developed by experimental computing using lists of equations and expressions im-
plemented in four procedures of 185 code lines in total. Afterwards, the computational method is proved by
theoretical computing with symbolic gene ral terms implemented in 16 procedures with 923 code lines in to tal.
Figure 3. Spatiotemporal plots of the two-wave solution for U = 1. 8, V = 1, a = 8. 4, b = 18, h = 1/8 (left) and h = 1/8
(right).
V. A. Miroshnikov
269
The invariance of the ZHT structure with respect to differentiation and multiplication is shown by using 2d
differentiation cas cade of binomial stru ctures and mathematical indu ction. Contrary to the asymmetric differen-
tiation cascade in one dimension [18], the 2d differentiation cascade spreads in symmetric square wav es. Com-
pared with the 2d series expansion, the ZHT structure considerably saves computational resources and simplifies
results since it implies a low-order polynomial in one dimension and a series expansion in another dimension.
The invariance of the ZHT structure enables other computational applications in nonlinear PDEs with solutions
approaching a constantor vanishing at infinity.
The ZHT structural approximation and remainder are computed theoretically to any algebraic order. Summa-
tion of the two-wave solution in the invariant ZHT structures is implemented and presented both through regular
and singular hyperbolic functions. When a reference level vanishes, the two-wave solution is reduced to the two-
soliton solution. Negative reference levels considerably increase amplitudes and decrease dispersions of pulsa-
tory waves compared with those for solitons with a vanishing reference level. The effects of positive reference
levels are opposite, i.e. amplitudes of pulsatory waves are decreased and dispersions are increased.
Acknowledgements
The author thanks I. Tari for the stimulating discussion at the 2013 SIAM Annual Meeting. Support of the Col-
lege of Mount Saint Vincent and CAAM is gratefully acknowledged.
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