Trilinearization and Localized Coherent Structures and Periodic Solutions for the (2+1) dimensional K-dV and NNV equations

0.19 MB·English

by  C. Senthil Kumar

Trilinearization and Localized Coherent
Structures and Periodic Solutions for the
(2+1) dimensional K-dV and NNV equations
7 C. Senthil Kumara, R. Radhab, M. Lakshmanan a,∗
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0 aCentre for Nonlinear Dynamics, School of Physics, Bharathidasan University,
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Tiruchirapalli -620 024, India.
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a bDepartment of Physics, Government College for Women,
J
Kumbakonam - 612 001, India.
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I Abstract
S
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In this paper, using a novel approach involving the truncated Laurent expansion
i
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n in the Painlev´e analysis of the (2+1) dimensional K-dV equation, we have trilin-
[ earized the evolution equation and obtained rather general classes of solutions in

1 termsofarbitraryfunctions.Thehighlightofthismethodisthatitallows ustocon-
v struct generalized periodic structures corresponding to different manifolds in terms
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of Jacobian elliptic functions, and the exponentially decaying dromions turn out
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0 to be special cases of these solutions. We have also constructed multi-elliptic func-
1 tion solutions and multi-dromions and analysed their interactions. The analysis is
0
also extended to the case of generalized Nizhnik-Novikov-Veselov (NNV) equation,
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0 which is also trilinearized and general class of solutions obtained.
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i
l
n
:
v
i
X 1 Introduction
r
a
The recent interest in the investigation of integrable models in (2+1) di-
mensions can be attributed to the identification of dromions in the Davey-
Stewartson I equation [1,2]. These dromions which originate at the crosspoint
of the intersection of two nonparallel ghost solitons decay exponentially in
all directions and are driven by lower dimensional boundaries or velocity po-
tentials (arbitrary functions) in the system. It must be mentioned that the
presence of these so called boundaries enriches the structure of (2+1) dimen-
sional integrable models leading to the formation of localized solutions like
∗
Corresponding Author. Fax:[phone]
Email address: [email protected](M. Lakshmanan ).
Preprint submitted to Elsevier 8 February 2008
dromions, lumps, breathers, positons, etc.[2,3]. Thus, a judicious harnessing
of these lower dimensional arbitrary functions [4,5,6,7,8,9] of space and time
may give rise to new spatio-temporal patterns in higher dimensions besides
throwing more light on their integrability.
In this direction, we have recently identified a simple procedure to solve the
(2+1) dimensional systems, namely the Painlev´e truncation approach (PTA)
[7]. This approach converts the given evolution equation into a multilinear (in
general) equation in terms of the noncharacteristic manifold. This multilin-
ear equation can be solved in terms of lower dimensional arbitrary functions
of space and time. Using this approach, one can generate various generalized
periodic and localized solutions. This was earlier demonstrated for the case
of long wave-short wave resonance interaction equation in (2+1) dimensions
by converting it into a trilinear equation [7]. In this paper, we wish to show
that through the PTA the (2+1) dimensional K-dV equation equation can
be trilinearized. A four parameter special class of solutions involving three
arbitrary functions g(y), h(y) and f(x,t) in the indicated variables naturally
arises for this trilinear equation. This in turn leads to the universal form of
solution obtained earlier by Tang, Lou and Zhang [5] using a variable separa-
tion approach as a special case. However, there exists a possibility of obtaining
more general solution to the trilinear equation, which remains to be explored.
A large class of Jacobi elliptic function periodic solutions, multidromion and
bound state solutions are also obtained. It is also shown that the procedure
can be extended to the case of Nizhnik-Novikov-Veselov (NNV) equation as
well.
The plan of the paper is as follows. In section 2, we show how the (2+1) di-
mensional K-dV equation can be trilinearized using the PTA and the solution
obtained in terms of three arbitrary functions and four arbitrary parameters.
Sections 3 and 4 contain two broad classes of solutions of the (2+1) dimen-
sional K-dVequation, both periodic and exponentially localized ones, through
judicious choices of the arbitrary functions. In section 5, we point out how the
generalized NNV equation can be trilinearized through the PTA and how var-
ious periodic and localized solutions can be obtained. Finally in section 6,
we summarize our results. The appendix contains the one dromion solution of
the (2+1) dimensional K-dV equation obtained through Hirota bilinearization
approach for comparison.
2
2 The (2+1) dimensional K-dV equation and construction of solu-
tions
The (2+1) dimensional K-dV equation introduced by Boiti et al. [10] has the
form
q +q = 3(q∂ 1q ) . (1)
t xxx y− x x
This nonlocal equation (1) reduces to the K-dV equation if x = y. Introducing
a potential function v(x,y,t) defined by the equation
v = q , (2)
y x
Eq.(1) can be converted into a set of coupled equations of the form
q +q = 3(qv) , (3a)
t xxx x
v = q . (3b)
y x
The nature of (3b) permits the presence of an arbitrary function v (x,t) as
′
y
v(x,y,t) = q dy +v (x,t). (4)
x ′ ′
Z−∞
Expanding the physical field q and the potential v in the form of a Laurent
series in the neighbourhood of the noncharacteristic manifold φ(x,y,t) = 0,
(φ = φ 6= 0) and utilising the results of the Painlev´e analysis [11], we obtain
x y
the following B¨acklund transformation by truncating the Laurent series at the
constant level term
q = q φ 2 +q φ 1 +q , (5a)
0 − 1 − 2
v = v φ 2 +v φ 1 +v , (5b)
0 − 1 − 2
where (q,v) and (q ,v ) are two different sets of solutions of the (2+1) di-
2 2
mensional K-dV equation. In ref. [4] certain basic localized dromion solutions
have been obtained using (5) and Hirota bilinearization. Here, we make use
of the full power of Eq. (5) by treating it as a variable transformation, and
obtain rather general classes of solutions. This new approach leads to a wide
class of solutions not only for Eq. (1) but for other (2+1) dimensional systems
as well [7].
Considering now a vacuum solution of the form
q = 0,v = v (x,t) (6)
2 2 2
and substituting (6) with (5) in Eq. (3), we obtain the following set of equa-
tions by equating the coefficients of (φ 5,φ 3) to zero,
− −
3
−24q φ3 +12q v φ = 0, (7a)
0 x 0 0 x
−v φ +q φ = 0. (7b)
0 y 0 x
Solving the above system of equations, we obtain
q = 2φ φ , (8a)
0 x y
v = 2φ2. (8b)
0 x
Again, collecting the coefficients of (φ 4,φ 2), we have
− −
18q φ2 +18q φ φ −6q φ3 −3(q v −2q v φ
0x x 0 x xx 1 x 0x 0 0 1 x
−q φ v )−3(v q −2v q φ −v φ q )=0, (9a)
1 x 0 0x 0 0 1 x 1 x 0
v −v φ −(q −q φ )=0. (9b)
0y 1 y 0x 1 x
Solving the above system of equations, we obtain
q = −2φ , (10a)
1 xy
v = −2φ . (10b)
1 xx
Next, collecting the coefficients of (φ 3,φ 1), we have
− −
−2q φ +(−6q φ −6q φ −2q φ +6q φ2
0 t 0xx x 0x xx 0 xxx 1x x
+6q φ φ )−3(q v −2q φ v +q v −q v φ )
1 x xx 0x 1 0 x 2 1x 0 1 1 x
−3(q v +q v −q v φ )=0, (11a)
0 1x 1 0x 1 1 x
v −q =0. (11b)
1y 1x
Here (11b) is an identity as may be verified using (10). Solving (11a) by using
(8) and (10), we get the form of v as
2
φ +φ φ φ −φ φ
t xxx x xxy xx xy
v = + . (12)
2
3φ φ φ
x x y
Now, we collect the coefficients of next order (φ 2,φ0) to obtain
−
(q −q φ )+(q −3q φ −q φ −3q φ )
0t 1 t 0xxx 1x xx 1 xxx 1xx x
−3(q v +q v −q v φ )−3(q v +q v )=0, (13a)
0x 2 1x 1 1 2 x 0 2x 1 1x
v =0. (13b)
2y
4
On the other hand, substituting the expression for the quantities q , v , q , v
0 0 1 1
and v in Eq. (13a), we obtain the trilinear equation
2
φ (φ φ −φ φ )+φ (φ φ −φ φ ) = 0. (14)
xx y xyy yy xy x yy xxy y xxyy
Substituting (12) in (13b), we obtain
(φ +φ )φ φ2 −(φ +φ )φ φ2 +3(φ φ −φ φ )φ φ
ty xxxy x y t xxx xy y x xxyy xx xyy x y
−3(φ φ −φ φ )φ φ −3(φ φ −φ φ )φ φ = 0. (15)
x xxy xx xy y xy x xxy xx xy x yy
Using Eq. (14) into (15), the later reduces to the form
(φ +φ )φ φ −(φ +φ )φ φ −3(φ φ −φ φ )φ = 0. (16)
ty xxxy x y t xxx xy y x xxy xx xy xy
which is again in a trilinear form. Thus equations (14) and (16) may be con-
sidered as the equivalent trilinear forms of Eqs. (3).
One can immediately observe that the set of trilinear equations (14) and (16)
admit a set of two arbitrary functions and their products as solutions
φ = f(x,t), (17a)
φ = g(y) (17b)
and
φ = f(x,t)g(y), (17c)
where f and g are arbitrary functions in the indicated variables. In fact a
more general solution involving three arbitrary functions and four arbitrary
constant parameters can be easily identified:
φ(x,y,t) = c +c f(x,t)+c h(y)+c f(x,t)g(y), (18)
1 2 3 4
where h(y) and g(y) are in general different arbitrary functions of y. Here c ,
1
c , c , and c are arbitrary parameters. The problem of identifying even more
2 3 4
general solution to (14) and (16) remains to be investigated, while we will
concentrate on the special solution (18) in this paper.
Collecting the coefficients (φ 1,φ), we obtain
−
q +q −3q v −3q v = 0. (19)
1t 1xxx 1x 2 1 2x
It can be checked that equation (19) is compatible with the earlier results.
Now substituting the above form (18) for the manifold φ(x,y,t) into the trun-
cated Painlev´e series (5) for the functions q(x,y,t) and v(x,y,t), along with
5
the expressions for the coefficient functions q , v , q , v and v given above,
0 0 1 1 2
we finally obtain the solution to Eqs. (3) as
2f [(c +c g)c h −(c +c h)c g ]
x 2 4 3 y 1 3 4 y
q(x,y,t)= , (20)
[c +c f(x,t)+c h(y)+c f(x,t)g(y)]2
1 2 3 4
2(c +c g)2f2
v(x,y,t)= 2 4 x
[c +c f(x,t)+c h(y)+c f(x,t)g(y)]2
1 2 3 4
2(c +c g)f f +f
2 4 xx t xxx
− + . (21)
c +c f(x,t)+c h(y)+c f(x,t)g(y) 3f
1 2 3 4 x
It may be noted that the expressions given in (20) and (21) coincide exactly
with the forms obtained by Tang, Lou and Zhang [5] using the method of
variableseparationforthespecialcaseg(y) = h(y)inEq.(18).Anotherspecial
case c = c = 0, c = c = 1 was identified by Peng recently [9]. Also, in our
1 2 3 4
case it is clear that finding any solution to (14) and (16) which is more general
than (18) will lead to more general solution than (20)-(21). This problem is
being investigated further. In the following, we will investigate the nature of
solutions (20)-(21) corresponding to periodic and localized solutions. For this
purpose, we shall consider the special cases (i) c = 0 and (ii) c = c = 0 in
4 2 3
(18)or(20)-(21),corresponding tothesumandproduct ofarbitraryfunctions,
respectively, and investigate the nature of periodic and localized solutions. It
may be noted that there is no difficulty in proceeding with the general form
(20)-(21) also except for the fact that the expression will be more lengthy
which we desist from presenting here.
3 Periodic and localized solutions corresponding to sum of arbi-
trary functions
In this section, we will assume c = c, c = c = 1, c = 0 in (18) or (20)-(21).
1 2 3 4
Consequently, we have
2f h
x y
q= , (22a)
(f +h+c)2
2f2 2f f +f
v= x − xx + t xxx. (22b)
(f +h+c)2 (f +h+c) 3f
x
6
3.1 Harnessing of arbitrary functions and novel solutions of (2+1) dimen-
sional K-dV equation
Let us now choose the arbitrary functions f(x,t) and h(y) to be Jacobian el-
liptic functions, namely sn or cn functions. The motivation behind this choice
of arbitrary function stems from the fact that the limiting forms of these func-
tions happen to be localized functions. Hence, a choice of cn and sn functions
can yield periodic solutions which are more generalized than exponentially
localized solutions (dromions). We choose, for example,
f = αsn(ax+c t+d ;m ), h = βsn(by +d ;m ) (23)
1 1 1 2 2
so that
2αβabcn(u ;m )dn(u ;m )cn(u ;m )dn(u ;m )
1 1 1 1 2 2 2 2
q(x,y,t) = , (24)
(c+αsn(u ;m )+βsn(u ;m ))2
1 1 2 2
where u = ax+c t+d and u = by+d . In Eqs. (23) and (24), the quanti-
1 1 1 2 2
ties m and m are moduli of the respective Jacobian elliptic functions while
1 2
α, β, a, b, c, c , d and d are arbitrary constants. We choose the constant
1 1 2
parameters c, α and β such that |c| > |α + β| for nonsingular solution. The
profile of the above solution for the parametric choice α = β=1, a=0.5, b=0.4,
c=-4, c =-1, d =d =0, m = 0.2, m = 0.3 at time t=0 is shown in Fig.1(a).
1 1 2 1 2
Note that the periodic wave travels along the x-direction only.
3.1.1 (1,1) dromion solution
As a limiting case of the periodic solution given by Eq. (24), when m , m
1 2
→ 1, the above solution degenerates into an exponentially localized solution
(dromion). Noting that cn(u;1) = dn(u;1) = sechu and sn(u;1) = tanhu, the
limiting form corresponding to (1,1) dromion takes the expression
2abαβsech2(ax+c t+d )sech2(by +d )
1 1 2
q = , (|c| > |α+β|) (25)
(c+αtanh(ax+c t+d )+βtanh(by +d ))2
1 1 2
and the variable v then takes the form (using expression (22b))
2α2a2sech4(ax+c t+d )
1 1
v=
(c+αtanh(ax+c t+d )+βtanh(by +d ))2
1 1 2
4αa2sech2(ax+c t+d )tanh(ax+c t+d )
1 1 1 1
+ +
(c+αtanh(ax+c t+d )+βtanh(by +d ))
1 1 2
1
[c +4a3tanh2(ax+c t+d )−2a3sech2(ax+c t+d )]. (26)
1 1 1 1 1
3a
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Schematic form of the (1,1) dromion for the parametric choice α = β=1,
a=0.5, b=0.4, c=-4, c =-1, d =d =0 at time t=0 is shown in Fig.1(b). Again
1 1 2
note that the dromion travels along the x-direction. One can obtain the (1,1)
dromion given by Radha and Lakshmanan in ref. [4] by fixing the parameters
α, β, a, b, c , d , d of equation (25) suitably (see also the Appendix).
1 1 2
3.1.2 More general periodic solution and (2,1) dromion
Next we obtain a more general periodic solution by choosing further general
forms for the arbitrary functions. As an example, we choose
f =α sn(a x+c t+d ;m )+α sn(a x+c t+d ;m ),
1 1 1 1 1 2 2 2 2 2
h=βsn(by +d ;m ), (27)
3 3
where b, β, α , a , c and d are arbitrary constants (i = 1,2; j = 1,2,3) and
i i i j
m ’s are modulus parameters. Then
j
q
1
q = , (28)
q
2
where, q = 2[(α a cn(u ;m )dn(u ;m )+α a cn(u ;m )dn(u ;m ))
1 1 1 1 1 1 1 2 2 2 2 2 2
βbcn(u ;m )dn(u ;m )],q = (c+α sn(u ;m )+α sn(u ;m )+βsn(u ;m ))2,
3 3 3 3 2 1 1 1 2 2 2 3 3
u = a x+c t+d , u = a x+c t+d and u = by +d with corresponding
1 1 1 1 2 2 2 2 3 3
expressions for v(x,y,t). We choose the constant c such that |c| > |α +
1
α + β| for nonsingular solutions. The profile of the above solution for the
2
parametric choice α = α = β=1, a =0.5, a =0.5, b =0.4, c=-4, c =-1,
1 2 1 2 1
c =-2, d =d =d =0, m = 0.2, m = 0.3, m = 0.4 at time t=0 is shown in
2 1 2 3 1 2 3
Fig.2a. As m , m , m → 1, the above solution given by Eq. (28) degenerates
1 2 3
into a (2,1) dromion solution given by
2(α a sech2u +α a sech2u )bsech2u
1 1 1 2 2 2 3
q = , (|c| > |α +α +β|) (29)
(c+α tanhu +α tanhu +βtanhu )2 1 2
1 1 2 2 3
where u = a x+c t+d , u = a x+c t+d and u = by+d . The solution
1 1 1 1 2 2 2 2 3 3
is plotted for the parametric choice α = α = β=1, a =0.5, a = 0.8, b =0.4,
1 2 1 2
c=-4, c =-1, c =-2.5, d =d =d =0, for various values of t in Figs. 2(b)-2(d)
1 2 1 2 3
in order to bring out the dromion interaction clearly.
3.1.3 Asymptotic analysis for (2,1) dromion solution
Since both the dromions in Figs. 2(b)-2(d) corresponding to the solution (29)
are travelling along the x-direction, following one another, it is enough to do
this analysis fory = 0 (and d = 0 so that u = 0). Similar analysis holds good
3 3
for any other value of y and d 6= 0. This restriction corresponds to the cross
3
8
section of dromions, which are essentially solitons. We analyse the limits t →
−∞ and t → +∞ separately so as to understand the interaction of dromions
centered around u ≈ 0 or u ≈ 0. Without loss of generality, let us assume
1 2
c > c and a < a . Then, we find in the limit t → ±∞, u = a x+c t+d
1 2 1 2 1 1 1 1
and u = a x+c t+d take the following limiting values.
2 2 2 2
1) As t → −∞:
u ≈ 0,u → +∞
1 2
u ≈ 0,u → −∞
2 1
2) As t → +∞:
u ≈ 0,u → −∞
1 2
u ≈ 0,u → +∞
2 1
1. Before interaction (as t → −∞):
For u ≈ 0,u → +∞ and α = α = β = 1, the (2,1) dromion solution (29)
1 2 1 2
becomes (soliton solution corresponding to dromion 1)
2a b c+2
q = 1 sech2(u +δ ), δ = . (30)
1 1 1
c(c+2) s c
For u ≈ 0,u → −∞, (29) becomes (soliton solution corresponding to
2 1
dromion 2)
2a b c
q = 2 sech2(u +δ ), δ = . (31)
2 2 2
c(c−2) sc−2
2. After interaction (as t → +∞):
For u ≈ 0,u → −∞, (29) becomes (soliton solution corresponding to
1 2
dromion 1)
2a b
q = 1 sech2(u +δ ). (32)
1 2
c(c−2)
For u ≈ 0,u → +∞, (29) becomes (soliton solution corresponding to
2 1
dromion 2)
2a b
q = 2 sech2(u +δ ). (33)
2 1
c(c+2)
The above results can be interpreted in the following way. Before interaction,
the dromion 1 has a larger amplitude, travelling slower and the dromion 2 has
a shorter amplitude, travelling faster. During interaction (at time t = 0), an
exchange of energy between the dromions take place. This results in the gain
9
in amplitude of dromion 2 and fall in amplitude of dromion 1. But there is no
change in velocity of the dromions and there is only a change in phase.
3.1.4 Bounded multiple solitary waves
One can also construct bounded two solitary waves by choosing
f =αsn(ax+c t+d ;m ),
1 1 1
h=β sn(b y +d ;m )+β sn(b y +d ;m ), (34)
1 1 2 2 2 2 3 3
where α, a, c , β , b , and d (i = 1,2; j = 1,2,3) are arbitrary constants and
1 i i j
m ’s are modulus parameters. Then
j
q
3
q = , (35)
q
4
whereq = 2αacn(u ;m )dn(u ;m )[β b cn(u ;m )dn(u ;m )+β b cn(u ;m )
3 1 1 1 1 1 1 2 2 2 2 2 2 3 3
dn(u ;m )], q = (c + αsn(u1;m1) + β sn(u2;m2) + β sn(u3;m3))2, u =
3 3 4 1 2 1
a x+c t+d , u = b y+d and u = b y+d with corresponding expressions
1 1 1 2 1 2 3 2 3
for v(x,y,t). Here |c| > |α + β + β |. The profile of the above solution for
1 2
the parametric choice α = β = β = 1, a =0.5, b =2.5, b =1.7, c=-4, c =1,
1 2 1 2 1
c =-2, d =d =d =0, m = 0.2, m = 0.3, m = 0.4, t=0 is shown in Fig.3(a).
2 1 2 3 1 2 3
As m , m , m → 1, the above solution, Eq. (35), degenerates into a bounded
1 2 3
two dromion solution given by
2αasech2u (β b sech2u +β b sech2u )
1 1 1 2 2 2 3
q = , (36)
(c+αtanhu +β tanhu +β tanhu )2
1 1 2 1 3
where u = ax+c t+d , u = b y+d and u = b y+d . The above solution
1 1 1 2 1 2 3 2 3
for the parametric choice α = β = β = 1, a =0.5, b =2.5, b =1.7, c=-
1 2 1 2
4, c =1, c =-2, d =d =d =0, is shown in Fig.3(b). Here both the dromions
1 2 1 2 3
travel with equal velocity along the x direction. Since they move parallel to
each other there is no interaction between them.
3.2 (N,M) dromion solution
Proceedinginasimilarwayasabove,onecangenerate(N,M)dromionsolution
by choosing
10