Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system

Multisolitonic solutions from a Bäcklund transformation for a parametric coupled Korteweg-de Vries system

Abstract

We introduce a parametric coupled KdV system which contains, for particular values of the parameter, the complex extension of the KdV equation and one of the Hirota-Satsuma integrable systems. We obtain a generalized Gardner transformation and from the associated - deformed system we get the infinite sequence of conserved quantities for the parametric coupled system. We also obtain a Bäcklund transformation for the system. We prove the associated permutability theorem corresponding to such transformation and we generate new multi-solitonic and periodic solutions for the system depending on several parameters. We show that for a wide range of the parameters the solutions obtained from the permutability theorem are regular solutions. Finally we found new multisolitonic solutions propagating on a non-trivial regular static background.

*Department of Mathematics

Antofagasta University

**Physics Department

Antofagasta University

**Physics Department

Simón Bolívar University

Keywords: partial differential equations, integrable systems, symmetry and conservation laws, solitons.

Pacs: 02.30.Jr , 02.30.Ik, 11.30.-j, 05.45. Yv.

1 Introduction

Coupled Korteweg-de Vries (KdV) systems were extensively analyzed since Hirota and Satsuma [1]. In that work the authors proposed a model that describes interactions of two long waves with different dispersion relations. Later, Gear and Grimshaw [2] derived a coupled KdV system for linearly stable internal waves in a density stratified fluid and Lou, Tong, Hu and Tang [3] derived systems for two-layer fluids models used in the description of the atmospheric and oceanic phenomena. An interesting classification of coupled KdV systems was presented in [4]. In [5] coupled KdV systems of Hirota-Satsuma type were studied in more detail.

Among the properties of these systems one has the existence of multisolitonic solutions [1] obtained using the Hirota bilinear method [6] or Bäcklund transformations [7], the symmetries and conserved quantities [8], the existence of Lax pairs and Painlevé property [5, 9], the satisfactory analysis of well posed problems [10] as well as stability properties of the solutions [11].

In this work we consider a parametric coupled KdV system. For some values of the parameter, , the system corresponds to the complexification of KdV equation. For the system corresponds to one of the Hirota-Satsuma coupled KdV systems, while for the system is equivalent to two decoupled KdV equations. Although some of the properties of the complexification of KdV equation arise directly from the corresponding ones on the space of solutions of the KdV equation, there are new properties of the system which do not have an analogous on the original real equation. Among them one has solutions of the complex KdV equation which present blow up properties [12] not present in the solutions of the KdV equation. In general the space of solutions of the complex KdV equation is much richer than the corresponding one on KdV equation and the construction of regular solutions arise naturally from the permutability theorem we will introduce in this work. We provide in this work new solitonic solutions with no counterpart on the real case.

We obtain a Bäcklund transformation for the parametric coupled KdV system. We prove the permutability theorem corresponding to the Bäcklund transformation and generate new multi-solitonic and periodic solutions of the system. In particular we found a new solution of the coupled system describing the propagation of a solitonic solution on a static background solution. We introduce a generalized Gardner transformation and obtained from the associated integrable -deformed system the infinite sequence of conserved quantities for the parametric coupled KdV system.

In section 2 we present the parametric coupled KdV system analyzing some of its properties like symmetries, we obtain a generalized Gardner transformation which yields infinite local conserved quantities and also we give two associated lagrangians. In sections 3 and 4 we give a Bäcklund transformation for the system and we prove the permutability theorem. In section 5 we obtain explicitly new parametric periodic, multi-solitonic and stationary solutions for the system using the Bäcklund transformation presented before. In section 6 we prove that the solutions obtained from the permutability theorem are regular for a wide range of values of the given parameters. In section 7 we discuss and present figures of the soliton evolution on a non-trivial stationary background. Finally, in section 8 we present our conclusions.

2 Lagrangian and Gardner transformation

We consider a coupled Korteweg-de Vries (KdV) system, formulated in terms of two real differentiable functions and , given by the following partial differential equations:

 ut+uux+uxxx+λvvx=0 (1) vt+uxv+vxu+vxxx=0 (2)

where is a real parameter.

Here and in the sequel and belong to the real Schwartz space defined by

 C∞↓(R)={u∈C∞(R)/limx→±∞xp∂q∂xqw=0;p,q≥0}.

By a redefinition of given by we may reduce the values of to be and to be . The systems for and are not equivalent. The case corresponds to the complex KdV equation in terms of :

 Ut+UUx+Uxxx=0. (3)

The case corresponds to two decoupled KdV equations, one for and the other for :

 (u+v)t+(u+v)(u+v)x+(u+v)xxx=0 (4) (u−v)t+(u−v)(u−v)x+(u−v)xxx=0. (5)

The system (1),(2) for describes a two-layer liquid model studied in references [2, 3, 13]. It is a very interesting evolution system. It is known to have solutions developing singularities on a finite time [12]. Also, a class of solitonic solutions was reported in [14] via the Hirota approach [6].

The system (1),(2) for corresponds to the ninth Hirota-Satsuma [1] coupled KdV system given in [5] (for the particular value of ) (see also [4]) and is also included in the interesting study which relates integrable hierarchies with polynomial Lie algebras [15].

(1),(2) were introduced in [16] and also considered independently in [17].

They remain invariant under Galileo transformations. In fact, if

 x→x+ctt→tu→u+cv→v (6)

(where is a real constant) the transformed fields defined by

 ^u(x+ct,t)=u(x,t)+c,^v(x+ct,t)=v(x,t) (7)

satisfy (1),(2) in terms of the new coordinates . That is, (1),(2) remain invariant under the above transformation, for any value of .

(1),(2) are also invariant under translations on and under the anysotropic rescaling

 x→bx t→b3t u→b−2u v→b−2v

which is the same rescaling, on the spatial coordinates and time coordinate, as in anisotropic Hořava-Lifschitz gravity, a renormalizable field theory [18, 19].

The system (1) and (2) may be derived from the following two parametric lagrangian densities,

 L1(w,y)=−12wxwt−16wx3+12wxx2−12λwx(yx)2+12λyxyt−12λ(yxx)2L2(w,y)=−12wxyt−12wtyx−12w2xyx−yxwxxx−λ6y3x (8)

formulated in terms of two real differentiable fields related to the original fields by the following relations:

 u=wxv=yx. (9)

In the case of one has to assume while for , is any real including

We notice that and belong to the space defined by

 C∞+(R)={w∈C∞(R)/∂xw∈C∞↓(R)}.

The field equations obtained from each of the lagrangians (8), by taking independent variations with respect to and , are equations (1) and (2). They can be integrated out to give

 wt+12(wx)2+wxxx+12λ(yx)2+C(t)=0 (10) yt+wxyx+yxxx+˜C(t)=0, (11)

where and are integration constants which depend only on the variable. These two integration constants may be eliminated by a redefinition of and :

 w(x,t)→w(x,t)+∫t0C(τ)dτy(x,t)→y(x,t)+∫t0˜C(τ)dτ. (12)

The system (10),(11) may then be reduced, without loss of generality, to the system

 Q1(w,y)≡wt+12(wx)2+wxxx+12λ(yx)2=0 (13) Q2(w,y)≡yt+wxyx+yxxx=0. (14)

We notice that the lagrangians given by (8) are not invariant under the Galileo transformations (6) and (7), which in terms of and take the form

 x→x+ct,t→t ^w(x+ct,t)=w(x,t)+cx+f(t) ^y(x+ct,t)=y(x,t).

However, the field equations (13) and (14) become invariant under the above transformation when

The three cases have infinite conserved quantities. In the particular case , which corresponds to the complex KdV equation, the infinite local conserved quantities are obtained by replacing the real field , in the expression of the local KdV conserved quantities, by the complex field . The real and pure imaginary parts of the expression are conserved quantities of the system (1) and (2) for . The system (1) and (2) for any value of has associated to it a generalized Gardner transformation and a corresponding Gardner system.

Lemma 1

Let be a solution of the following -parameter partial differential equations (called the Gardner system)

Then defined through the relations (called the Gardner transformation)

 u=r+εrx−16ε2(r2+λs2) v=s+εsx−13ε2rs

are solutions of the system (1),(2).

Proof of Lemma 1

The two component vector defined by the left hand members of (1),(2) is equal to the Frechet derivative of the Gardner transformation, with respect to and , applied to the two component vector defined by the left hand member of the Gardner system. It then follows that a solution of the Gardner equations define through the Gardner transformation a solution of the system (1),(2).

Theorem 1

The system (1),(2) has infinite conserved quantities and they are explicitly obtained from the first two conserved quantities of the Gardner system.

Proof of Theorem 1
 ∫+∞−∞rdxand∫+∞−∞sdx

are conserved quantities of the Gardner system. From the Gardner transformation we can expand and as formal series on with coefficients which are polynomials on and their derivatives with respect to .

After replacing the formal series on the first two conserved quantities, we obtain an infinite sequence of local conserved quantities in terms of and their spatial derivatives.

The first few of them are

 ∫+∞−∞udx,∫+∞−∞vdx, ∫+∞−∞(13u3+λuv2−λ(v′)2−(u′)2)dx,∫+∞−∞(12u2v−u′v′+16λv3)dx.

In the following section we introduce a Bäcklund transformation for this system, which among other important issues will allow us to obtain multisolitonic as well as periodic solutions to the system (1),(2).

3 The Bäcklund transformation

We propose, following [7, 16], a Bäcklund transformation, which maps a solution of (13), (14) to a new solution of (13), (14).

Theorem 2

If and satisfy the following equations (Bäcklund transformation)

 wx+w′x = 2η−112(w−w′)2−λ12(y−y′)2, (15) wt+w′t = 16(w−w′)(wxx−w′xx)+λ6(y−y′)(yxx−y′xx)−13w2x− − 13w′x2−13wxw′x−λ3y2x−λ3y′x2−λ3yxy′x, yx+y′x = 2μ−16(w−w′)(y−y′), (17) yt+y′t = 16(w−w′)(y−y′)xx+16(w−w′)xx(y−y′)− − (23wxyx+23w′xy′x+13wxy′x+13w′xyx)

on an open set and

 (w−w′)2−λ(y−y′)2≠0

on , then and are two (different) solutions of (13), (14).

Proof of Theorem 2

From +(16) we obtain

 Q1(w,y)+Q1(w′,y′)=0. (19)

From +(18) we get

 Q2(w,y)+Q2(w′,y′)=0. (20)

A solution of (15), (16), (17), (18) satisfy the integrability conditions and .

Calculating and using (15) and (17) to express second order derivatives and in terms of first order derivatives we obtain

 (w−w′)[Q1(w,y)−Q1(w′,y′)]+λ(y−y′)[Q2(w,y)−Q2(w′,y′)]=0. (21)

Analogously, calculating and using (15) and (17) to express second order derivatives and in terms of first order derivatives we get

 (y−y′)[Q1(w,y)−Q1(w′,y′)]+(w−w′)[Q2(w,y)−Q2(w′,y′)]=0. (22)

Under the assumptions of the theorem, (21) and (22) imply

 Q1(w,y)−Q1(w′,y′)=0,Q2(w,y)−Q2(w′,y′)=0 (23)

(19),(20) and (23) ensure that and are solutions of (13),(14).

Theorem 3

If and satisfy (15),(16),(17),(18) and is a solution of (13),(14) then is a solution of (13),(14).

Proof of Theorem 3

If we take +(16) we obtain (19). If we take +(18) we get (20). Consequently if is a solution of (13), (14) then a solution of (15), (16), (17), (18) is also a solution of (13), (14).

4 The permutability theorem for the coupled KdV system

We prove in this section the permutability theorem for the system (13), (14). That is, we start from a solution of (13), (14), we define and from the Bäcklund transformations associated to the parameters and respectively. We then perform a new Bäcklund transformation starting from with parameter and obtain the new solution . While, if we start with and perform the Bäcklund transformation with parameter we obtain the solution The theorem states that and

The theorem was proven for the KdV equation in [7].

Theorem 4

(Permutability theorem) Let be the solution of (13),(14) obtained from the Bäcklund transformation following the sequence

 (w0,y0)→η1(w1,y1)→η2(w12,y12)

and the solution following the sequence

 (w0,y0)→η2(w2,y2)→η1(w21,y21).

Then and

 w12−w0=24(η1−η2)(w1−w2)(w1−w2)2−λ(y1−y2)2 y12−y0=−24(η1−η2)(y1−y2)(w1−w2)2−λ(y1−y2)2.
Proof of Theorem 4

We have

 w1x+w0x=2η1−112(w1−w0)2−λ12(y1−y0)2 (24) y1x+y0x=−16(w1−w0)(y1−y0) (25) w2x+w0x=2η2−112(w2−w0)2−λ12(y12−y0)2 (26) y2x+y0x=−16(w2−w0)(y2−y0) (27) w12x+w1x=2η2−112(w12−w1)2−λ12(y12−y1)2 (28) y12x+y1x=−16(w12−w1)(y12−y1) (29) w21x+w2x=2η1−112(w21−w2)2−λ12(y21−y2)2 (30) y21x+y2x=−16(w21−w2)(y21−y2). (31)

The strategy will be to assume , then to obtain an expression for them and finally to verify that they satisfy (28), (29) and (30), (31).

Assuming and we obtain from (24)-(26)+(30)-(28),

 4(η1−η2)+16(w1−w2)(w0−w12)+λ6(y1−y2)(y0−y12)=0. (32)

From (25)-(27)+(31)-(29) we get

 (y0−y12)(w1−w2)+(w0−w12)(y1−y2)=0 (33)

Finally from (32) and (33) we have

 w12−w0=24(η1−η2)(w1−w2)(w1−w2)2−λ(y1−y2)2 (34) y12−y0=−24(η1−η2)(y1−y2)(w1−w2)2−λ(y1−y2)2. (35)

We notice that under the interchange

 η1↔η2w1↔w2y1↔y2 (36)

the expressions on the right hand members of (34), (35) are invariants. This is a necessary condition in order to have

Finally, we may use these formulas to verify that (28), (29) and (30), (31) as well as (16), (18) are satisfied.

We conclude that and that they have the nice explicit expressions (34), (35).

We notice that the denominator in (34), (35) is the same expression which appears in the assumptions of theorem 2. This condition is necessary in order to have regular solutions. In the case , the denominator becomes

 (w1−w2)2+(y1−y2)2. (37)

5 Explicit solutions from the Bäcklund transformation

In this section we obtain new solutions to the system (1),(2). We use the Bäcklund transformation. We will explicitly consider .

For we can solve first KdV equation (1) in and using (2), which results linear in , we can integrate it directly to obtain . The case can be solved directly in terms of and . The solutions for and can be obtained by redefinition of the solutions for and respectively.

We start by considering We introduce the field ,

 (38)

From (17), when ,

 yxy=−γ6κxκ,assumingy≠0. (39)

Replacing into (15), and choosing we obtain

 κxx=η6κ+(ρ12)2κ−3 (40)

and from (39)

 y=ρκ−2,ρ≠0 (41)

where is an integration constant, there is no restriction on its sign.

The differential equation may be integrated once, to obtain

 (κx)2=η6κ2−(ρ12)2κ−2+C (42)

where is another integration constant.

Proposition 1

For any

 u(x,t)=4η[1−3CηAcosh(ax+b)][cosh(ax+b)−3ηCA]2,v(x,t)=−ρAasinh(ax+b)[cosh(ax+b)−3ηCA]2

are solutions of (1),(2) with for any set of parameters

Proof of Proposition 1

The most general solution of (42) is

 κ2=[(3Cη)2+6η(ρ12)2]12cosh(2√η6x+b)−3Cη, (43)

where are integration constants (which may be functions of ), .

We notice that the right hand member of (43) is always positive.

The corresponding expressions for and are

 w=12√η6sinh(ax+b)[cosh(ax+b)−3CηA],y=ρA1[cosh(ax+b)−3CηA] (44)

where

 A=[(3Cη)2+6η(ρ12)2]12,a=2√η6

and a function of which is determined by using (16). We obtain +constant.

The expressions for and are

 u(x,t)=4η[1−3CηAcosh(ax+b)][cosh(ax+b)−3ηCA]2,v(x,t)=−ρAasinh(ax+b)[cosh(ax+b)−3ηCA]2. (45)

We notice that the denominator in both expressions is always strictly positive (because ). These solutions are the same as the ones obtained in [14] following the Hirota method.

In the following figures we plot the solutions for particular values of the parameters.

Proposition 2

For any

 u=−4|η|(1+3Cϵ|η|^Acos(ax+b))(ϵcos(ax+b)+3C|η|^A)2,v=\-ρa^Aϵsin(ax+b)(ϵcos(ax+b)+3C|η|^A)2

are solutions of (1),(2) for any set of parameters satisfiyng

 ρ≠0, δ=32C2|η|−(ρ12)2>0, C>0.
Proof of Proposition 2

We obtain

 κ2=±(6δ|η|)12cos(ax+b)+3C|η| (46)

where and + constant.

are integration constants which must satisfy the conditions

 ρ≠0,δ>0,C>0. (47)

The condition ensures that the right hand member in (46) is strictly positive.

The final expressions for and are

 w=−ϵ6asin(ax+b)ϵcos(ax+b)+3C|η|^A,y=ρ^A1(ϵcos(ax+b)+3C|η|^A) (48)

where

 ^A=(6δ|η|)12=⎡⎣(3C|η|)2−(ρ6a)2⎤⎦12

and

Finally we have the expressions of and ,

 u=−4|η|(1+3Cϵ|η|^Acos(ax+b))(ϵcos(ax+b)+3C|η|^A)2,v=\-ρa^Aϵsin(ax+b)(ϵcos(ax+b)+3C|η|^A)2. (49)

We notice that the denominator in both expressions is strictly positive, the requirement implies that must be different from zero. This restriction is not present on the solutions (45). The time dependence is obtained using the remaining equations of the Bäcklund transformations, as before we obtain +constant. We notice that in the above expressions the factor may be be omitted by adding to .

Proposition 3

For any value of the parameters and

 u=wx=C2ρ212−12C4(x+H)2[(ρ12)2+C2(x+H)2]2,v=yx=−2C3ρ(x+H)[(ρ12)2+C2(x+H)2]2

are solutions to (1),(2) with corresponding to on the Bäcklund transformation.

Proof of Proposition 3

Following the same approach as in the case we obtain

 w=12C2(x+H)(ρ12)2+C2(x+H)2,u=wx=C2ρ212−12C4(x+H)2[(ρ12)2+C2(x+H)2]2 y=Cρ(ρ12)2+C2(x+H)2,v=yx=−2C3ρ(x+H)[(ρ12)2+C2(x+H)2]2

where and are integration constants, .

It is quite interesting that it is a regular solution and hence it may be interpreted as a static background for the coupled system. In section 7 we will study the propagation of solitonic solutions on this background.

6 Regularity of the solutions

We can use the expressions for and in (34) and (35) respectively, obtained from the permutability theorem, to construct multisolitonic solutions for the system (1) and (2).

In distinction with what occurs with the Bäcklund transformation for the KdV equation, we can use directly and in terms of the regular solutions obtained in this section to find new regular solutions of the coupled KdV system. In fact, the denominator in (34),(35) is manifestly positive and . Moreover, we are going to show that if we choose adequately the parameters in (44) the denominator in (34), (35) is strictly positive and the new solutions are regular. We take and two solutions with the expression (44) and parameters and respectively. We assume and any satisfying

 C1η1ρ1=C2η2ρ2. (50)

We remind that we have assumed and in order to obtain (44). The result is that under these conditions, the denominator satisfies

 (w1−w2)2+(y1−y2)2>d>0. (51)

Let us define

 C=−3Cη

and use, as before, and

Then the equations

 w1−w2=0,y1−y2=0

imply

 sinh(a1x+b1) = a2a1ρ1A2ρ2A1sinh(a2x+b2) cosh(a1x+b1) = ρ1A2ρ2A1(cosh(a2x+b2)+C2A2)−C1A1.

We then obtain

 (ρ1A2ρ2A1)2(1−(a2a1)2)cosh2(a2x+b2)+2ρ1A2ρ2A1(ρ1C2ρ2A1−C1A1)cosh(a2x+b2)+ + (a2ρ1A2a1ρ2A1)2+(ρ1C2ρ2A1−C1A1)2−1=0.

Under the assumption (50), the second and fourth terms of the previous equation are zero. After some calculations we get

 cosh2(a2x+b2)=6a2C2ρ21+6a2C2ρ2<1,

which does not have a solution.

Consequently, under assumption (50) the denominator for all . Moreover, since are and asymptotically the denominator approaches a positive constant different from zero we conclude that relation (51) is always satisfied for any value of and . The solution is then a regular solution of the coupled system, it describes a two-solitonic solution of the system.

From the above argument we conclude the following,

Theorem 5

For any value of the parameters and