Symmetry Reductions of A Nonisospectral Lax Pair for A (2+1)-Dimensional Breaking Soliton System

Vol. 78 (2016)

REPORTS ON MATHEMATICAL PHYSICS

No. 1

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR FOR A (2 + 1)-DIMENSIONAL BREAKING SOLITON SYSTEM NA LV, DATIAN N IU* , X UEGANG Y UAN and X UDONG Q IU School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China; School of Science, Dalian Minzu University, Dalian 116600, China (e-mails: [email protected], [email protected], [email protected], [email protected]) (Received September 23, 2015) In this paper, we use the classical Lie group method to seek the symmetry algebras of the nonisospectral Lax pair for a (2 + 1)-dimensional breaking soliton system by considering the spectral parameter as an additional field. Based on the obtained symmetries, four reduced (1 + 1)-dimensional equations with their new Lax pairs are presented. After studying one of the reduced Lax pairs, we obtain an explicit solution of the breaking soliton system by a Darboux transformation. Keywords: symmetry reduction, breaking soliton system, nonisospectral Lax pair, Darboux transformation.

1.

Introduction As we all know, symmetry analysis plays an important role in nonlinear mathematical physics. Since Lie symmetries and many properties of differential equations are closely related, lots of scholars are interested in finding symmetries, symmetry groups of transformation, symmetry reductions, construction group invariant solutions and so on. Most calculations for symmetries of differential equations are done with the classical methods, such as the classical Lie group method [1, 2], the nonclassical Lie group method [3], and the Clarkson and Kruskal direct method [4]. Then Lou improved the direct method [5], which was based on Lax pairs. Some of the authors [6–9] applied the classical Lie group method not only to (2 + 1)-dimensions partial differential equations (PDEs), but also to their Lax pairs. In this paper, we consider a (2 + 1)-dimensional breaking soliton system [10], qt = qxy − 2[q∂x−1 (qr)y ]x ,

rt = −rxy − 2[r∂x−1 (qr)y ]x ,

(1)

where x and y are the scaled space coordinates, t is the scaled time coordinate, q and r are functions of (x, y, t) which represent the wave profiles. Eq. (1) has the * Corresponding

author. [57]

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N. LV, D. NIU, X. YUAN and X. QIU

following nonisospectral Lax pair, 1 81x = − λ81 + λr82 , 2 1 82x = q81 + λ82 , 2 81t = λ81y + λ∂x−1 (qr)y 81 − λ[ry + 2r∂x−1 (qr)y ]82 , 82t = λ82y + [qy − 2q∂x−1 (qr)y ]81 − λ∂x−1 (qr)y 82 ,

(2)

with λt = λλy , λx = 0. Eq. (1) concerns the Kaup–Newell soliton hierarchy closely. As we know the breaking soliton equations are a kind of nonlinear evolution equations which can be used to describe the (2 + 1)-dimensional interaction of Riemann wave propagation along the y-axis with long-wave propagation along the x-axis. The plan of the present paper is as follows: by the classical Lie group method in Section 2 we obtain the classical symmetry algebras and symmetry reductions of the (2 + 1)-dimensional nonisospectral Lax pair for Eq. (1) and analyze the reductions. Section 3 presents an explicit solution of the system via a Darboux’s transformation. A short summary is in Section 4. 2.

Symmetry reductions of the breaking soliton system and its reduced Lax pairs In this section, we will use the classical Lie group method to seek the symmetry algebras and symmetry reductions of the Lax pair for Eq. (1). Eq. (1) is equivalent to the following system

which possesses the Lax pair

qt = qxy − 2[qu]x , rt = −rxy − 2[ru]x , ux = (qr)y ,

1 81x = − λ81 + λr82 , 2 1 82x = q81 + λ82 , 2 81t = λ[81y + u81 − (ry + 2ru)82 ], 82t = λ82y + (qy − 2qu)81 − λu82 ,

(3)

(4)

with λt = λλy , λx = 0. The Lie point symmetry algebras admitted by its corresponding Lax pair (4) are ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ X = ξ1 + ξ2 + ξ3 + η1 + η2 + η3 + τ1 + τ2 + τ3 , (5) ∂x ∂y ∂t ∂q ∂r ∂u ∂81 ∂82 ∂λ where ξi , ηi and τi , i = 1, . . . , 3, are functions of x, y, t, q, r, u, 81 , 82 , λ.

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR. . .

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With the aid of Maple, we obtain the following infinitesimals: ξ1 = C2 x + g(t), ξ3 = (C1 + C2 )t + C3 , η2 = −2f (y)r, τ1 = (−f (y) + h(y, t, λ))81 ,

ξ2 = C1 y + C4 , η1 = (2f (y) − C2 )q, 1 dg(t) df (y) η3 = − C1 u + , dy 2 dt τ2 = (f (y) + h(y, t, λ))82 , τ3 = −C2 λ, (6)

where Cj , j = 1, . . . , 4, are arbitrary constants, f (y) and g(t) are arbitrary functions, and h(y, t, λ) satisfies ht = λhy . (7) Also, we can obtain the Lie point symmetry algebras admitted by Eq. (3), and we find that the breaking soliton system and its Lax pair admit the same symmetry transformations of the independent variables except for τi , i = 1, 2, 3. Having determined the infinitesimals in Eq. (6), the symmetry variables are found by solving the corresponding characteristic equations dx dy dt dq dr du d81 d82 dλ = = = = = = = = . ξ1 ξ2 ξ3 η1 η2 η3 τ1 τ2 τ3

(8)

While solving the above characteristic equations one has to distinguish whether the constants Cj , j = 1, . . . , 4, are identical to zero or not. This leads to different relations between the symmetry variables (x, ˜ t˜, Q, R, U, 91 , 92 , λ˜ ) and the original variables (x, y, t, q, r, u, 81 , 82 , λ). As a result we obtain the following cases. Case 1. C1 6 = 0, C2 6 = 0. In this case, there is no restriction in setting C3 implies a trivial translation in t and y. Also we can C1 and C2 , for simplicity, we set C1 = C2 = 1. Integrating Eq. (8), we have Z x 1 g(t) x˜ = 1 − dt, t˜ = 3 2 t2 t2 q = y −1 e

R 2f (y) dy y

Q(x, ˜ t˜),

r = e−

R 2f (y) dy y

R(x, ˜ t˜),

= C4 = 0 because it only freely choose the values of

y2 , t u=

f (y) + U (x, ˜ t˜) g(t) + , y 2t (9)

where Q, R and U are symmetry reduction fields with respect to the group invariants x, ˜ t˜. Now substituting Eq. (9) into Eq. (3), we immediately obtain the reduction equations, which read   3 1 1 x˜ t˜ 2 − 1 Qx˜ + t˜ 2 Qt˜ = −2t˜Qx˜ t˜ + 2(QU )x˜ , 2 3 1 1 x˜ t˜ 2 Rx˜ + t˜ 2 Rt˜ = 2t˜Rx˜ t˜ + 2(RU )x˜ , 2

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N. LV, D. NIU, X. YUAN and X. QIU 1

t˜ 2 Ux˜ = 2t˜(QR)t˜ − QR.

(10)

˜ t˜), λ = y −1 λ(

(11)

From Eq. (8), the reduction of the spectral parameter is obtained ˜ where Eq. (11) yields the following equation for λ, λ˜ t˜ =

λ˜ 2

. (12) t˜2 + 2t˜λ˜ Therefore, the reduced linear problem is nonisospectral. And the eigenfunctions are 81 = e

R −f (y)+h dy

2

y

91 (x, ˜ t˜),

82 = e

R f (y)+h dy y

92 (x, ˜ t˜),

(13)

where h = h(y, t = yt˜ , λ = y −1 λ˜ (t˜)). On substitution of the reductions into the (2 + 1)-dimensional spectral problem (4) gives us the following (1 + 1)-dimensional Lax pair 1 1 t˜ 2 91x˜ = − λ˜ (91 − 2R92 ), 2 1 1 t˜ 2 92x˜ = Q91 + λ˜ 92 , 2

1 x˜ t˜91x˜ + t˜(t˜ + 2λ˜ )91t˜ = λ˜ [−U 91 + (2t˜Rt˜ + 2RU )92 ], 2 1 x˜ t˜92x˜ + t˜(t˜ + 2λ˜ )92t˜ = (−2t˜Qt˜ + Q + 2QU )91 + λ˜ U 92 , 2

(14)

where λ˜ is given by Eq. (12). It is easy to check that the reduced system (10) is just the compatibility condition of the reduced Lax pair (14). Case 2. C1 = 0, C2 6 = 0, C4 6 = 0. Since the values of C2 and C4 can be freely chosen, for simplicity, we set C2 = C4 = 1. In this case, there is no restriction in setting C3 = 0 because it only implies a trivial translation in t. Integrating Eq. (8) with C1 = 0, C3 = 0, C2 = C4 = 1 leads to Z x g(t) ey ˜= , x˜ = − dt, t t t2 t R R g(t) q = e (2f (y)−1)dy Q(x, ˜ t˜), r = e−2 f (y)dy R(x, ˜ t˜), u = f (y) + + U (x, ˜ t˜), 2t R R ˜ t˜), (15) 81 = e (−f (y)+h)dy 91 (x, ˜ t˜), 82 = e (f (y)+h)dy 92 (x, ˜ t˜), λ = e−y λ( y

˜ t˜)). where h = h(y, t = et˜ , λ = e−y λ( On substitution of the symmetry variables (15) into the (2 + 1)-dimensional breaking system (3) and its spectral problem (4) gives us the following system and

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR. . .

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its (1 + 1)-dimensional Lax pair

(1 − x)Q ˜ x˜ − t˜Qt˜ = t˜Qx˜ t˜ − 2(QU )x˜ , xR ˜ x˜ + t˜Rt˜ = t˜Rx˜ t˜ + 2(RU )x˜ , t˜Ux˜ = t˜(RQ)t˜ − QR,

(16)

with

x˜ t˜91x˜

1 t˜91x˜ = − λ˜ (91 − 2R92 ), 2 1 t˜92x˜ = Q91 + λ˜ 92 , 2 ˜ ˜ + t˜(t˜ + λ)91t˜ = λ[−U 91 + (t˜Rt˜ + 2RU )92 ],

x˜ t˜91x˜ + t˜(t˜ + λ˜ )91t˜ = (−t˜Qt˜ + Q + 2QU )91 + λ˜ U 92 ,

where λ˜ t˜ =

λ˜ 2 , t˜2 +t˜λ˜

(17)

and the reduced problem is nonisospectral.

Case 3. C1 6 = 0, C2 = 0. In this case, there is no restriction in setting C3 = C4 = 0 because it only implies a trivial translation in t and y. From Eq. (8), the following symmetry variables are obtained: Z g(t) y x˜ = x − dt, t˜ = , C1 t t R 2f (y) R 2f (y) g(t) U (x, ˜ t˜) f (y) dy − C y dy 1 q = e C1 y Q(x, + + . ˜ t˜), r=e R(x, ˜ t˜), u= C1 y 2C1 t y (18) Substituting Eq. (18) into Eq. (3) yields the second type of symmetry reductions t˜2 Qt˜ = −t˜Qx˜ t˜ + 2(QU )x˜ , t˜2 Rt˜ = t˜Rx˜ t˜ + 2(RU )x˜ , Ux˜ = t˜(RQ)t˜.

(19)

The eigenfunctions and spectral parameter are R −f (y)+h dy

R f (y)+h dy

91 (x, ˜ t˜), 82 = e C1 y where h = h(y, t = yt˜ , λ = λ˜ ). Then the reduced Lax pair turns to be 81 = e

C1 y

92 (x, ˜ t˜),

1 ˜ 1 − 2R92 ), 91x˜ = − λ(9 2 1 92x˜ = Q91 + λ˜ 92 , 2

λ = λ˜ = const,

(20)

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˜ 1t˜ = λ[U ˜ 91 − (t˜Rt˜ + 2RU )92 ], −t˜(t˜ + λ)9 ˜ 2t˜ = (t˜Qt˜ − 2QU )91 − λU ˜ 92 , −t˜(t˜ + λ)9

(21)

whose compatibility condition is just Eq. (19), λ˜ is a constant, and therefore the reduced problem is isospectral. Case 4. C1 = C2 = 0, C3 6 = 0, C4 6 = 0. Integrating Eq. (8), we have the following symmetry variables: Z g(t) dt, t˜ = C3 y − C4 t, x˜ = x − C3 R 2f (y) R f (y) g(t) dy − 2fC(y) dy 4 q = e C4 Q(x, ˜ t˜), r=e R(x, ˜ t˜), u= + + U (x, ˜ t˜). C4 2C3 (22) The reduced eigenfunctions and the spectral parameter are 81 = e

R −f (y)+h dy C4

91 (x, ˜ t˜),

82 = e

R f (y)+h dy C4

92 (x, ˜ t˜),

λ = λ˜ = const,

(23)

˜ where h = h(y, t = C3Cy−t , λ = λ˜ ). 4 Substitution of symmetry variables (22) in Eq. (3) and its spectral problem (4) gives us the following system

−C4 Qt˜ = C3 Qx˜ t˜ − 2(QU )x˜ , C4 Rt˜ = C3 Rx˜ t˜ + 2(RU )x˜ , Ux˜ = C3 (RQ)t˜,

(24)

and its (1 + 1)-dimensional Lax pair

1 91x˜ = − λ˜ (91 − 2R92 ), 2 1 92x˜ = Q91 + λ˜ 92 , 2 ˜ ˜ −(C4 + C3 λ)91t˜ = λ[U 91 − (C3 Rt˜ + 2RU )92 ], −(C4 + C3 λ˜ )92t˜ = (C3 Qt˜ − 2QU )91 − λ˜ U 92 ,

(25)

λ˜ 6 = − CC4 is a constant, and therefore the reduced linear problem is isospectral. 3 Now we consider a special case of C1 = C2 = 0, C3 = C4 = 1, g(t) = 1, h = 0, and obtain the following symmetry variables: q = e2

R

81 = e−

f (y)dy R

Q(x, ˜ t˜),

f (y)dy

91 (x, ˜ t˜),

x˜ = x − y, r = e−2

R

82 = e

t˜ = y − t,

f (y)dy R

R(x, ˜ t˜),

f (y)dy

92 (x, ˜ t˜),

u = f (y) + U (x, ˜ t˜), λ = λ˜ = const.

(26)

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR. . .

63

On substitution of symmetry variables (26) into Eq. (3) and its spectral problem (4) gives us the following system and its (1 + 1)-dimensional Lax pair, Qt˜ = Qx˜ x˜ − Qx˜ t˜ + 2(QU )x˜ , Rt˜ = −Rx˜ x˜ + Rx˜ t˜ + 2(RU )x˜ , Ux˜ = (RQ)t˜ − (RQ)x˜ ,

with

1 91x˜ = − λ˜ (91 − 2R92 ), 2 1 92x˜ = Q91 + λ˜ 92 , 2     1 (1 + λ˜ )91t˜ = λ˜ − λ˜ + U 91 + (Rt˜ − Rx˜ + λ˜ R + 2RU )92 , 2 1 (1 + λ˜ )92t˜ = (Qx˜ − Qt˜ + λ˜ Q + 2QU )91 + λ˜ ( λ˜ + U )92 , 2

(27)

(28)

where λ˜ is a constant, and therefore the reduced linear problem is also isospectral. 3.

Darboux transformations and explicit solutions of the breaking soliton system This section is devoted to an explicit solution of the (2 + 1)-dimensional breaking soliton system (3). We shall first construct a Darboux transformation for the reduced Lax pair (28). By a transformation λ˜ = µ − 1, Eq. (28) is equivalent to the following system     91 91 =V , (29) 92 92 x˜     91 91 =W , (30) 92 t˜ 92 where V =



− 12 µ +

1 2

Rµ − R

Q

1 µ 2

−

1 2



,

and  − 12 µ−U +1−( 21 −U )µ−1 Rµ+Rt˜+2RU −Rx˜ −2R+(R+Rx˜ −Rt˜−2RU )µ−1 W= . 1 µ+U −1+( 12 −U )µ−1 Q+(Qx˜ +2QU −Qt˜−Q)µ−1 2 

We introduce a gauge transformation ϕ¯ = T ϕ,

(31)

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N. LV, D. NIU, X. YUAN and X. QIU

which transforms the spectral problems (29) and (30) into the new spectral problems ϕ¯x˜ = V¯ ϕ, ¯ V¯ = (Tx˜ + T V )T −1 , ¯ W¯ = (Tt˜ + T W )T −1 , (32) ϕ¯t˜ = W¯ ϕ, It means that we have to find a matrix T such that the old potentials (Q, R, U ) ¯ R, ¯ U¯ ). in V and W are replaced by the new ones (Q, Following [11–14], we choose ϕ = (ϕ1 , ϕ2 ) and χ = (χ1 , χ2 ) to be the basic solutions of the spectral problem (28) and define a 2 × 2 matrix T by   A(µ2 − 1) B(µ2 − 1) , (33) T = C(µ + 1) A−1 (µ2 − 1) with A = −Bδ, where B and γ are constants. Therefore, it holds that

C=

µ−1 , B

δ=

ϕ2 − γ χ2 , ϕ1 − γ χ1

det T (µ) = 0.

(34)

In a way similar to the proof in [11–14], it is easy to show the following theorem. T HEOREM 1. Suppose that A fulfils a differential equation At˜ − Ax˜ t˜ + Ax˜ x˜ − 2Ax˜ U − 2A2x˜ A−1 + 2Ax˜ At˜A−1 − 2BQt˜ + 2BQx˜

− 2BAx˜ A−1 Q + 2BAt˜A−1 Q = 0.

The matrices V¯ and W¯ determined by Eq. (32) have the same forms as V and W that is ¯ R, ¯ U¯ , µ), ¯ R, ¯ U¯ , µ), V¯ = V (Q, W¯ = W (Q, (35)

where the transformation formulae from the old potentials Q, R and U into the new ones are given by ¯ = A−1 (A−1 Q − C), Q R¯ = A(AR + B), U¯ = U − A−1 At˜ + A−1 Ax˜ . (36)

Now we start from a trivial solution Q = R = 1, U = 0 of Eq. (27). Substitute it into the Lax pairs (29) and (30), then we choose two basic solutions of (29) and (30),   3√ √ √ 1 2 µ+3 √   t˜   − µ+32 µ−1 x˜ − (µ−1)2µ ϕ = 1 e , (37) µ + 3 e 1− √ 2 µ−1   3√ √ √ 1 2 µ+3 √   t˜   µ+32 µ−1 x˜ (µ−1)2µ χ = 1 e . (38) µ + 3 e 1+ √ 2 µ−1

65

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR. . .

According to (33), we have √ √ √ √ ( µ + 3 µ − 1 − µ + 1)e−ν + γ ( µ + 3 µ − 1 + µ − 1)eν δ= , 2(e−ν − γ eν )(1 − µ) and

√ √ √ √ ( µ + 3 µ − 1 − µ + 1)e−ν + γ ( µ + 3 µ − 1 + µ − 1)eν A = −B , 2(e−ν − γ eν )(1 − µ)

where

(39)

(40)

√ √ µ + 3 µ − 1(µx˜ + µt˜ − t˜) . ν= 2µ

Substituting (39) and (40) into (36), we obtain a solution of Eq. (27) √ √ √ √ 2 −2ν 2 2γ (1−µ )+(µ−1)(µ+1− µ+3 µ−1)e +γ (µ+1+ µ+3 µ−1)e2ν ¯ = Q , √ √ √ √ [4γ +(µ+1− µ+3 µ−1)e−2ν +γ 2 (µ+1+ µ+3 µ−1)e2ν ]B 2 (γ µ + γ + e−2ν + γ 2 e2ν )B 2 R¯ = , (1 − µ)(2γ − e−2ν − γ 2 e2ν )

U¯ =

µ(e−ν

−

γ eν )[(1

−µ+

√

2γ (µ + 3)(µ − 1) . √ √ √ µ + 3 µ − 1)e−ν − γ (1 − µ − µ + 3 µ − 1)eν ] (41)

Combining (26) and (41), we obtain an explicit solution of the (2+1)-dimensional breaking soliton system (3), q(x, y, t) = e2

R

) + (µ − 1)(µ + 1 − τ )e−2ν + γ 2 (µ − 1)(µ + 1 + τ )e2ν , [4γ + (µ + 1 − τ )e−2ν + γ 2 (µ + 1 + τ )e2ν ]B 2

f (y)dy 2γ (1 − µ

2

+ γ + e−2ν + γ 2 e2ν )B 2 , (1 − µ)(2γ − e−2ν − γ 2 e2ν ) 2γ (µ + 3)(µ − 1) u(x, y, t) = f (y) + , µ(e−ν − γ eν )[(1 − µ + τ )e−ν − γ (1 − µ − τ )eν ] r(x, y, t) = e−2

where ν=

√

R

f (y)dy (γ µ

√ µ + 3 µ − 1(µ(x − t) + t − y) 2µ

and τ =

(42)

p p µ + 3 µ − 1.

Because of the arbitrary function f (y), the solution that we present above may be more meaningful under special conditions. Then we consider the evolutions of the explicit solution (42) of Eq. (3). In the case of f (y) = cos y and t = 1, B = 3, µ = 6, γ = 2, Figs. 1, 2 and 3 display the evolutions of the solutions q, r, u, respectively. And when x = 1 the graph of u can be seen in Fig. 4.

66

4.

N. LV, D. NIU, X. YUAN and X. QIU

Fig. 1. An explicit solution of q.

Fig. 2. An explicit solution of r.

Fig. 3. An explicit solution of u.

Fig. 4. u in the case of x = 1.

Summary

With the classical Lie group method, we sought the symmetry algebras of the nonisospectral Lax pair for Eq. (3). We obtained these symmetries by considering the spectral parameter as an additional field except for the field q, r, u and the eigenfunctions 81 , 82 . Reducing both their equations and the Lax pair by the obtained symmetries, we got four interesting reductions. Since the reduced equations and their Lax pairs are much simpler than the original ones, it is easy to obtain some group-invariant solutions of the (2 + 1)-dimensional breaking soliton system. By one of the reduced Lax pairs, we presented an explicit solution of Eq. (3) via Darboux transformation.

SYMMETRY REDUCTIONS OF A NONISOSPECTRAL LAX PAIR. . .

67

Acknowledgements The work is supported by National Natural Science Foundation of China (No. 11326162), General Program of Department of Education of Liaoning Province (No. L2013507), and the Fundamental Research Funds for the Central Universities (Nos. DC201502050203, DC201502050302, DC201502050403). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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