Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

Xiaorui Hu; Yong Chen

doi:10.1515/zna-2015-0254

Publicly Available Published by De Gruyter August 4, 2015

Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

Xiaorui Hu and Yong Chen

From the journal Zeitschrift für Naturforschung A

https://doi.org/10.1515/zna-2015-0254

Abstract

Applying the truncated Painlevé expansion to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system, some Bäcklund transformations (BTs) including auto BT and non-auto BT are obtained. The auto BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion and the related nonlocal symmetry group is presented with the help of the localization procedure. Further, it is shown that the ANNV system has a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, some complex interaction solutions between soliton and arbitrary other seed waves of the ANNV system are readily constructed, such as bight-dark soliton solution, dark-dark soliton solution, soliton-cnoidal interaction solutions, solitoff solutions and so on.

Keywords: Bäcklund Transformation; Exact Interaction Solutions; Nonlocal Symmetry; Painlevé Expansion

1 Introduction

To find exact solutions of nonlinear systems is a difficult and tedious but very important and meaningful work. One knows that the symmetry method established by Lie and developed by Olver [1] and Bluman and Kumei [2] is very effective for constructing explicit solutions of both integrable and nonintegrable systems. Because the procedure for computing Lie point symmetry is standard and thorough, more and more attention is paid to the nonlocal symmetry which is known to be difficult found and applied. In latter studies, starting from the Darboux transformation (DT) [3–7] and Bäcklund transformation (BT) [8], kinds of nonlocal symmetries are obtained and successfully localized to construct abundant novel solutions, such as the exact interaction solutions amongst solitons and other complicated waves. In Ref. [9], Bluman and Yang introduce a new and complementary method for constructing nonlocally related PDE systems, which is on the basis of each point symmetry.

To study the integrability of nonlinear systems, Painlevé analysis is also one of the best approaches. Recently Lou [10, 11] finds that the residues with respect to the singular manifold of the truncated Painlevé expansions, which are ignored for a long time, may be just the nonlocal symmetries of the original system. The localization of this type of nonlocal symmetry, which is called residual symmetry, seems easily performed than that coming from DT and BT. Moreover, based on the truncated Painlevé expansion, a consistent Riccati expansion (CRE) method [12] is proposed to identify CRE solvable systems. It is found that the CRE method is valid to find exact solutions of nonlinear systems and provide strong signals to clarify possible integrable models. The author [12] points that although not all the integrable systems are CRE solvable, it is strongly indicated that the CRE solvable systems are integrable. It is clear that various integrable systems are CRE solvable [12], such as the Korteweg de-Vries (KdV) equation, the fifth-order KdV equation, the Sawada–Kortera eqquation, the Kaup–Kupershmidt equation, the Kadomtsev–Petviashvili equation, the Boussinesq equation, the AKNS (Ablowitz–Kaup–Newell–Segur) system, the Sine–Gordon equation, the modified asymmetric Veselov–Novikov equation, the dispersive water wave equation and the Burgers equation.

In this article, we concentrate on investigating the residual symmetry and the CRE solvability of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system [also named the (2+1)-dimensional KdV equation or Boiti–Leon–Manna–Pempinelli equation]:

(1)

ut + uxxx − 3uvx − 3vux = 0,ux = vy, (1)

which was firstly derived by Boiti et al. [13] using the idea of the weak Lax pair and may be considered as a model for an incompressible fluid where u and v are the components of the (dimensionless) velocity [14]. The spectral transformation for this system has been investigated in Refs. [13, 15]. The nonclassical symmetries, Painlevé property and similarity solutions of the system have been studied by Clarkson and Mansfield [16]. This system has also been considered in [17] as a generalization to (2+1)-dimensions of the results from Hirota and Satsuma [18]. Lou and Hu [19] reobtained this system from the inner parameter-dependent symmetry constraint of the KP equation and found it to be an asymmetric part of the Nizhnik–Novikov–Veselov equation. The conditional similarity reductions are studied in [20] and abundant localized excitations such as dromion solutions [21] and ring solitons [22] are presented by the variable separation approach.

This article is organized as follows. In Section 2, two types of BT and the nonlocal symmetry of the ANNV system are obtained by the Painlevé expansion approach. Then the nonlocal symmetry is localized by introducing another four dependent variables and the corresponding finite group transformation is found. In Section 3, the CRE property and the CTE (which is a special form of CRE) property for the ANNV system are investigated. Based on the CTE method, luxuriant explicit solutions including bight–dark soliton, dark–dark soliton, soliton–cnoidal waves and solitoff solutions are constructed. The last section contains a summary and discussion.

2 Nonlocal Symmetry from the Painlevé Expansion

2.1 BTs and Nonlocal Symmetry from the Painlevé Expansion

For the (2+1)-dimensional ANNV system (1), there exists a truncated Painlevé expansion

(2)

u = u0 + u1ϕ + u2ϕ2, v = v0 + v1ϕ + v2ϕ2, (2)

with u₀, u₁, u₂, v₀, v₁, v₂, ϕ being the functions of x, y and t.

Substituting (2) into (1), we have

(3)

u0t + u0xxx − 3u0xv0 − 3u0v0x + (u1t + u1xxx − 3u1xv0 − 3u0xv1 − 3u1v0x − 3u0v1x)1ϕ + [(3v1ϕx − 3v2x)u0+(3u1ϕx − 3u2x)v0 − 3u0xv2 − 3u2v0x − (ϕt + ϕxxx + 3v1x)u1 − 3(v1 + ϕxx)u1x − 3ϕxu1xx + u2t + u2xxx]1ϕ2−[3(u1v2 + u2v1)x + 2u2ϕt − 6(u0v2 + u2v0 + u1v1 − u2xx + u1ϕxx)ϕx − 6u1x(ϕx)2 + 6u2xϕxx]1ϕ3−3[(u2v2)x − 3(u1v2 + u2v1)ϕx − 6u2x(ϕx)2 + 2u1(ϕx)3 − 6u2ϕxϕxx]1ϕ4 + 12u2ϕx[v2 − 2(ϕx)2]1ϕ5 = 0, (3)

and

(4)

u0x − v0y + (u1x − v1y)1ϕ + (u2x − v2y − u1ϕx + v1ϕy)1ϕ2 − 2(u2ϕx − v2ϕy)1ϕ3 = 0. (4)

By vanishing all the coefficients of different powers of 1ϕ in (3) and (4), we have

(5)

u1 = − 2ϕxy, v1 = − 2ϕxx, u2 = 2ϕxϕy, v2 = 2(ϕx)2. (5)

(6)

u0 = [H − C − 13HyKx + 12(ϕxxϕx)2]K + ϕxxϕxKx + Kxx, v0 = H3 + 13HyKx + 12(ϕxxϕx)2, (6)

and the function ϕ satisfies

(7)

(HyKx)y + K(HyKx)x + 3Cy − Hy = 0, (7)

(8)

(HyKx)t + (HyKx)xxx − 3(CHyKx)x − [(HyKx)2]x + (HyKx)Hx + 2H(HyKx)x = 0. (8)

with

(9)

C ≡ ϕtϕx, K ≡ ϕyϕx, S ≡ ϕxxxϕx − 32(ϕxxϕx)2, H ≡ S + C. (9)

Here C, K and S are the usual Schwartzian variables, which keep the Möbious (conformal) invariance property

(10)

ϕ → a + bϕc + dϕ(ad ≠ bc). (10)

Due to the aforementioned Möbious invariance, (7) and (8) possess the point symmetries in the form of

(11)

σϕ = κ0 + κ1ϕ + κ2ϕ2 (11)

with κ₀, κ₁ and κ₂ being constants. Furthermore, one can verify that (7) and (8) are consistent because H_yyt=H_yty is identically satisfied.

On the one hand, from the standard truncated Painlevé expansion (2), we have the following BT theorems of (1).

Theorem 1(a) (auto-BT theorem). If the function ϕ satisfies (7) and (8), then

(12)

u = −2(ln ϕ)xy + u0, v = − 2(ln ϕ)xx + v0, (12)

is an auto-BT between the solutions {u, v} and {u₀, v₀} of the ANNV system (1).

Theorem 1(b) (non-auto-BT theorem). If the function ϕ satisfies (7) and (8), then the formula (6) is a non-auto BT between the ϕ and the solution {u₀, v₀} of the ANNV system (1).

On the other hand, one knows that if σ^u and σ^ν are the symmetry of u and v in (1) respectively, it requires

(13)

σtu + σxxxu − 3σxuv − 3uxσv − 3σuvx − 3uσxv = 0, (13)

(14)

σxu = σyv. (14)

Comparing the coefficients of 1ϕ in (3) and (4) with (13) and (14), one can easily see that if u₀ and v₀ satisfy (1), u₁ and v₁ are just the symmetry with respect to u₀ and v₀, which are called “residual symmetries”. Hence, the ANNV system possesses a symmetry

(15)

σu = − 2ϕxy, σv = − 2ϕxx, (15)

with

(16)

u = [H − C − 13HyKx + 12(ϕxxϕx)2]K + ϕxxϕxKx + Kxx, v = H3 + 13HyKx + 12(ϕxxϕx)2. (16)

By solving ϕ from (16), one knows that the residual symmetries (15) of u and v are nonlocal.

2.2 Localization of the Nonlocal Symmetry

To find out the group transformation

{u, v} → {u¯(ϵ), v¯(ϵ)} = {u, v} + ϵ{σu, σv}

corresponding to the nonlocal symmetry (15), we have to solve the following initial value problem

(17)

du¯(ϵ)dϵ = − 2ϕ¯xy(ϵ), dv¯(ϵ)dϵ = − 2ϕ¯xx(ϵ),u¯(ϵ)|ϵ = 0 = u, v¯(ϵ)|ϵ = 0 = v. (17)

with ϵ being the infinitesimal parameter.

However, it is difficult to solve (17) about u̅ and v̅ due to the intrusion of the function ϕ and its differentiations. For this, we need to prolong (1) such that the nonlocal symmetry (15) becomes Lie point symmetry of a larger system. To realize this localization, we introduce four new dependent variables g₁, g₂, h₁ and h₂ by

(18)

g1 = ϕx, g2 = ϕy, h1 = g1x, h2 = g2x. (18)

Now one can easily verify that the nonlocal symmetry (15) of the original system (1) becomes a Lie point symmetry of the prolonged system including (1), (16) and (18):

(19)

σu = − 2h2, σv = − 2h1, σg1 = − 2ϕg1 σg2 = − 2ϕg2,σh1 = − 2(g12 + ϕh1), σh2 = − 2(g1g2 + ϕh2), σϕ = − ϕ2. (19)

Correspondingly, the initial value problem (17) is changed as

(20)

Then the solution of the initial value problem (17) leads to the following group theorem for the enlarged system.

Theorem 2 (group). If {u, v, g₁, g₂, h₁, h₂, ϕ} is a solution of the prolonged system (1), (16) and (18), so is {u̅, v̅, g̅₁, g̅₂, h̅₁, h̅₂, ϕ̅} with

(21)

u¯(ϵ) = u − 2ϵh21 + ϵϕ + 2ϵ2g1g2(1 + ϵϕ)2, v¯(ϵ) = v − 2ϵh11 + ϵϕ + 2ϵ2g12(1 + ϵϕ)2, g¯1(ϵ) = g1(1 + ϵϕ)2, g¯2(ϵ) = g2(1 + ϵϕ)2, h¯1(ϵ) = h1(1 + ϵϕ)2 − 2ϵg12(1 + ϵϕ)3, ϕ¯(ϵ) = ϕ1 + ϵϕ, h¯2(ϵ) = h2(1 + ϵϕ)2 − 2ϵg1g2(1 + ϵϕ)3. (21)

Theorem 2 shows us an interesting result that the nonlocal symmetry (15) coming from the truncated Painlevé expansion is just the infinitesimal form of the group (21). Furthermore, we notice that the BT theorem 1(a) and group theorem 2 are equivalent because the singularity manifold equations (1), (16) and (18) are form invariant under the transformation

1 + ϵϕ → ϕ (with ϵg1 → ϕx, ϵg2 → ϕy, ϵh1 → ϕxx, ϵh2 → ϕxy).

3 CRE Solvable

3.1 CRE Solvable

For the ANNV system, we aim to look for the following possible truncated Painlevé expansion solution

(22)

u = u0 + u1R(w) + u2R2(w), v = v0 + v1R(w) + v2R2(w), (22)

where R(w) is a solution of the Riccati equation

(23)

Rw = r0 + r1R + r2R2, (23)

and r₀, r₁, r₂ are arbitrary constants. It requires all the expansion coefficient functions u_i and v_i be determined by vanishing all the coefficients of the like powers of R(w) after substituting (22) with (23) into the ANNV system. We write down the final results

(24)

u2 = 2r22wxwy, v2 = 2r22wx2, u1 = 2r2(r1wxwy + wxy), v1 = 2r2(r1wx2 + wxx),u0K1x = − 16δwxK1(wxK1x − 2wxxK1) − 13K1(S1y + K1t + K1C1x) + [12r12wx2K1 + r1(wxK1)x + K1xx + K1S1+ 13C1K1 + wxxwxK1x + 12K1(wxxwx)2]K1x,v0K1x = − 16δwx(2wxxK1 + 3wxK1x) + 13(S1y + K1t + K1C1x) + [12r12wx2 + r1wxx + 13S1 + 12(wxxwx)2]K1x (24)

with

(25)

δ ≡ r12 − 4r0r2, C1 ≡ wtwx, K1 ≡ wywx, S1 ≡ wxxxwx − 32(wxxwx)2, H1 ≡ S1 + C1, (25)

and the function w should satisfy

(26)

K1x(H1yy + K1H1xy + 3C1yK1x) − (K1xy + K1K1xx + K1x2)H1y − δ[wx2K1x(K1K1xx + K1x2 + 2K12S1)− wxwxx(K12K1xx + K1K1xy − K1xK1y − 7K1Kx2) + 5wxx2K12K1x] = 0, (26)

(27)

4wx2{K1x2[K1xH1yt + K1xH1xxxy − 3K1xxH1xxy] + K1xH1xy[(8H1 − 9C1)K1x2 + 3(K1S1x + C1y)K1x + 6K1xx2]+ 8K1xK1xxH1y2 + H1y[(4H1x − 6C1x)K1x3 + (K1H1xx − 6H1xy − 12K1xxS1)K1x2 − 6K1xK1xx(K1S1x + C1y)− 6K1xx3]} − 2δ2wx3K1x{2wx3K1K1x2S1 + 2wx2wxxK1x(2K12S1 − K1K1xx + 2K1x2) + wxwxx2K1(15K1x2 − 4K1K1xx)+ 10wxx3K12K1x} − δ{4wx4K1x[6K1x3S1x − (S1xy − 15K1S12 + C1xy − K1H1xx)Kx2 − 3K1x2K1xxS1 − (3K1S1x− H1y)K1xK1xx + K1K1xS1(5S1y + 2C1y − 3K1S1x) + 6K1K1xx2S1] + wx3wxx[180S1K1x4 + (60K1S1x − 28S1y− 16C1y)K1x3 + (12K1xx2 − 132K1K1xxS1 + 4K12H1xx − 12K1H1xy)Kx2 + 8K1K1xK1xx(5S1y − 3K1S1x + 2C1y)− 24K1K1xx3] − 10wx2wxx2K1x[3(K1xx − 8K1S1)K1x2 + K1K1x(5S1y − 3K1S1x + 2C1y) − 6K1K1xx2]+ 30wxwxx3K1x2(5K1x2 − 3K1K1xx) + 105wxx4K1K1x3} = 0. (27)

One can see that (26) and (27) are more complicated than (7) and (8). Luckily (26) and (27) are also consistent because H_1yyt=H_1yty is identically satisfied. At this point, we call the (2+1)-dimensional ANNV system be CRE solvable.

3.2 CTE solvable and BT

Obviously, the Riccati equation (23) has a special solution

(28)

R(w) = tanh(w), (28)

while the truncated expansion solution (22) becomes

(29)

u = u′0 + u′1tanh(w) + u′2tanh2(w), v = v′0 + v′1tanh(w) + v′2tanh2(w), (29)

where u′0, u′1, u′2, v′0, v′1, v′2 and w are determined by (24), (26) and (27) with r₀=1, r₁=0 and r₂=– 1.

In view of the consistent tanh-function expansion (CTE) (29), which is a special case of CRE, the ANNV system is called CTE solvable. It is quite clear that a CRE solvable system must be CTE solvable, and vice versa. If a system is CTE solvable, one may directly construct some important explicit solutions, such as the soliton and the interactions between a soliton and a cnoidal wave. To leave this clear, we write down the following non-auto Bäcklund transformation which comes from the aforementioned CTE theorem and use it to find exact solutions.

Theorem 3 (BT). If w is a solution of (26) and (27) with r₀=1, r₁=0 and r₂=– 1, then

(30)

u = u0 − 2wxytanh(w) + 2wxwytanh2(w), v = v0 − 2wxxtanh(w) + 2wx2tanh2(w) (30)

is a solution of the ANNV system (1), where {u₀, v₀} is given by (24) with δ=4 and r₁=0.

3.3 Exact solutions from Theorem 3

Obviously, it seems more difficult to solve (26) and (27). However, it is interesting that one can derive some nontrivial solutions of the ANNV system from some quite trivial solutions of (26) and (27) by means of theorem 3. Here are some interesting examples.

Example 1.Soliton solutions. In theorem 3, we take a quite trivial straight-line solution for w, saying

(31)

w = kx + ly + dt + d0 (31)

with k, l, d, d₀ being arbitrary constants. Substituting (30) with the line solution (31) into (1) yields the following soliton solution

(32)

u = 2kltanh2(kx + ly + dt + d0) − 2kρ32ltanh2(− kρ3lx + ρ3y + ρ2t + ρ1) + 4kρ323l + l(d − 8k3)6k2 + ρ2l26ρ3k2, (32)

(33)

v = 2k2tanh2(kx + ly + dt + d0) + 2k2ρ32l2tanh2(− kρ3lx + ρ3y + ρ2t + ρ1) − 4k2ρ323l2 + d − 8k36k − ρ2l6ρ3k. (33)

Figure 1 displays a special bright–dark soliton solution for u and a dark–dark soliton solution for v shown by (32) and (33) at t=1 with the parameter selected as

Figure 1:

Two-soliton solutions of the ANNV system shown by (32) and (33) at t=1 with the parameter selections (34): (a–b) the bright–dark soliton for u and its corresponding density plot; (c–d) the dark–dark soliton for v and the corresponding density plot.

(34)

{k, l, d, d0, ρ1, ρ2, ρ3} = {1, −1, 1, 0, 0, 1, 1}. (34)

Example 2.Soliton-cnoidal waves. In Ref. [12], it is shown that many CTE solvable systems possess the interaction solutions between solitons and cnoidal periodic waves. Here, to find this type of interaction solutions, let

(35)

w = k1x + l1y + d1t + W(ξ), (ξ ≡ k2x + l2y + d2t) (35)

where W₁ ≡ W₁(ξ)=W_ξ satisfy

(36)

W1ξ2 = a0 + a1W1 + a2W12 + a3W13 + a4W14 (36)

with a₀, a₁, a₂, a₃, a₄ being constants. Substituting (35) with (36) into theorem 3, it requires

(37)

a0 = 5k1l2 + k2l16k2l2a1 − k1(2k1l2 + k2l1)3k22l2a2 + k12(k1l2 + k2l1)2k23l2a3 − 4k143k24 − 8l1k133l2k23 + d2k123k25 − (l1d2 + l2d1)k13l2k24 + d1l13l2k23 (37)

and

(38)

a4 = 4. (38)

Then the system (1) has the explicit solution expressed as

(39)

u = 2(k1 + k2W1)(l1 + l2W1)tanh2(k1x + l1y + d1t + W) − 2k2l2W1ξtanh(k1x + l1y + d1t + W)+ 2k2l2W12 − (2k1l2 + 2k2l1 − 12k2l2a3)W1 − k2l22(k2a1 − 2k1a2)3(k1l2 − k2l1) − k1l2a3(k1l2 + k2l1)2(k1l2 − k2l1)+ 4k13k2l22 + 6k12k22l1l2 − k1(l22d2 − 6k23l12) + k2l22d13k22(k1l2 − k2l1).v = 2(k1 + k2W1)2tanh2(k1x + l1y + d1t + W) − 2k22W1ξtanh(k1x + l1y + d1t + W)+ 2k22W12 + 12k2(k2a3 − 8k1)W1 + k23l2a13(k1l2 − k2l1) − k2(k1l2 + k2l1)(2k2a2 − 3k1a3)6(k1l2 − k2l1)− 10k13k2l2 + 6k12k22l1 + k2(l1d2 + l2d1) − 2k1l2d23k2(k1l2 − k2l1). (39)

It is clear that (36) has abundant explicit solutions in terms of Jacobi elliptic functions. Hence, the explicit solution (39) exhibits the interactions between a soliton and a cnoidal periodic wave.

To show this soliton+cnoidal wave solutions more intuitively, we just take a a simple solution of (36) as

(40)

W1 = − a316 + nm2sn(mξ, n), (40)

which leads (39) to a dark soliton residing on a cnoidal wave background, saying

(41)

u = k2128(− a3 + 8mnsn(mξ, n))(16l1 − a3l2 + 8l2mnsn(mξ, n))tanh2[a316ξ − l1y − d1t + k0 − mn2∫ξ0ξsn(mY, n)dY]+ k2l2m2ncn(mξ, n)dn(mξ, n)tanh[a316ξ − l1y − d1t + k0 − mn2∫ξ0ξsn(mY, n)dY] + 12k2l2m2n2sn2(mξ, n)+ 18k2mn(l2a3 − 8l1)sn(mξ, n) + 13k2l22a1 + 18k2l12a3 − 3128k2l1l2a32 − l22d13k22, (41)

(42)

v = 1128k22(− a3 + 8mnsn(mξ, n))2tanh2[a316ξ − l1y − d1t + k0 − mn2∫ξ0ξsn(mY, n)dY]+ k22m2ncn(mξ, n)dn(mξ, n)tanh[a316ξ − l1y − d1t + k0 − mn2∫ξ0ξsn(mY, n)dY] + 12k22m2n2sn(mξ, n)2+ 18k22a3mnsn(mξ, n) − k22l2a13l1 + 13k22a2 − 3128k22a32 + l2d13k2l1 + d23k2. (42)

The parameters in (41) and (42) should satisfy

(43)

a1 = 1256a33 − 18m2(1 + n2)a3, a2 = 332a32 − m2(1 + n2),d1 = 3k23l2a3416384l1 − k23a33512 − 3(1 + n2)256l1k23l2m2a32 + 116k23m2(1 + n2)a3 + 3k23l2m4n24l1. (43)

In Figure 2, we plot a dark soliton coupled to a cnoidal wave background expressed by (41) and (42) with

Figure 2:

Plot of the dark soliton on a cnoidal wave background expressed by (41) and (42) at t=0 with the parameter selections (44) of the ANNV system: (a–c) one-dimensional image for u at x=0, y=1 and its density plot, respectively; (d–f) one-dimensional image for v at x=0, y=1 and its density plot, respectively.

(44)

{a3, l1, l2, d2, k2, k0, ξ0, m, n} = {0, 1, 1, 0, 1, 0, 0, 1, 12}. (44)

This kind of solution describing solitons moving on a cnoidal wave background instead of on the plane continuous wave background is very important in the real world and can be easily applicable to the analysis of physically interesting processes; see Refs. [4, 5].

Example 3.Solitoff solutions. In theorem 3, if we let

(45)

w = α1x + β1y + γ1t + P(X), (X ≡ α2x + β2y + γ2t) (45)

where P₁ ≡ P₁(X)=P_X satisfy

(46)

P1X = b0 + b1P1 − 2P12, (46)

one can obtain the following explicit solution of the ANNV system (1)

(47)

u = 18(4β1 + b1β2 + Aβ2T)(4α1 + b1α2 + Aα2T)tanh2[α1x + β1y + γ1t + c2 + b14X − 12ln sechA(X + c1)2]+ 14α2β2A2(T2 − 1)tanh[α1x + β1y + γ1t + c2 + b14X − 12ln sechA(X + c1)2] + 18α2β2A2T2− 2α1β2 + 2α2β1 + b1α2β24AT − 18(α1β2 − α2β1)2[8α23β23b02 − 32α1β1α22β22b0 − 8α22β22(α1β2 + α2β1)b0b1+ α2β2(3α22β12 + 2α1α2β1β2 + 3α12β22)b12 + 4(α1β2 + α2β1)3b1 + 16α1β1(α12β22 + α2β12)], (47)

(48)

v = 18(4α1 + b1α2 + Aα2T)2tanh2[α1x + β1y + γ1t + c2 + b14X − 12ln sechA(X + c1)2]+ 14α22A2(T2 − 1)tanh[α1x + β1y + γ1t + c2 + b14X − 12ln sechA(X + c1)2] + γ13α1 + 18α22A2T2− 14α2(4α1 + b1α2)AT + 124α1(α1β2 − α2β1)2[24α24β2(2α1β2 − α2β1)b02 − 96α13α22β22b0− 8α23(8α12β22 − α22β12 − α1α2β1β2)b0b1 + 32α1α22(5α12β22 − 3α22β12 + 6α1α2β1β2)b12+ 24α12α2(α12β22 − α22β12 + 4α1α2β1β2)b1 − 16α13(α12β22 + α22β12 − 8α1α2β1β2)] (48)

with

A = 8b0 + b12, T = tanhA(X + c1)2,γ2 = α2α1[γ1 + (α22b0 − α1α2b1 − 2α12)((3b0β2 − b1β1)α22 − 2α1α2(2β1 + b1β2) − 2β2α12)α1β2 − α2β1].

Figure 3 shows the structures of the two-solitoff solution and three-solitoff solution for the quantity u and v shown by (47) and (48), respectively, with

Figure 3:

(a, b) Special two-solitoff solution for the quantity u and v with the parameter selections (49) at time t=0; (c–d). Three-solitoff solution for the quantity u and v with the parameter selections (50) at time t=0.

(49)

{α1, α2, β1, β2, b0, b1, γ1, c1, c2} = {− 12, 1, 1, 1, 0, 1, 0, 0, 0} (49)

and

(50)

{α1, α2, β1, β2, b0, b1, γ1, c1, c2} = {1, 1, − 1, 1, 0, 1, 0, 0, 0}. (50)

4 Summary and Discussion

In this article, some Bäcklund transformations, the nonlocal symmetry and different types of exact solutions for the ANNV system are presented. On the one hand, applying the Painlevé expansion method to the ANNV system, two BTs including auto BT and non-auto BT are obtained. It is shown that the residual of the truncated Painlevé expansion in the non-auto BT is just a nonlocal symmetry of the ANNV system, which is also called residual symmetry. Because the residual symmetry is closely related to the Schwartzian variable, we can readily localize it to Lie point symmetries by introducing other auxiliary dependent variables. By enlarging the ANNV system to a prolonged system of eight equations, the corresponding transformation group of the nonlocal symmetry is found, which manifests that the residual symmetry is just the infinitesimal form (or the generator) of the group.

On the other hand, the ANNV system is proved integrable under the mean of consistent Riccati expansion (CRE). As a special form of CRE, the consistent tanh-function expansion (CTE) method for the ANNV system leads to another non-auto BT theorem from which various exact explicit solutions are obtained, such as bight–dark soliton solution, dark-dark soliton solution, soliton–cnoidal interaction solutions, solitoff solutions. The CTE integrability of the ANNV system leaves it clear that the interaction solutions between soliton and arbitrary other seed waves can be constructed simply by plus a straight line solution on a general solution in the non-auto BT.

To find interaction solutions between different types of nonlinear excitations which may display new physical applications is an interesting and meaningful work. With regard to the CRE (or CTE) method, it is worthwhile to detect more associated w equations, such as (31), (35) with (36) and (45) with (46). How to combine the truncated Painlevé expansion approach and other methods to construct more residual symmetries and BTs for the ANNV system and other integrable systems, based on which one can obtain more exact solutions, will be further studied in our future researches.

Corresponding author: Xiaorui Hu, Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China, E-mail: lansexiaoer@163.com

Acknowledgments

The project is supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13A010014, the National Natural Science Foundation of China (Grant Nos. 11326164, 11401528, 11435005, 11375090), Global Change Research Program of China (No.2015CB953904), Research Fund for the Doctoral Program of Higher Education of China (No. 20120076110024), Innovative Research Team Program of the National Natural Science Foundation of China (Grant No. 61321064), Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No. ZF1213.

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Received: 2015-6-4

Accepted: 2015-7-9

Published Online: 2015-8-4

Published in Print: 2015-9-1

Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

Abstract

1 Introduction

2 Nonlocal Symmetry from the Painlevé Expansion

2.1 BTs and Nonlocal Symmetry from the Painlevé Expansion

2.2 Localization of the Nonlocal Symmetry

3 CRE Solvable

3.1 CRE Solvable

3.2 CTE solvable and BT

3.3 Exact solutions from Theorem 3

4 Summary and Discussion

Acknowledgments

References

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