Painlevé Integrability and a New Exact Solution of the Multi-Component Sasa-Satsuma Equation

1 Introduction

In this article, we consider the following multi-component Sasa-Satsuma (MSS) equation

which is a generalization of the SS equation introduced recently [1]

pt + pxxx + 3pux + 6pxu = 0, uy = (|p|2)x.

This equation is usually used to describe the propagation of an ultra-short pulse through optical fibres. The integrability of the two-component MSS equation is confirmed via the standard Painlevé analysis, and some special coherent structures with periodic properties have been obtained by means of the local Laurent expansion approach in [2, 3].

In the past four decades, integrable equations have been widely studied and applied in many natural sciences such as biology, chemistry, mathematics, communication, and especially in almost all the physics branches, like condense matter physics, quantum field theory, fluid dynamics, and nonlinear optics. It is well known that integrable equations have nice properties such as Lax-pair, infinite number of conservation laws, infinite number of symmetries, and Painlevé property. One of the most important methods for the investigation of integrable equations is the Painlevé analysis. There are two main ways: one is the Ablowitz–Ramani–Segur (ARS) test [4]; the other is the Weiss–Tabor–Carnevale (WTC) test [5–7]. By using the ARS test, authors have discussed the possible connection between the integrability of partial differential equations (PDEs) and the Painlevé property of ordinary differential equations (ODEs) which are obtained by some exact reductions, Here, an ODE is said to have Painlevé property if all movable singularities of all solutions are poles [4]. However, the main drawback in applying the ARS test to a given PDE is that one has to first obtain all possible reduced ODEs and then determine whether each ODE is of Painlevé type (after allowing for possible transformations). Fortunately, the WTC test can be viewed as a natural extension of the ARS test. We can apply this test directly to a given PDE without having to reduce it to ODEs. Here, a PDE has Painlevé property if its solutions are single-valued about a movable singularity manifold. If one only needs to prove the Painlevé property of a model, one may use the Kruskal simplification for the WTC approach. In Section 2, we combine the standard WTC approach and the Kruskal simplification to study the Painlevé property of the MSS equation (1)–(2).

The multi-linear variable separation (MLVS) approach [8–15] is an important and effective method to construct exact solutions for nonlinear evolution equations, which has been used to solve a diversity of (2+1)-dimensional integrable equations including the DS equation, the NNV equation, the dispersive long wave equation, the Broer–Kaup–Kupershmidt (BKK) equation, the general (M+N)-component AKNS system, the symmetric sine–Gordon equation, the differential-difference (DΔ) asymmetric NNV equation and the special Toda lattice. One of the most important things may be that rich coherent structures such as dromions, lumps, ring solitons, breathers, instantons, peakons, compactons, and foldons can be constructed, because MLVS solutions (or a universal formula) include some lower-dimensional arbitrary functions. In Section 3, we show whether the MLVS approach can be applied to (1)–(2). The last section contains the conclusions and discussion.

2 Painlevé Integrability of the MSS Equation

According to the standard WTC approach, if (1)–(2) can pass the Painlevé test, all the possible solutions can be written as

with 3N arbitrary functions among p_ik, u_k in addition to f, where a_i and b are negative integers to be determined. Namely, all the solutions of (1)–(2) are single valued about an arbitrary movable singularity manifold f.

To determine the constants a_i and b, one may use the standard leading order analysis. Substituting p_i=p_i0f^ai, u=u₀f^b into (1)–(2) and balancing the most dominant terms, we have

Then, substituting the Laurent expansion of the solutions (20)–(21) with (5)–(6) into (1)–(2) and balancing the coefficients of the powers (f^{k− 4}, f^{k− 3}), respectively, for different k, we could get the recursion relations to fix the functions p_ik and u_k as follows:

where

E = Afx3(k − 2)k(k − 4),Fi = − (k − 2)fx(∑m = 1iamipm0 + ∑m = i + 1Naimpm0),  i = 1, 2, ⋯, N,

Pik = −(pi(k − 3), t + (k − 3)pi(k − 2)ft + Api(k − 3), xxx+ 3A(k − 3)pi(k − 2), xxfx + 3A(k − 3)pi(k − 2), xfxx+ A(k − 3)pi(k − 2)fxxx + 3(k − 2)(k − 3)Api(k − 1), xfxfxx+ 2Aβ∑m = 1k − 1(m − 1)pimuk − mfx + Aβ∑m = 1k − 1(k − m − 2)pimuk − mfx+ 2Aβ∑m = 0k − 1pim, xuk − 1 − m + Aβ∑m = 0k − 1pimuk − 1 − m, x), i = 1, 2, ⋯, N,Uk = −uk − 1, y + ∑i = 1N∑j = iNaij(∑m = 1k − 1(k − 2)pimpj(k − m)fx+ ∑m = 0k − 1(pimpj(k − 1 − m), x)x).

with p_ik=u_k=0 for k<0. It is easy to know that if the determinant

Δ ≡ detJ = AN(k + 1)kN − 1(k − 2)N(k − 3)(k − 4)Nfx3Nfy

of matrix J in (7) is not equal to zero, the functions p_ik and u_k can be solved uniquely by Cramer’s rule. Hence, we can obtain the resonances

k = −1, 0, 0, ⋯, 0︸N − 1, 2, 2, ⋯, 2︸N, 3, 4, 4, ⋯, 4︸N.

The resonance at k=−1 corresponds to the arbitrary singularity manifold f. If the MSS equations (1) and (2) have Painlevé property, we require that 3N resonance conditions totally at k=0 (N − 1 resonance conditions), at k=2 (N resonance conditions), at k=3 (one resonance conditions), and at k=4 (N resonance conditions), are satisfied identically so that the general series expansion (20)–(21) includes 3N arbitrary functions among p_ik and u_k. From (6), we know that the N − 1 resonance conditions at k=0 are satisfied identically because only one condition exists among N functions p_i0. For the computational complexity, we use the Kruskal’s simplification

f = x + ϕ(y, t),  pik = pik(y, t),  uk = uk(y, t)

without loss of the generality to prove the Painlevé property, where ϕ (y, t) is an arbitrary function. In this case, the resonance conditions are simplified to

and the result at k=0 becomes

u0 = −32β,  ∑i = 1N∑j = iNaijpi0pj0 = −3fy2β

with N − 1 arbitrary functions.

For k=1, (8)–(9) become to

3Api1 − 3Aβpi0u1 = 0,  i = 1, 2, ⋯, N,− u1ϕy = −∑i = 1N∑j = iNaij(pi0pj1 + pi1pj0).

Through the Cramer’s rule, we obtain

u1 = 0,  pi1 = 0,  i = 1, 2, ⋯, N.

For k=2, we can get

u2 = −ϕt2Aβ

and the functions p_i2, (i=1, 2, …, N) are arbitrary. In other words, N resonance conditions satisfy identically at k=2. For k=3, by using (8)–(9), we have

pi0, t − 3Api3 − Aβpi0u3 = 0,  i = 1, 2, ⋯, N,u2, y + u3ϕy = ∑i = 1N∑j = iNaij(pi0pj3 + pi3pj0).

This means that

pi3 = pi0, t − Aβpi0u33A,  i = 1, 2, ⋯, N

and the function u₃ is arbitrary. For k=4, we arrive at the following system of simplified equations

pi2ϕt + 2Aβpi2u2 = 0,u3, y + 2u4ϕy = 2∑i = 1N∑j = iNaij(pi0pj4 + pi2pj2 + pi4pj0).

After calculation, we can see that u₄ can be expressed as

u4 = 12ϕy(2∑i = 1N∑j = iNaij(pi0pj4 + pi2pj2 + pi4pj0) − u3, y)

and p_i4, (i=1, 2, …, N) are arbitrary. It means that N resonance conditions satisfy identically at k=4. Therefore the Painlevé property of the MSS equation (1)–(2) is proved, namely, the MSS equation (1)–(2) has Painlevé integrability.

3 Multi-Linear Variable Separation Approach

In this section, we construct a MLVS solution which is based on the corresponding Bäcklund transformation for the MSS equation (1)–(2). It is well known that Bäcklund transformation can be obtained by using the truncated Painlevé expansion usually. See (20)–(21) with (5)–(6). So we begin to discuss the problem from the following Bäcklund transformation:

where {0, 0, ⋯, 0︸N, u2(x, t)} is a seed solution of (1)–(2) and p_i0 ≡ p_i0(x, y, t), f ≡ f(x, y, t) need be determined. To consider furthermore, by taking the prior variable separation ansatz

f = F(x, t) + G(y, t),

we have

∑i = 1N∑j = iNaijpi0pj0 = −32βFxGy,

from (2). So, we can let

pi0 = Hi(y)FxGy,  i = 1, 2, ⋯, N,

and this restrict equation is reduced to

Then, a direct computation gives

−2Ft − 2Gt − 4Aβu2Fx + 32AFxx2Fx − 2AFxxx + (F + G)[2Aβu2x + FxtFx+ GytGy + 2Aβu2FxxFx + 34A(FxxxFx)3− 32AFxxFxxxFx2 + AF4xFx] = 0

from (1). For solving this equation, we change it to the following form,

[−2 + F + GGy∂y]Gt + [−2 + F + GFx∂x] (Ft + 2Aβu2Fx + AFxxx − 34AFxx2Fx) = 0.

Because F, u₂ are only functions of {x, t} and G is a function of {y, t}, this equation can be solved by the following variable separated equations:

with the arbitrary functions c₀ ≡ c₀(t), c₁ ≡ c₁(t) and c₂ ≡ c₂(t). To solve the Riccati equation (13) is also quite easy because of the arbitrariness of the functions c₀, c₁, and c₂. The result reads

where C₀ ≡ C₀(t), C₁ ≡ C₁(t), C₂ ≡ C₂(t), and K(y) can all be considered as arbitrary functions of the indicated variables while c₀, c₁, and c₂ are related to C₀, C₁, and C₂ by

c0 = 1C0(C0C′2 − C2C′0 − C22C′1), c1 = 1C0(C′0 + 2C2C′1), c2 = −C′1C0.

For (14), we may treat it alternatively. Namely, we consider F is an arbitrary function while the function u₂ can be determined (14), that is

u2(x, t) = −c0 + c1F − c2F2 − Ft2AβFx − 12βFxxxFx + 38β(FxxFx)2.

Therefore, the MSS equation (1)–(2) has an exact solution

where F(x, t) is an arbitrary function, G(y, t) is determined by (15) and H_i(y), (i=1, 2, …, N) need to satisfy the restrict equation (12).

For the quantity U = ∑i = 1N∑j = iNaijpipj, we have

∑i = 1N∑j = iNaijpipj = −32βFxGy(F + G)2

with F being an arbitrary function of {x, t} and G being (15). It is interesting that this expression is valid for many (2+1)-dimensional integrable equations, such as the DS equation, the NNV equation, the dispersive long wave equation, the Broer–Kaup–Kupershmidt (BKK) equation, the general (M+N)-component AKNS system, and the symmetric sine–Gordon equation. If we select the functions F and G appropriately, we can obtain many kinds of new coherent structures, and some of them are listed as follows.

Since the pioneering work of Camassa and Holm (CH) [16], a special type of (1+1)-dimensional weak solutions has attracted the attention of scientists. These types of solitary waves are called peakons because they are discontinuous at their crest, and the collisions among them are completely elastic. Furthermore, Rosenau and Hyman [17] introduced a class of (1+1)-dimensional solitary waves with compact support (called compacton) in fully nonlinear KdV equation K(m, n) for understanding the role of nonlinear dispersion. Although many soliton equations, such as the CH equation [18] have been extended to (2+1)-dimensions in several ways, one does not know anything on the (2+1)-dimensional peakons and compactons which are localized in all directions. In fact, the entrance of the arbitrary functions F and G in (1) tells that the (2+1)-dimensional peakons and compactons can exist by selecting the arbitrary functions as some suitable piecewise continuous functions [8]. Thus, if setting

where P_i(ξ)=P_i(x + e_i), i=1, 2, …, M are differentiable functions and possess the boundary conditions P_i(±∞)=E_i, i=1, 2, …, M with E_i being constants and

where Q_j, j=1, 2, …, N are differentiable functions and satisfy the conditions Q_jy|_y=y1j=Q_jy|_y=_y2j=0, j=1, 2, …, N, we have a new mixed coherent structure which possesses peakons at x-axis and compactons at y-axis.

In Figures 1–5, we plot the interaction property between two peakon-compacton structures for the quantity U = 1000FxGy(F + G)2 with

Figure 1:

Two peakon-compacton structures at t=−3.

Figure 2:

Interaction at t=−1.

Figure 3:

Interaction at t=0.

Figure 4:

Interaction at t=1.

Figure 5:

Position is exchanged after interaction at t=3.

where |λ| = 12 means the amplitude of the right structure in Figure 1 and if λ>0, this structure is a bright soliton.

In Figure 6, for simplification, we only plot the four peakon-compacton structures for the quantity U = 1000FxGy(F + G)2 with the above F and

Figure 6:

Four peakon-compacton structures at t=−3.

4 Conclusions and Discussion

Painlevé test and the variable separation method are two useful methods in the study of nonlinear evolution equations. In this article, Painlevé integrability of the MSS equation (1)–(2) is confirmed by means of the WTC approach. Then, a MLVS solution (16)–(17) including some lower-dimensional arbitrary functions is constructed by using the MLVS approach. For the physical quantity U = ∑i = 1N∑j = iNaijpipj = −32βFxGy(F + G)2, new coherent structure which possesses peakons at x-axis and compactons at y-axis is illustrated both analytically and graphically.

It is known that the fast diffusion equation ut = (u− 12ux)x has a MLVS solution

u(x, t) = 2(ln[F(x) + G(t)])xt = −2F′(x)G′(t)(F(x) + G(t))2

with constraint conditions

F′ = (c0 + c1F + c2F2 +c3F3)23,G′ = c4(c0 − c1G + c2G2 − c3G3)23,

which is obtained by means of the generalized conditional symmetry approach (or the nonlinear differential constraint method) [19]. Thus, the classification problem of integrable and non-integrable equations admitting MLVS solutions by using the generalized conditional symmetry approach is worthy of study.