Oscillatory Solutions for Lattice Korteweg-de Vries-Type Equations

1 Introduction

Integrable partial difference equations are discrete time and discrete space analogues of integrable partial differential equations, which often admit classical integrable partial differential equations as continuum limits. The difference analogue of the potential Korteweg de-Vires (pKdV) equation was first introduced by Nijhoff et al. in [1], which is also notably the permutability condition of the Bäcklund transformation for the pKdV equation [2]. Together with the lattice potential Korteweg de-Vires (lpKdV) equation, the lattice potential modified Korteweg de-Vires (lpmKdV) equation [1], [3] and the lattice Schwarzian Korteweg-de Vries (lSKdV) equation [4] have also been shown. The lpmKdV and lSKdV equations are, respectively, discrete versions of the pmKdV and SKdV equations, where the lSKdV equation is also referred to as the cross-ratio equation. Similar to the continuous case [5], there are Miura transformations among these three lattice equations [6]. These three lattice equations appear as distinct parameter choices for a general lattice equation called the Nijhoff-Quispel-Capel equation [1].

Searching for various exact solutions to the discrete integrable systems has become a topic that generates considerable interest and significant progress has been made in this regard. The fixed-point idea was proposed to construct seed solutions [7] (see also [8], [9]) and the Bäcklund transformation was used to generate 1-soliton solutions [10]. Subsequently, the Cauchy matrix approach [8] and the bilinear method [9] were introduced to yield multisoliton solutions. Both these methodologies are constructive and have been extended to derive limit solutions [11], [12]. At the same time, an algebro-geometric method was also developed to obtain finite genus solutions [13]. Recently, the Bäcklund transformation and degenerating approach were used to derive rational solutions for discrete integrable systems [14], [15]. In addition, the Cauchy matrix approach was applied to elliptic integrable systems [16]. Among these methods, the Cauchy matrix approach is a purely algebraic procedure that enables one to obtain discrete integrable equations together with their explicit solutions. This approach is actually a by-product of the linearisation approach which was first proposed by Fokas and Ablowitz [17] and developed to discrete integrable systems by Nijhoff, Quispel et al. One can also refer to [18], [19] and, in particular, to [20], [21], [22] for (2+1)−D discrete systems.

Recent work [23] has shown that the lpKdV equation possesses oscillatory solutions. Without loss of generality, starting from two different seed solutions, two kinds of 1-oscillatory solutions for the lpKdV equation were derived using the Bäcklund transformation. The oscillatory properties, respectively, come from the plane wave factors

and

where ρ⁰ is a constant. In (1), the plane wave factor ρ_n,_m involves an exponential part (p+kp−k)n and an oscillatory part q+(−1)mkq+k, whereas in (2) the plane wave factor ρ_n,m contains two oscillatory parts, p+(−1)nkp+k and q+(−1)mkq+k. Owing to the complicated calculation involved, multi-oscillatory solutions to the lpKdV equation were not presented in [23]. Another type of oscillation of quadrilateral equations is the oscillation of dependent variables (see [24]).

In this article, motivated by plane wave factors (1) and (2), we apply the generalised Cauchy matrix approach to systematically construct oscillatory solutions for the lattice KdV-type equations, including the lpKdV, lpmKdV, and lSKdV equations. The article is organised as follows. In Section 2, we briefly review the Sylvester equation and some properties of the master function S^(i,j), such as recurrence relations, the symmetric property, and invariance. In Section 3, by imposing shift relations on r related with (1) and (2), we derive the lattice KdV-type equations as closed forms. In Section 4, we discuss the oscillatory solutions. Section 5 concludes. In addition, an appendix is given as a complement to the article.

2 The Sylvester Equation and Master Functions

As a preliminary step, we briefly review the Sylvester equation and some properties of the master function S^(i,j). For more details one can refer to [5], [11].

The Sylvester equation (in matrix form)

appears frequently in many areas of applied mathematics and plays a central role particularly in systems and control theory, signal processing, filtering, model reduction, image restoration, and so on. For the solution of the Sylvester equation (3), the following a well-known result [25].

Proposition 1:Let us denote the eigenvalue sets of A and B by ℰ(A) and ℰ(B), respectively. For known A, B, and C, the Sylvester equation (3) has a unique solution M if and only ifℰ(A)∩ℰ(B)=∅.

When ℰ(A) and ℰ(B) satisfy certain conditions, the solution to the Sylvester equation (3) can be expressed via a series or by integration. (See [26] and the references therein.)

We now consider the following Sylvester equation

in which M, K∈ℂ_N×N , r∈ℂ_N×1, and ^tc∈ℂ_1×N . Proposition 1 implies that (4) has a unique solution for M provided ℰ(K)∩ℰ(−K)=∅, where ℰ(K) denotes the eigenvalue sets of K. We assume that K satisfies such a condition.

Using the Sylvester equation (4) we introduce the master function

where I represents the N^th-order unit matrix. This master function possesses the following properties.

3 The Lattice KdV-Type Equations

Motivated by the form of the plane wave factor (1), we consider the following shift relations

where p is the spacing parameter of along the direction of n and q is that along the direction of m. The dispersion relations (12) and (13) lead to the following proposition.

Proposition 4:For the master function S^(i,j)defined by (5) with M, K, r, ^tc satisfying the Sylvester equation (4) and the dispersion relations(12)and(13), the following relations involving the shifts ˜ and ˆ can be derived:

Proof. We begin with the shift relations of M. Subtracting (4) from the ˜-shift of (4) and, respectively, the ˆ-shift of (4), i.e., (4)˜ and (4)ˆ, and using (12) and (13), we obtain

and

in the light of Proposition 1. Subsequently, by using the Sylvester equation (4) and shift relations (16) and (17), one can arrive at

Next, we introduce an auxiliary vector function

Taking a ˜-shift on (20) and using (18), we have

(I+M)(pI−K)u˜(i)=pKir+Ki+1r−rt​cu˜(i),

and then

where we have used the connection

Similarly, we derive the shift of u⁽ⁱ⁾ with respect to the direction of m

With the shift relations (21) and (23) in hand, left-multiplication of ^tcK^j and using (22) immediately yields the shift relations (14) and (15).□

Using the shift relations (14) and (15) together with the symmetric property S^(i,j)=S^(j,i), one can derive the lpKdV, lpmKdV, and lSKdV equations.

We start with the variable S^(0,0). On setting i=j=0 and denoting S^(0,0)=u, (14) and (15) yield

Eliminating S^(0,1) leads to

Introducing the point transformation

we arrive at

which is the lpKdV equation.

Next, we consider S^(−1,0). Taking i=−1, j=0 and introducing the variable v=S^(−1,0)−1, one obtains

Eliminating S^(−1,1) gives the relation

Alternatively, in (14) and (15) we set i=0, j=−1. Noting that S^(0,−1)=S^(−1,0), an easy calculation yields

Compared with (31) and (32), one has

which becomes the lpmKdV equation

under the transformation

The symmetric relation of (14) and (15) reads

Using these relations together with (14) and (15) and by similar analysis, one has

which also gives rise to (33). By transformations (27) and (35), system (31), (32), (38) and (39) become

The system (40)–(43) constitutes the Miura transformation between (28) and (34).

Finally, let us consider i=j=−1. Introducing the object z=S(−1,−1)−np−(−1)m2q+z0 with z₀∈ℂ, from (14) and (15) one derives the equations

Substituting (44) and (45) into the equality

νν˜νν^=(νν^)∼(νν˜)∧

gives the lattice equation

which is nothing but the lSKdV equation. Naturally, system (44) and (45) constitutes the Miura transformation between (34) and (46).

In conclusion, based on the shift relations (12) and (13) we have derived the shift relations for S^(i,j), from which lattice KdV-type equations were obtained as closed forms. Let us summarise the above results in Table 1, where M, K, r, and ^tc satisfy the Sylvester equation (4) and the shift relations (12) and (13).

With regard to the plane wave factor (2), the shifts of r should be formulated by

By an analysis similar to that in the previous discussion, one has the following proposition.

Proposition 5:For the master function S^(i,j)defined by (5) with M, K, r, ^tc satisfying the Sylvester equation (4) and dispersion relations(47)and(48), the following relations involving the shift ˜ and ˆ can be derived:

The lpKdV, lpmKdV, and lSKdV equations can be derived by repeating, this procedure, and their solutions are listed in Table 2, where M, K, r, and ^tc satisfy the Sylvester equation (4) and shift relations (47) and (48).

4 Oscillatory Solutions

In this section, we investigate the oscillatory solutions for the resulting lattice KdV-type equations given in Section 3. According to the analysis in Section 3, we know that lattice KdV-type equations are given by the scalar function S^(i,j)=^tcK^j(I+M)⁻¹Kⁱr, where M, K, r, and ^tc satisfy the Sylvester equation (4) and dispersion relations (12) and (13) [or dispersion relations (47) and (48)]. Therefore, for deriving exact solutions to these lattice equations, we just need to solve the following equation set

or

We solve the equation set (51)–(53) as an example. Noting that in (51)–(53) ^tc and K are known, we just need to present explicit expressions for r and M. In terms of the invariance of S^(i,j) under similar transformations (9) and (10), we turn to solve the Jordan canonical form of (51)–(53), which reads as follows:

where Γ is the Jordan canonical form of the matrix K and satisfies condition ℰ(Γ)∩ℰ(−Γ)=∅.

The Sylvester equation (59) has been exactly solved by factorising M into the triplet FGH, which can be found in [11]. The most general mixed-solutions for r and M to (57)–(59) can be described as follows (A set of notations is given in the Appendix).

Theorem 1:For the equation set (57)–(59) with generic

and

we have solutions

where

and G is a symmetric matrix with the block structure

with

Besides, for Γ, ^tc, r, and M mentioned above, the pair

also solve the equation set (57)–(59) with same Γ and ^tc, where

A=Diag(IN1,A2,A3,…,As),

in which IN1 is the N₁^th-order unit matrix and 𝒜_j is an N_j^th-order constant lower triangular Toeplitz matrix.

Some special solutions can be derived from the mixed solution by setting the order. The first one comes from N₁=N and N_i=0 (i=2, 3, …, N_s), that is, Γ=ΓD[N]({kj}1N). In this case, the corresponding solution can be described by

with F=FD [N] ({kj}1N), G=GD [N] ({kj}1N), and H=HD [N] ({cj}1N). Such a solution gives the N-oscillatory solutions. The second solution is led by N₂=N and N_i=0 (i=1, 3, …, N_s), that is, Γ=ΓJ [N] (k) with k=kN1+1. In this case,

with F=FJ [N] (k), G=GJ [N] (k), and H=HJ [N] ({cj}1N) lead to limit solutions.

The solutions for the equation set (54)–(56) can still be expressed by Theorem 1. The slight difference is that the plane wave factor (A1) is replaced by

where ρi0 is constant.

5 Conclusions

In this article, starting from the plane wave factor (1) we have shown two kinds of oscillatory solutions for the lattice KdV-type equations by utilising the generalised Cauchy matrix approach. A well-known matrix equation, the Sylvester equation, was introduced to define the object S^(i,j), whose shift relations lead to the lattice KdV-type equations. By solving the equation set involving the Sylvester equation (4) together with the dispersion relations (12) and (13), we derived the oscillatory solutions. As, in the continuous case, oscillatory factors (−1)ⁿ and (−1)^m break differentiability and do not appear in analytic solutions, there is no continuum limit for such solutions. The main difference between the soliton and oscillatory solutions focuses on the following aspects:

linear function:pn+qm→pn+(−1)m2q→(−1)n2p+(−1)m2q,plane wave factor:(p−kp+k)n(q−kq+k)m→(p−kp+k)nq+(−1)mkq+k→p+(−1)nkp+kq+(−1)mkq+k.

We believe that many other types of discrete integrable systems, such as the Adler-Bobenko-Suris list (scalar affine-linear quadrilateral equations) [27], lattice Boussinesq-type equations (multi-component equations) [28] and lattice Kadomtsev-Petviashvili-type equations (scalar octahedral equations) [29] should also possess oscillatory solutions. Besides, how to extend the standard symmetry reduction method [30], [31] and the fractional steps domain decomposition method [32] to construct the oscillatory solutions for integrable partial difference equations is an interesting topic and worth considering.