Abstract
In this paper, we derive the Grammian determinant solutions to the modified two-dimensional Toda lattice, and then we construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure. We show the integrability of the modified two-dimensional Toda lattice with self-consistent sources by presenting its Casoratian and Grammian structure of the N-soliton solution. It is also demonstrated that the commutativity between the source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice.
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1 Introduction
The two-dimensional Toda lattice, which can be regarded as a spatial discretization of the KP equation, takes the following form:
where \(V_{n}\) denotes \(V(n,x,s)\). We use the above notation throughout the paper. Under the dependent variable transformation
equation (1) is transformed into the bilinear form [1, 2]:
where the bilinear operators are defined by [2]
It is shown in [2, 3] that the two-dimensional Toda lattice equation possesses the following bilinear Bäcklund transformation:
where λ, μ, ν are arbitrary constants. Equations (4)-(5) are transformed into the following nonlinear form:
through the dependent variable transformation \(u_{n}=\frac{\partial}{ \partial s}\ln(\frac{f_{n}}{f'_{n}})\), \(v_{n}=-\frac{\partial}{\partial x}\ln(\frac{f_{n}}{f'_{n-1}})\). Equations (4)-(5) or (6)-(7) are called the modified two-dimensional Toda lattice [2, 3]. The solutions \(V_{n}\) of the two-dimensional Toda lattice (1) and \(u_{n}\), \(v_{n}\) of the modified two-dimensional Toda lattice (6)-(7) are connected through a Miura transformation [2].
The soliton equations with self-consistent sources can model a lot of important physical processes. For example, the KdV equation with self-consistent sources describes the interaction of long and short capillary-gravity waves [4]. The KP equation with self-consistent sources describes the interaction of a long wave with a short-wave packet propagating on the \(x,y\) plane at an angle to each other [5, 6]. Since the pioneering work of Mel’nikov [7], lots of soliton equations with self-consistent sources have been studied via inverse scattering methods [7–11], Darboux transformation methods [12–17], Hirota’s bilinear method and the Wronskian technique [18–24].
In [25], a new algebraic method, called the source generation procedure, is proposed to construct and solve the soliton equations with self-consistent sources both in continuous and discrete cases. The source generation procedure has been successfully applied to many \((2+1)\)-dimensional continuous and discrete soliton equations such as the Ishimori-I equation [26], the semi-discrete Toda equation [27], the modified discrete KP equation [28], and others. The purpose of this paper is to construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure and clarify the determinant structure of N-soliton solution for the modified two-dimensional Toda lattice with self-consistent sources.
The paper is organized as follows. In Section 2, we derive the Grammian solution to the modified two-dimensional Toda lattice equation and then construct the two-dimensional Toda lattice equations with self-consistent sources. In Section 3, the Casoratian formulation of N-soliton solution for the modified two-dimensional Toda lattice with self-consistent is given. Section 4 is devoted to showing that the commutativity of the source generation procedure and Bäcklund transformation is valid for two-dimensional Toda lattice. We end this paper with a conclusion and discussion in Section 5.
2 The modified two-dimensional Toda lattice equation with self-consistent sources
The N-soliton solution in Casoratian form for the modified two-dimensional Toda lattice equation (4)-(5) is given in [2] and [29]. In this section, we first derive the Grammian formulation of the N-soliton solution for the modified two-dimensional Toda lattice equation, and then we construct the modified two-dimensional Toda lattice equation with self-consistent sources via the source generation procedure.
If we choose \(\lambda=1\), \(\nu=\mu=0\), then the modified two-dimensional Toda lattice (4)-(5) becomes
Proposition 1
The modified two-dimensional Toda lattice (8)-(9) has the following Grammian determinant solution:
where
in which the \(\phi_{i}(n)\) denote \(\phi_{i}(n,x,s)\) and the \(\psi_{i}(-n)\) denote \(\psi_{i}(-n,x,s)\) for \(i=1,\ldots,N+1\). In addition, \(c_{ij}\) (\(1\leq i,j \leq N+1\)) are arbitrary constants and \(\phi_{i}(n)\), \(\psi_{i}(-n)\) (\(i=1,\ldots,N+1\)) satisfy the following dispersion relations:
Proof
The Grammian determinants \(f_{n}\) in (10) and \(f'_{n}\) in (11) can be expressed in terms of the following Pfaffians:
where the Pfaffian elements are defined by
in which \(i,j=1,\ldots,N+1\) and k, m are integers.
Using the dispersion relations (14)-(15), we can compute the following differential and difference formula for the Pfaffians (16)-(17):
Substituting equations (21)-(24) into the modified two-dimensional Toda lattice (8)-(9) gives the following two Pfaffian identities:
□
In order to construct the modified two-dimensional Toda lattice with self-consistent sources, we change the Grammian determinant solutions (10)-(11) into the following form:
where Nth column vectors \(\Phi(n)\), \(\Psi(n)\) are given in (12)-(13) and \(\phi_{i}(n)\), \(\psi_{i}(-n)\) (\(i=1,\ldots, {N+1}\)) also satisfy the dispersion relations (14)-(15). In addition, \(\gamma_{ij}(s)\) satisfies
with \(\gamma_{i}(s)\) being an arbitrary function of s and K being a positive integer.
The Grammian determinants \(f_{n}\) in (25) and \(f'_{n}\) in (26) can be expressed by means of the following Pfaffians:
where the Pfaffian elements are defined by
in which \(i,j=1,\ldots,N+1\) and k, m are integers.
It is easy to show that the functions \(f(n,x,s)\), \(f'(n,x,s)\) given in (28)-(29) still satisfy equation (8). However, they will not satisfy (9), and they satisfy the following new equation:
where the new functions \(g_{n}^{(j)}\), \(h_{n}^{(j)}\) are given by
where \(j=1,\ldots,K\) and the dot denotes the derivative of \(\gamma_{j}(t)\) with respect to t. Furthermore, we can show that \(f_{n}\), \(f'_{n}\), \(g_{n}^{(j)}\), \(h_{n}^{(j)}\) (\(j=1,\ldots,K\)) satisfy the following 2K equations:
In fact, we have the following differential and difference formula for \(f_{n}\) in (28), \(f'_{n}\) in (29) and \(g_{n}^{(j)}\), \(h _{n}^{(j)}\) (\(j=1,\ldots,K\)) by employing the dispersion relations (14)-(15):
where \(\hat{\ }\) indicates deletion of the letter under it.
Substitution of equations (38)-(47) into equations (33), (36)-(37) gives the following Pfaffian identities:
and
respectively. Therefore, equations (8), (33), (36)-(37) constitute the modified two-dimensional Toda lattice with self-consistent sources, and it possesses the Grammian determinant solution (28)-(29), (34)-(35).
Through the dependent variable transformation
the bilinear modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37) can be transformed into the following nonlinear form:
When we take \(G_{n}^{(j)}=H_{n}^{(j)}=0\), \(j=1,\ldots,K\) in (49)-(52), the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) is reduced to the nonlinear modified two-dimensional Toda lattice (6)-(7) with \(\lambda=1\), \(\nu=\mu=0\).
If we choose
where \(i=1,2,\ldots,N+1\) in the Grammian determinants (25)-(26), (34)-(35), then we obtain the N-soliton solution of the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37). Here \(q_{i}\), \(Q_{i}\) (\(i=1,2, \ldots,N+1\)) are arbitrary constants.
For example, if we take \(K=1\), \(N=1\) and
where \(a(t)\) is an arbitrary function of t, then we have
Therefore, the one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) is given by
If we take \(K=1\), \(N=2\) and
we derive
Substituting functions (63)-(66) into the dependent variable transformations (48), we obtain two-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52).
3 Casorati determinant solution to the modified two-dimensional Toda lattice equation with self-consistent sources
In Section 2, we derived that the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) possess the Grammian determinant solution (25), (26), (34), (35). In this section, we derive the Casoratian formulation of the N-soliton for the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37).
Proposition 2
The modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) has the following Casorati determinant solution:
where \(\psi_{i}(n+m)=\phi_{i1}(n+m)+(-1)^{i-1}C_{i}(s)\phi_{i2}(n+m)\) (\(m=0, \ldots,N\)) and
with \(\gamma_{i}(s)\) being an arbitrary function of s and K, N being positive integers. In addition, \(\phi_{i1}(n)\), \(\phi_{i2}(n)\) satisfy the following dispersion relations:
and the Pfaffian elements are defined by
in which \(i,j=1,\ldots,N+1\) and m, l are integers.
Proof
We can derive the following dispersion relation for \(\psi_{i}(n)\) (\(i=1, \ldots,N+1\)) from equations (72):
Applying the dispersion relation (75)-(76), we can calculate the following differential and difference formula for the Casorati determinants (67)-(70):
By substituting equations (77)-(88) into the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37), we obtain the following Pfaffian identities, respectively:
and
respectively. □
In order to obtain the one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52), we take \(N=1\), \(K=1\) and
in the Casoratian determinants (67)-(70). Here \(\xi_{i}\), \(\eta_{i}\) (\(i=1,2\)) are given in (53) and \(a(t)\) is an arbitrary function of t. Hence we obtain
Substituting functions (89)-(92) into the dependent variable transformations (48), we get a one-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) given in (59)-(62).
If we take \(N=2\), \(K=1\) and
in the Casoratian determinants (67)-(70), we get
We introduce five constants \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\epsilon _{1}\), \(\epsilon_{2}\) satisfying
and take
then equations (93)-(96) become
We rederive the two-soliton solution of the nonlinear modified two-dimensional Toda lattice with self-consistent sources (49)-(52) obtained in Section 2, substituting the above functions in equations (97)-(100) into the dependent variable transformation (48).
4 Commutativity of the source generation procedure and Bäcklund transformation
In this section, we show that the commutativity of the source generation procedure and Bäcklund transformation holds for the two-dimensional Toda lattice. For this purpose, we derive another form of the modified two-dimensional Toda lattice with self-consistent sources which is the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources given in [25].
We have shown that the Casorati determinants \(f_{n}\), \(f'_{n}\), \(g^{(j)} _{n}\), \(h^{(j)}_{n}\) given in (67)-(70) satisfy the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37). Now we take
and we introduce two new fields
where the Pfaffian elements are defined in (67)-(74).
In [25], the authors prove that the Casorati determinant \(F_{n}\), \(G^{(j)}_{n}\), \(H^{(j)}_{n}\) solves the following two-dimensional Toda lattice with self-consistent sources [25]:
It is not difficult to show that the Casorati determinant with \(F'_{n}\), \(G^{\prime(j)}_{n}\), \(H^{\prime(j)}_{n}\) is another solution to the two-dimensional Toda lattice with self-consistent sources (107)-(109).
Furthermore, we can verify that the Casorati determinants \(F_{n}\), \(F'_{n}\), \(G ^{(j)}_{n}\), \(G^{\prime(j)}_{n}\), \(H^{(j)}_{n}\), \(H^{\prime(j)}_{n}\) given in (101)-(106) satisfy the following bilinear equations:
which is another form of the modified two-dimensional Toda lattice with self-consistent sources. It is proved in [30] that equations (110)-(115) constitute the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources (107)-(109). Therefore, the commutativity of source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice.
5 Conclusion and discussion
In this paper, Grammian solutions to the modified two-dimensional Toda lattice are presented. From the Grammian solutions, the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) are produced via the source generation procedure. We show that the modified two-dimensional Toda lattice with self-consistent sources (8), (33), (36)-(37) are resolved into the determinant identities by presenting its Grammian and Casorati determinant solutions. We also construct another form of the modified discrete KP equation with self-consistent sources (110)-(115) which is the Bäcklund transformation for the two-dimensional Toda lattice with self-consistent sources derived in [25].
Now we show that the modified two-dimensional Toda lattice has a continuum limit into the mKP equation [2, 31], and the modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37) yields the mKP equation with self-consistent sources derived in [32] through a continuum limit. For this purpose, we take
in the modified two-dimensional Toda lattice (8)-(9), and compare the \(\epsilon^{2}\) order in (8), and the \(\epsilon^{3}\) order in (9), then we obtain the mKP equation [2, 31]:
where F, \(F'\) denote \(F(X,Y,T)\), \(F'(X,Y,T)\), respectively.
By taking
for \(j=1,\ldots,K\) in the modified two-dimensional Toda lattice with self-consistent sources (8, 33, 36)-(37), and comparing the \(\epsilon^{2}\) order in (8), (36)-(37), and the \(\epsilon^{3}\) order in (33), we obtain the mKP equation with self-consistent sources [32]:
where F, \(F'\), \(G_{j}\), \(H_{j}\) denote \(F(X,Y,T)\), \(F'(X,Y,T)\), \(G_{j}(X,Y,T)\), \(H_{j}(X,Y,T)\) for \(j=1,\ldots,K\), respectively.
Recently, generalized Wronskian (Casorati) determinant solutions are constructed for continuous and discrete soliton equations [33–39]. Besides soliton solutions, a broader class of solutions such as rational solutions, negatons, positons and complexitons solutions are obtained from the generalized Wronskian (Casorati) determinant solutions [33–38]. In [39], a general Casoratian formulation is presented for the two-dimensional Toda lattice equation from which various examples of Casoratian type solutions are derived. It is interesting for us to construct the two-dimensional Toda lattice equation with self-consistent sources having a generalized Casorati determinant solution via the source generation procedure. This will bring us a broader class of solutions such as negatons, positons, and complexiton type solutions of the two-dimensional Toda lattice equation with self-consistent sources.
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Acknowledgements
This work was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (Grant no. 2016MS0115), the National Natural Science Foundation of China (Grants no. 11601247 and 11605096).
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Gegenhasi The modified two-dimensional Toda lattice with self-consistent sources. Adv Differ Equ 2017, 277 (2017). https://doi.org/10.1186/s13662-017-1347-3
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DOI: https://doi.org/10.1186/s13662-017-1347-3