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- Open Access
On a class of Darboux-integrable semidiscrete equations
- Kostyantyn Zheltukhin^{1}Email author,
- Natalya Zheltukhina^{2} and
- Ergun Bilen^{1}
https://doi.org/10.1186/s13662-017-1241-z
© The Author(s) 2017
- Received: 27 January 2017
- Accepted: 13 June 2017
- Published: 27 June 2017
Abstract
We consider a classification problem for Darboux-integrable hyperbolic semidiscrete equations. In particular, we obtain a complete description for a special class of equations admitting four-dimensional characteristic x-rings and two-dimensional characteristic n-rings. For all described equations, the corresponding x- and n-integrals are constructed.
Keywords
- semidiscrete equations
- Darboux integrability
- characteristic rings
1 Introduction
Classification problems play an important role in the study of integrable equations. For classification of hyperbolic equations, it is convenient to define integrability in terms of characteristic rings. The notion of a characteristic ring was introduced by Shabat for integrable hyperbolic equations of exponential type (see [1, 2]) and then used by Zhiber to study general integrable hyperbolic equations (see [3–7]). Later, Habibullin extended this notion to the case of semidiscrete and discrete equations (see [8–16]). For more details on characteristic rings, see survey paper [17].
We consider semidiscrete hyperbolic equations that admit nontrivial x- and n-integrals, so-called Darboux-integrable equations [18]. It was proved in [9] that a semidiscrete hyperbolic equation is Darboux integrable if and only if its characteristic x- and n-rings are finite-dimensional. Description of all equations with characteristic x- and n-rings of finite dimensions is a very difficult classification problem. The majority of known Darboux-integrable semidiscrete equations possess x- and n-rings of dimensions not exceeding five (see [14, 16, 19]). Necessary and sufficient conditions for a characteristic x-ring to be four-dimensional were obtained in [20] (also see [21] for a characterization of five-dimensional characteristic x-rings). In [12] the conditions for a two-dimensional characteristic n-ring were obtained. We use these conditions to explicitly derive integrable equations with four-dimensional characteristic x-rings and two-dimensional characteristic n-rings.
Theorem 1
Theorem 2
The paper is organized as follows. First, we give proofs of Theorems 1 and 2, and in the last section, we provide x- and n- integrals for equations found in Theorems 1 and 2.
2 Proofs of Theorem 1 and Theorem 2
2.1 Preliminary results
In what follows, all calculations are done on the set of solutions of equation (1), that is, we consider \(\dots, t_{-1}, t_{0}, t_{1}, \dots\) and \(t_{x}, t_{xx}, t_{xxx}, \dots\) as independent dynamical variables. The derivatives of \(\dots, t_{-1}, t_{0}, t_{1}, \dots\) and shifts of \(t_{x}, t_{xx}, t_{xxx}, \dots\) are expressed in terms of the dynamical variables using (1).
Let us formulate necessary and sufficient conditions so that equation (3) has a characteristic x-ring of dimension four and a characteristic n-ring of dimension two. First, we consider the n-ring. The following theorem was proved in [12].
Theorem 3
For the characteristic x-ring, we have to consider two cases: \(f_{t_{x}t_{x}}= 0\), that is, f is a linear function of \(t_{x}\), and \(f_{t_{x}t_{x}}\ne0\), that is, f is a nonlinear function of \(t_{x}\).
The following theorems were proved in [20].
Theorem 4
Theorem 5
For convenience of the reader, let us give definitions of x- and n-integrals and of Darboux-integrable semidiscrete equations.
Definition 6
Equation (1) is called Darboux integrable if it admits a nontrivial x-integral and a nontrivial n-integral.
2.2 Proof of Theorem 1
The proof of the Theorem 1 is based on the following lemmas.
Lemma 7
Proof
In the next lemma, we give conditions for equation (20) to have a four-dimensional characteristic x-ring.
Lemma 8
Proof
Returning to the original variable t in equation (20) with Q given by equation (21), we get Theorem 1.
2.3 Proof of Theorem 2
The proof of the Theorem 2 is based on the following lemmas.
Lemma 9
Proof
We use expressions (25) and (29) for the derivative and shift of M in the next lemma.
Lemma 10
Let equation (24) have a characteristic n-ring of dimension two. Then M has either of the forms \(M = \frac{1}{t_{x} + P}\), or \(M = \sqrt{t_{x}^{2} + Pt_{x} + Q}\), or \(M = t_{x}^{2}\).
Proof
Now we consider each value of M obtained in the lemma, separately. We start with the simple case \(M =t_{x}^{2}\).
Lemma 11
Equation (24) cannot have a four-dimensional characteristic x-ring if \(M=t_{x}^{2}\).
Proof
Let us consider the case \(M = {\frac{1}{t_{x} + P}}\).
Lemma 12
Proof
Let us consider the case \(M = \sqrt{t_{x}^{2} + Pt_{x} + Q}\).
Lemma 13
Proof
The proof of Theorem 2 easily follows from the above lemmas.
3 Examples
The functions f given in the Theorem 1 lead to the following examples.
Example 14
Example 15
The functions f given in the Theorem 2 lead to the following examples.
Example 16
Example 17
Example 18
In all examples, we can check that F is an x-integral and I is an n-integral by direct calculations.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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