The Korteweg-de Vries equation and a Diophantine problem related to Bernoulli polynomials
© Pintér and Tengely; licensee Springer. 2013
Received: 19 April 2013
Accepted: 29 July 2013
Published: 14 August 2013
Some Diophantine equations related to the soliton solutions of the Korteweg-de Vries equation are resolved. The main tools are the connection with Bernoulli polynomials and the application of certain computational number-theoretical results.
MSC:11D41, 14H45, 11Y50.
Dedicated to Professor Hari M Srivastava
The KdV equation possesses a remarkable family of so-called n-soliton solutions corresponding to the initial profile . For some recent generalizations and applications of the Korteweg-de Vries equation, we refer to [3, 4] and  and the references given therein.
where k and l are fixed distinct integers? Of course, one can consider the much more general problem, when k and l are also unknown integers; however, in this case, the solution of the corresponding equation seems beyond the reach of current techniques.
Applying some recent results by Rakaczki, see  and , it is not too hard to give some ineffective and effective finiteness statements for the solutions m and n to equation (1). However, the purpose of this note is to resolve (1) for certain values of m and n, including an infinite family of the parameters.
Theorem 1 For and , equation (1) has only one solution, namely, .
Theorem 2 Assume that and l is a positive integer such that is prime. Then (1) has no solution in positive integers m and n.
2 Auxiliary results
In our first lemma, we summarize some classical properties of Bernoulli polynomials. For the proofs of these results, we refer to .
, , ,
(von Staudt-Clausen) ,
where the set is stable under the action of Galois, all are non-zero and , where is an algebraic integer and . If , are conjugate, then we may suppose that , and so, , are conjugate. We have the following lemma (Lemma 3.1 in ).
As an application of his theory of lower bounds for linear forms in logarithms, Baker  gave an explicit upper bound for the size of integral solutions of hyperelliptic curves. This result has been improved by many authors (see, e.g., [12–18] and ).
The following result will be used to get an upper bound for the size of the integral solutions of our equations. It is Theorem 3 in .
To obtain a lower bound for the possible unknown integer solutions, we are going to use the so-called Mordell-Weil sieve. The Mordell-Weil sieve has been successfully applied to prove the non-existence of rational points on curves (see, e.g., [26–28] and ).
The next lemma [, Lemma 12.1] gives a lower bound for the size of rational points, whose images are not in the set W.
The following lemma plays a crucial role in the proof of Theorem 1.
Bound for log|x|
−20 − α
This contradicts the bound for that we obtained by Baker’s method. □
3 Proofs of the theorems
where and . By Lemma 5, we have that . Therefore, we have that , a contradiction and there is no solution in positive integers of (7). □
Suppose that , and let d be the smallest positive integer such that , and let . Since is divisible by p for and , we have that p is not a divisor of d. The constant term of the polynomial is , and, by von Staudt-Clausen theorem, it is not divisible by p. On the other hand, p is a divisor of m and , we have a contradiction. If , then we can repeat the previous argument using the fact , cf. Lemma 1. □
The work is supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0010 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund. Research was supported in part by the Hungarian Academy of Sciences, OTKA grants K75566, K100339, NK101680, NK104208 (ÁP) and OTKA grants PD75264, NK104208, K100339 and János Bolyai Research Scholarship of the Hungarian Academy of Sciences (ST). The authors are grateful to the reviewers for their helpful remarks.
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