# The Korteweg-de Vries equation and a Diophantine problem related to Bernoulli polynomials

- Ákos Pintér
^{1}Email author and - Szabolcs Tengely
^{2}

**2013**:245

https://doi.org/10.1186/1687-1847-2013-245

© Pintér and Tengely; licensee Springer. 2013

**Received: **19 April 2013

**Accepted: **29 July 2013

**Published: **14 August 2013

## Abstract

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.

## Keywords

## Dedication

Dedicated to Professor Hari M Srivastava

## 1 Introduction

*etc.*, does not change as the system evolves) of the form

*u*and its

*x*-derivatives up to order

*m*, see [2]. For example,

The KdV equation possesses a remarkable family of so-called *n*-soliton solutions corresponding to the initial profile ${u}_{n}(x,0)=-2n(n+1){sech}^{2}x$. For some recent generalizations and applications of the Korteweg-de Vries equation, we refer to [3, 4] and [5] and the references given therein.

for $k=-1,0,1,\dots $ .

*k*and

*l*? In other words, what is the cardinality of the set of solutions

*m*,

*n*to the equation

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 [7] and [8], 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* $k=-1$ *and* $l\in \{0,1,2,3\}$, *equation* (1) *has only one solution*, *namely*, $(l,m,n)=(0,24,5)$.

**Theorem 2** *Assume that* $k=0$ *and* *l* *is a positive integer such that* $2l+3$ *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 [9].

**Lemma 1**

*Let*${B}_{j}(X)$

*denote the*

*jth Bernoulli polynomial and*${B}_{j}={B}_{j}(0)$, $j=1,2,\dots $ .

*Further*,

*let*${D}_{j}$

*be the denominator of*${B}_{j}$.

*Then we have*

- (A)
${B}_{j}(X)={X}^{n}+{\sum}_{i=1}^{j}\left(\genfrac{}{}{0ex}{}{j}{i}\right){B}_{i}{X}^{j-i}$,

- (B)
${S}_{j}(x)={1}^{j}+{2}^{j}+\cdots +{(x-1)}^{j}=\frac{1}{j+1}({B}_{j+1}(x)-{B}_{j+1})$,

- (C)
${B}_{1}=-\frac{1}{2}$, ${B}_{2j+1}=0$, $j=1,2,\dots $ ,

- (D)
(

*von Staudt*-*Clausen*) ${D}_{2j}={\prod}_{p-1|2j,p\phantom{\rule{0.25em}{0ex}}\mathrm{prime}}p$, - (E)
${X}^{2}{(X-1)}^{2}|{B}_{2j}(X)-{B}_{2j}$ (

*in*$\mathbb{Q}[X]$), - (F)
${B}_{j}(X)={(-1)}^{j}{B}_{j}(1-X)$.

*α*be a root of

*F*, and let $J(\mathbb{Q})$ be the Jacobian of the curve . We have that

*κ*comes from a finite set. By knowing the Mordell-Weil group of the curve , it is possible to provide a method to compute such a finite set. To each coset representative ${\sum}_{i=1}^{m}({P}_{i}-\mathrm{\infty})$ of $J(\mathbb{Q})/2J(\mathbb{Q})$, we associate

where the set $\{{P}_{1},\dots ,{P}_{m}\}$ is stable under the action of Galois, all $y({P}_{i})$ are non-zero and $x({P}_{i})={\gamma}_{i}/{d}_{i}^{2}$, where ${\gamma}_{i}$ is an algebraic integer and ${d}_{i}\in {\mathbb{Z}}_{\ge 1}$. If ${P}_{i}$, ${P}_{j}$ are conjugate, then we may suppose that ${d}_{i}={d}_{j}$, and so, ${\gamma}_{i}$, ${\gamma}_{j}$ are conjugate. We have the following lemma (Lemma 3.1 in [10]).

**Lemma 2** *Let*
*be a set of* *κ* *values*, *associated as above to a complete set of coset representatives of* $J(\mathbb{Q})/2J(\mathbb{Q})$. *Then*
*is a finite subset of* ${\mathcal{O}}_{K}$, *and if* $(x,y)$ *is an integral point on the curve* (2), *then* $x-\alpha =\kappa {\xi}^{2}$ *for some* $\kappa \in \mathcal{K}$ *and* $\xi \in K$.

As an application of his theory of lower bounds for linear forms in logarithms, Baker [11] 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 [19]).

*K*be a number field of degree

*d*, and let

*r*be its unit rank, and let

*R*be its regulator. For $\alpha \in K$, we denote by $\mathrm{h}(\alpha )$ the logarithmic height of the element

*α*. Let

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 [10].

**Lemma 3**

*Let*

*α*

*be an algebraic integer of degree at least*3,

*and let*

*κ*

*be an integer belonging to*

*K*.

*Denote by*${\alpha}_{1}$, ${\alpha}_{2}$, ${\alpha}_{3}$

*distinct conjugates of*

*α*

*and by*${\kappa}_{1}$, ${\kappa}_{2}$, ${\kappa}_{3}$

*the corresponding conjugates of*

*κ*.

*Let*

*and*

*In what follows*

*R*

*stands for an upper bound for the regulators of*${K}_{1}$, ${K}_{2}$

*and*${K}_{3}$,

*and*

*r*

*denotes the maximum of the unit ranks of*${K}_{1}$, ${K}_{2}$, ${K}_{3}$.

*Let*

*and let*

*and let*

*Define*

*and*

*If*$x\in \mathbb{Z}\mathrm{\setminus}\{0\}$

*satisfies*$x-\alpha =\kappa {\xi}^{2}$

*for some*$\xi \in K$

*then*

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 [29]).

*J*be its Jacobian. We assume the knowledge of some rational point on

*C*, so let

*D*be a fixed rational point on

*C*, and let

*ȷ*be the corresponding Abel-Jacobi map

*W*be the image in

*J*of the known rational points on

*C*and ${D}_{1},\dots ,{D}_{r}$ generators for the free part of $J(\mathbb{Q})$. By using the Mordell-Weil sieve, we are going to obtain a very large and smooth integer

*B*such that

*ϕ*is the free part of $J(\mathbb{Q})$. The variant of the Mordell-Weil sieve explained in [10] provides a method to obtain a very long decreasing sequence of lattices in ${\mathbb{Z}}^{r}$

for $j=1,\dots ,k$.

The next lemma [[10], Lemma 12.1] gives a lower bound for the size of rational points, whose images are not in the set *W*.

**Lemma 4**

*Let*

*W*

*be a finite subset of*$J(\mathbb{Q})$,

*and let*

*L*

*be a sublattice of*${\mathbb{Z}}^{r}$.

*Suppose that*$\u0237(C(\mathbb{Q}))\subset W+\varphi (L)$.

*Let*${\mu}_{1}$

*be a lower bound for*$h-\stackrel{\u02c6}{h}$

*and*

*Denote by*

*M*

*the height*-

*pairing matrix for the Mordell*-

*Weil basis*${D}_{1},\dots ,{D}_{r}$,

*and let*${\lambda}_{1},\dots ,{\lambda}_{r}$

*be its eigenvalues*.

*Let*

*and let*$m(L)$

*be the Euclidean norm of the shortest non*-

*zero vector of*

*L*.

*Then*,

*for any*$P\in C(\mathbb{Q})$,

*either*$\u0237(P)\in W$

*or*

The following lemma plays a crucial role in the proof of Theorem 1.

**Lemma 5**

*The integral solutions of the equation*

*are*

*Proof of Lemma 5*Let $J(\mathbb{Q})$ be the Jacobian of the genus two curve (3). Using MAGMA, we determine a Mordell-Weil basis, which is given by

*α*be a root of

*f*. We will choose for coset representatives of $J(\mathbb{Q})/2J(\mathbb{Q})$ the linear combinations ${\sum}_{i=1}^{2}{n}_{i}{D}_{i}$, where ${n}_{i}\in \{0,1\}$. Then

*e.g.*, [26, 30]). In our case, one can eliminate

**Bounds**

κ | Bound for log|x| |
---|---|

1 | 6.27⋅10 |

− | 4.48⋅10 |

−20 − | 1.89⋅10 |

*W*be the image of this set in $J(\mathbb{Q})$. Applying the Mordell-Weil sieve, implemented by Bruin and Stoll and explained in [10], we obtain that $\u0237(C(\mathbb{Q}))\subseteq W+BJ(\mathbb{Q})$, where

This contradicts the bound for $log|x|$ that we obtained by Baker’s method. □

## 3 Proofs of the theorems

*Proof of Theorem 1*For $k=-1$ and $l\in \{0,1,2,3\}$, we have the Diophantine equations

where $Y=375(2n+1)$ and $X=20{m}^{2}+20m-20$. By Lemma 5, we have that $X=0\text{or}-20$. Therefore, we have that $m\in \{-1,0\}$, a contradiction and there is no solution in positive integers of (7). □

*Proof of Theorem 2*Now $k=0$ and $p=2l+3\ge 3$ is a prime. From (1), we get

*m*and

*n*be an arbitrary but fixed solution. An elementary number theoretical argument and Lemma 1 yield that $p|m(m+1)$ and

Suppose that $p|m$, and let *d* be the smallest positive integer such that ${B}_{p+1}(m+1)-{B}_{p+1}=\frac{1}{d}f(m){m}^{2}{(m+1)}^{2}$, and let $f(X)\in \mathbb{Z}[X]$. Since $\left(\genfrac{}{}{0ex}{}{p+1}{k}\right)$ is divisible by *p* for $k=2,\dots ,p-1$ and ${B}_{1}=-1/2$, we have that *p* is not a divisor of *d*. The constant term of the polynomial $f(X)$ is $d\left(\genfrac{}{}{0ex}{}{p+1}{p-1}\right){B}_{p-1}$, and, by von Staudt-Clausen theorem, it is not divisible by *p*. On the other hand, *p* is a divisor of *m* and $f(m)$, we have a contradiction. If $p|m+1$, then we can repeat the previous argument using the fact $f(X)=f(-X-1)$, *cf.* Lemma 1. □

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

## References

- Fairlie DB, Veselov AP: Faulhaber and Bernoulli polynomials and solitons.
*Physica D*2001, 152–153: 47–50.MathSciNetView ArticleGoogle Scholar - Miura RM, Gardner CS, Kruskal WD: Korteweg-de Vries equation and generalizations. II.
*J. Math. Phys.*1968, 9: 1204–1209. 10.1063/1.1664701MathSciNetView ArticleMATHGoogle Scholar - Ismail MS: Numerical solution of complex modified Korteweg-de Vries equation by Petrov-Galerkin method.
*Appl. Math. Comput.*2008, 202: 520–531. 10.1016/j.amc.2008.02.033MathSciNetView ArticleMATHGoogle Scholar - Gai X-L, Gao Y-T, Yu X, Wang L: Painlevé property, auto-Bäcklund transformation and analytic solutions of a variable-coefficient modified Korteweg-de Vries model in a hot magnetized dusty plasma with charge fluctuations.
*Appl. Math. Comput.*2011, 216: 271–279.MathSciNetView ArticleGoogle Scholar - Rady ASA, Khater AH, Osman ES, Khafallah M: New periodic wave and soliton solutions for system of coupled Korteweg-de Vries equations.
*Appl. Math. Comput.*2009, 207: 406–414. 10.1016/j.amc.2008.10.064MathSciNetView ArticleMATHGoogle Scholar - Zakharov VE, Faddaev LD: KdV equation is completely integrable Hamiltonian system.
*Funct. Anal. Appl.*1971, 5: 18–27.Google Scholar - Rakaczki C:On the Diophantine equation ${S}_{m}(x)=g(y)$.
*Publ. Math. (Debr.)*2004, 65: 439–460.MathSciNetMATHGoogle Scholar - Rakaczki C:On some generalization of the Diophantine equation $s({1}^{k}+{2}^{k}+\cdots +{x}^{k})+r=d{y}^{n}$.
*Acta Arith.*2012, 151: 201–216. 10.4064/aa151-2-4MathSciNetView ArticleMATHGoogle Scholar - Rademacher H:
*Topics in Analytic Number Theory*. Springer, Berlin; 1973.View ArticleMATHGoogle Scholar - Bugeaud Y, Mignotte M, Siksek S, Stoll M, Tengely S: Integral points on hyperelliptic curves.
*Algebra Number Theory*2008, 2: 859–885. 10.2140/ant.2008.2.859MathSciNetView ArticleMATHGoogle Scholar - Baker A: Bounds for the solutions of the hyperelliptic equation.
*Proc. Camb. Philos. Soc.*1969, 65: 439–444. 10.1017/S0305004100044418View ArticleMATHGoogle Scholar - Bilu YF: Effective analysis of integral points on algebraic curves.
*Isr. J. Math.*1995, 90: 235–252. 10.1007/BF02783215MathSciNetView ArticleMATHGoogle Scholar - Bilu YF, Hanrot G: Solving superelliptic Diophantine equations by Baker’s method.
*Compos. Math.*1998, 112: 273–312. 10.1023/A:1000305028888MathSciNetView ArticleMATHGoogle Scholar - Brindza B: On
*S*-integral solutions of the equation ${y}^{m}=f(x)$ .*Acta Math. Hung.*1984, 44: 133–139. 10.1007/BF01974110MathSciNetView ArticleMATHGoogle Scholar - Bugeaud Y: Bounds for the solutions of superelliptic equations.
*Compos. Math.*1997, 107: 187–219. 10.1023/A:1000130114331MathSciNetView ArticleMATHGoogle Scholar - Poulakis D:Solutions entières de l’équation ${Y}^{m}=f(X)$.
*Sémin. Théor. Nr. Bordx.)*1991, 3: 187–199.MathSciNetView ArticleMATHGoogle Scholar - Schmidt WM: Integer points on curves of genus 1.
*Compos. Math.*1992, 81: 33–59.MATHGoogle Scholar - Sprindžuk VG: The arithmetic structure of integer polynomials and class numbers.
*Tr. Mat. Inst. Steklova*1977, 143: 152–174. Analytic number theory, mathematical analysis and their applications (dedicated to I.M. Vinogradov on his 85th birthday)Google Scholar - Voutier PM:An upper bound for the size of integral solutions to ${Y}^{m}=f(X)$.
*J. Number Theory*1995, 53: 247–271. 10.1006/jnth.1995.1090MathSciNetView ArticleMATHGoogle Scholar - Bugeaud Y, Győry K: Bounds for the solutions of unit equations.
*Acta Arith.*1996, 74: 67–80.MathSciNetMATHGoogle Scholar - Bugeaud Y, Mignotte M, Siksek S: Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers.
*Ann. Math.*2006, 163: 969–1018. 10.4007/annals.2006.163.969MathSciNetView ArticleMATHGoogle Scholar - Landau E: Verallgemeinerung eines Pólyaschen satzes auf algebraische zahlkörper.
*Nachr. Ges. Wiss. Gött., Math.-Phys. Kl.*1918, 1918: 478–488.MATHGoogle Scholar - Matveev EM: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II.
*Izv. Ross. Akad. Nauk, Ser. Mat.*2000, 64: 125–180. 10.4213/im314View ArticleGoogle Scholar - Pethö A, de Weger BMM: Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation.
*Math. Comput.*1986, 47: 713–727.View ArticleMATHGoogle Scholar - Voutier PM: An effective lower bound for the height of algebraic numbers.
*Acta Arith.*1996, 74: 81–95.MathSciNetGoogle Scholar - Bruin N, Stoll M: Deciding existence of rational points on curves: an experiment.
*Exp. Math.*2008, 17: 181–189. 10.1080/10586458.2008.10129031MathSciNetView ArticleMATHGoogle Scholar - Bruin N, Stoll M: The Mordell-Weil sieve: proving non-existence of rational points on curves.
*LMS J. Comput. Math.*2010, 13: 272–306.MathSciNetView ArticleMATHGoogle Scholar - Flynn EV: The Hasse principle and the Brauer-Manin obstruction for curves.
*Manuscr. Math.*2004, 115: 437–466. 10.1007/s00229-004-0502-9MathSciNetView ArticleMATHGoogle Scholar - Scharaschkin, V: Local-global problems and the Brauer-Manin obstruction. PhD thesis, University of Michigan (1999)Google Scholar
- Bruin N, Stoll M: Two-cover descent on hyperelliptic curves.
*Math. Comput.*2009, 78: 2347–2370. 10.1090/S0025-5718-09-02255-8MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.