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

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.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

In the paper [1], Fairlie and Veselov obtained a relation of the Bernoulli polynomials with the theory of the Korteweg-de Vries (KdV) equation

$$u_t-6u{u}_x+u_{xxx}=0.$$

This equation has infinitely many conservation laws (that is, certain laws, which show that a particular measurable property of an isolated physical system, like mass, energy, momentum, etc., does not change as the system evolves) of the form

$$I_m[u]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_m(u,u_x,u_{xx},\dots ,u_m)dx,$$

where $P_m$ are some polynomials of the function u and its x-derivatives up to order m, see [2]. For example,

$$I_{-1}[u]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}udx,I_0[u]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{u}^{2}dx,I_1[u]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(u_x^2+2u^3)dx$$

and

$$I_2[u]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(u_{xx}^2+10u{u}_x^2+5u^4)dx.$$

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.

Using the spectral theory of Schrödinger operators, see [6], Fairlie and Veselov [1] proved that there is a strong connection between the improver integrals related to the functions $u_n(x,0)$ above and the well-known discrete sums of power values, namely,

$$I_k[u_n]=\frac{{(-1)}^k{4}^{k+2}}{2k+3}\sum _{i=1}^n{i}^{2k+3}$$

for $k=-1,0,1,\dots$ .

Now, let $k\ne l$ be fixed integers with $k,l\in \{-1,0,1,2,\dots \}$, and suppose that

$$|I_k[u_n]|=|I_l[u_m]|.$$

One can ask how often can these integrals be equal for given k and l? In other words, what is the cardinality of the set of solutions m, n to the equation

$$\frac{4^k}{2k+3}\sum _{i=1}^n{i}^{2k+3}=\frac{4^l}{2l+3}\sum _{i=1}^m{i}^{2l+3},$$

(1)

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) $D_{2j}={\prod }_{p-1|2j,p\mathrm{prime}}p$,

• (E) $X^2{(X-1)}^2|B_{2j}(X)-B_{2j}$ (in $\mathbb{Q}[X]$),

• (F) $B_j(X)={(-1)}^j{B}_j(1-X)$.

Consider the hyperelliptic curve

$$\mathcal{C}:y^2=F(x):=x^5+b_4{x}^4+b_3{x}^3+b_2{x}^2+b_{1}x+b_0,$$

(2)

where $b_i\in \mathbb{Z}$. Let α be a root of F, and let $J(\mathbb{Q})$ be the Jacobian of the curve . We have that

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

where $\kappa ,\xi \in K=\mathbb{Q}(\alpha )$ and 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

$$\kappa =\prod _{i=1}^m({\gamma }_i-\alpha d_i^2),$$

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]).

In [10], an improved completely explicit upper bound were proved combining ideas from [15, 19–25]. Now we will state the theorem, which gives the improved bound. We introduce some notation. Let 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

$${\partial }_K=\{\begin{array}{cc}\frac{log2}{d}\hfill & \text{if }d=1,2,\hfill \\ \frac{1}{4}{(\frac{loglogd}{logd})}^3\hfill & \text{if }d\ge 3,\hfill \end{array}$$

and let

$${\partial }_K^{\mathrm{\prime }}={(1+\frac{{\pi }^2}{{\partial }_K^2})}^{1/2}.$$

Define the constants

$$\begin{array}{c}{c}_1(K)=\frac{{(r!)}^2}{2^{r-1}{d}^r},c_2(K)=c_1(K){\left(\frac{d}{{\partial }_K}\right)}^{r-1},\hfill \\ c_3(K)=c_1(K)\frac{d^r}{{\partial }_K},c_4(K)=rd{c}_3(K),c_5(K)=\frac{r^{r+1}}{2{\partial }_K^{r-1}}.\hfill \end{array}$$

Let

$${\partial }_{L/K}=max\{[L:\mathbb{Q}],[K:\mathbb{Q}]{\partial }_K^{\mathrm{\prime }},\frac{0.16[K:\mathbb{Q}]}{{\partial }_K}\},$$

where $K\subseteq L$ are number fields. Define

$$C(K,n):=3\cdot {30}^{n+4}\cdot {(n+1)}^{5.5}{d}^2(1+logd).$$

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

$$K_1=\mathbb{Q}({\alpha }_1,{\alpha }_2,\sqrt{{\kappa }_1{\kappa }_2}),K_2=\mathbb{Q}({\alpha }_1,{\alpha }_3,\sqrt{{\kappa }_1{\kappa }_3}),K_3=\mathbb{Q}({\alpha }_2,{\alpha }_3,\sqrt{{\kappa }_2{\kappa }_3})$$

and

$$L=\mathbb{Q}({\alpha }_1,{\alpha }_2,{\alpha }_3,\sqrt{{\kappa }_1{\kappa }_2},\sqrt{{\kappa }_1{\kappa }_3}).$$

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

$$c_j^{\ast }=\underset{1\le i\le 3}{max}{c}_j(K_i),$$

and let

$$N=\underset{1\le i,j\le 3}{max}{\left|{Norm}_{\mathbb{Q}({\alpha }_i,{\alpha }_j)/\mathbb{Q}}({\kappa }_i({\alpha }_i-{\alpha }_j))\right|}^2,$$

and let

$$H^{\ast }=c_5^{\ast }R+\frac{logN}{{min}_{1\le i\le 3}[K_i:\mathbb{Q}]}+\mathrm{h}(\kappa ).$$

Define

$$A_1^{\ast }=2H^{\ast }\cdot C(L,2r+1)\cdot {\left(c_1^{\ast }\right)}^2{\partial }_{L/L}\cdot {\left(\underset{1\le i\le 3}{max}{\partial }_{L/K_i}\right)}^{2r}\cdot R^2$$

and

$$A_2^{\ast }=2H^{\ast }+A_1^{\ast }+A_1^{\ast }log\{(2r+1)\cdot max\{c_4^{\ast },1\}\}.$$

If $x\in \mathbb{Z}\mathrm{\setminus }\{0\}$ satisfies $x-\alpha =\kappa {\xi }^2$ for some $\xi \in K$ then

$$log|x|\le 8A_1^{\ast }log\left(4A_1^{\ast }\right)+8A_2^{\ast }+H^{\ast }+20log2+13\mathrm{h}(\kappa )+19\mathrm{h}(\alpha ).$$

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]).

Let $C/\mathbb{Q}$ be a smooth projective curve (in our case a hyperelliptic curve) of genus $g\ge 2$. Let 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

$$ȷ:C\to J,P\mapsto [P-D].$$

Let 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

$$ȷ(C(\mathbb{Q}))\subseteq W+BJ(\mathbb{Q}).$$

Let

$$\varphi :{\mathbb{Z}}^r\to J(\mathbb{Q}),\varphi (a_1,\dots ,a_r)=\sum a_i{D}_i,$$

so that the image of 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 ${\mathbb{Z}}^r$

$$B{\mathbb{Z}}^r=L_0⊋L_1⊋L_2⊋\cdots ⊋L_k$$

such that

$$ȷ(C(\mathbb{Q}))\subset W+\varphi (L_j)$$

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 $ȷ(C(\mathbb{Q}))\subset W+\varphi (L)$. Let ${\mu }_1$ be a lower bound for $h-\hat{h}$ and

$${\mu }_2=max\{\sqrt{\hat{h}(w)}:w\in W\}.$$

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

$${\mu }_3=min\{\sqrt{{\lambda }_j}:j=1,\dots ,r\},$$

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 $ȷ(P)\in W$ or

$$h(ȷ(P))\ge {({\mu }_{3}m(L)-{\mu }_2)}^2+{\mu }_1.$$

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

Lemma 5 The integral solutions of the equation

$$\mathcal{C}:Y^2=X{(X+20)}^2(X^2+10X+400)+140\text{,}625$$

(3)

are

$$(X,Y)\in \{(0,\pm 375),(-20,\pm 375)\}.$$

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

$$\begin{array}{c}{D}_1=(0,375)-\mathrm{\infty },\hfill \\ D_2=(-20,375)-\mathrm{\infty }.\hfill \end{array}$$

Let $f=x{(x+20)}^2(x^2+10x+400)+140\text{,}625$, and let 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

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

where $\kappa \in \mathcal{K}$, and is constructed as described in Lemma 2. We have that $\mathcal{K}=\{1,-\alpha ,-20-\alpha ,\alpha (\alpha +20)\}$. By local arguments, it is possible to restrict the set further (see, e.g., [26, 30]). In our case, one can eliminate

$$\alpha (\alpha +20)$$

by local computations in ${\mathbb{Q}}_3$. We apply Lemma 3 to get a large upper bound for $log|x|$ in the remaining cases. A MAGMA code was written to obtain the bounds that appeared in [10]; they can be found at http://www.warwick.ac.uk/~maseap/progs/intpoint/bounds.m. We obtain that these bounds are as in Table 1.

Table 1 Bounds

The set of known rational points on the curve (3) is $\{\mathrm{\infty },(0,\pm 375),(-20,\pm 375)\}$. Let 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 $ȷ(C(\mathbb{Q}))\subseteq W+BJ(\mathbb{Q})$, where

$$B=2^8\cdot 5^3\cdot 7^2\cdot {11}^2\cdot {13}^2\cdot {17}^2\cdot 19\cdot 31\cdot 37\cdot 41\cdot 43\cdot 53\cdot 59\cdot 71\cdot 79\cdot 83\cdot 89$$

that is

$$B=46\text{,}128\text{,}223\text{,}306\text{,}000\text{,}188\text{,}203\text{,}435\text{,}897\text{,}312\text{,}000.$$

Now, we use an extension of the Mordell-Weil sieve due to Samir Siksek to obtain a very long decreasing sequence of lattices in ${\mathbb{Z}}^2$. After that, we apply Lemma 4 to obtain a lower bound for possible unknown rational points. We get that if $(x,y)$ is an unknown integral point, then

$$log|x|\ge 2.216448\times {10}^{782}.$$

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

$$\frac{n(n+1)}{2}=\frac{m^2{(m+1)}^2}{3},$$

(4)

$$\frac{n(n+1)}{8}=\frac{1}{15}{z}^2(2z-1)\text{with }z=m(m+1),$$

(5)

$$\frac{n(n+1)}{8}=\frac{2}{21}{z}^2(3z^2-4z+2)\text{with }z=m(m+1)$$

(6)

and

$$\frac{1}{4}\sum _{i=1}^{n}i=\frac{64}{9}\sum _{i=1}^m{i}^9,$$

(7)

respectively. One can see that the first three equations are elliptic Diophantine equations, thus using the program package MAGMA, subroutines IntegralPoints or IntegralQuarticPoints are just a straightforward calculation to solve them. In these cases, the unique solution is $(l,m,n)=(0,24,5)$. The forth equation can be written as follows

$${(2n+1)}^2=\frac{128}{45}(m^2+m-1){(m^2+m)}^2(2m^4+4m^3-m^2-3m+3)+1.$$

So, we easily obtain a hyperelliptic curve

$$Y^2=X{(X+20)}^2(X^2+10X+400)+140\text{,}625,$$

where $Y=375(2n+1)$ and $X=20m^2+20m-20$. 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

$$p\cdot n^2{(n+1)}^2=3\cdot 4^{l+1}(1^p+2^p+\cdots +m^p).$$

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

$${ord}_p\left(\frac{1^p+2^p+\cdots +m^p}{m^2{(m+1)}^2}\right)={ord}_p\frac{B_{p+1}(m+1)-B_{p+1}}{m^2{(m+1)}^2}\ne 0.$$

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. □