---
abstract: 'By the Cole-Hopf transformation, with any linear evolution equation in $1+1$ dimensions a generalized Burgers equation is associated. We describe local conservation laws of these equations. It turns out that any generalized Burgers equation has only one conservation law, while a linear evolution equation with constant coefficients has an infinite number of $(x,t)$-independent conservation laws iff the equation involves only odd order terms and, therefore, is bi-Hamiltonian.'
address: 'University of Twente, Faculty of Mathematical Sciences, P.O. Box 217, 7500 AE Enschede, the Netherlands'
author:
- Sergei Igonin
title: |
    Conservation laws of generalized higher Burgers\
    and linear evolution equations
---

It is well known that the classical Burgers equation $v_t=v_{xx}+2v_xv$ is obtained from the linear heat equation $u_t=u_{xx}$ by the Cole-Hopf transformation $v=u_x/u$, but it is less known that this construction can be applied to an arbitrary linear evolution equation $$\label{lin}
 u_t=\sum_{i=0}^m a^i(x,t)u_i,$$ where, as usual, $u_i={{\partial}^i u}/{{\partial}x^i}$ and $$\label{m}
  a^m(x,t)\neq 0.$$

Indeed, denote $v=v_0=u_1/u$ and $v_k=D_x^k(v)$, then $$v_t=D_tD_x(\ln|u|)=
D_x(\frac{u_t}u)=D_x\Bigl(\sum_{i=0}^m a^i(x,t)(D_x+v)^i(1)\Bigl),$$ where $D_x$ and $D_t$ are the total derivatives, because it is easily seen that $$\frac{u_i}{u}=(D_x+v)^i(1).$$ The equation $$\label{burg}
  v_t=D_x\Bigl(\sum_{i=0}^m a^i(x,t)(D_x+v)^i(1)\Bigl)$$ is called the [*generalized Burgers equation*]{} associated with linear equation . This is an example of nonlinear $\mathrm{C}$-integrable equations in the sense of Calogero [@C].

Consider linear equations with constant coefficients $$\label{lin1}
 u_t=\sum_{i=0}^m a^i u_i,\ a_m\neq 0.$$ Clearly, flows mutually commute for arbitrary $a_i\in{\mathbb{R}}$, hence flows with constant coefficients $a^i(x,t)=a^i$ also commute and form the integrable [*Burgers hierarchy*]{} (see, for example, [@burg]). It passes the Painlevé test [@P], but, as it follows from Theorem \[clB\] below, any generalized Burgers equation has only one local conservation law. Note that this seems to be the only nonlinear integrable hierarchy that allows complete description of conservation laws for all its members.

In the present article we study conservation laws of equations , . This question is trivial in the case of evolution equations of order $1$, and from now on we assume $m\ge 2$.

Recall [@rb; @olver] that a smooth function ${\varphi}(x,t,u,u_1,\dots,u_k)$ is called a [*conserved density*]{} of an equation $$\label{evol}
 u_t=F(x,t,u,u_1,\dots,u_m)$$ if $D_t({\varphi})=D_x(\psi(x,t,u,u_1,\dots,u_l))$ for some function $\psi$. Two conserved densities ${\varphi}_1,\,{\varphi}_2$ are said to be [*equaivalent*]{} if ${\varphi}_1-{\varphi}_2=D_x(\psi)$, and conserved densities of the form $D_x(\psi)$ are called [*trivial*]{}. Equivalence classes of conserved densities are called [*conservation laws*]{}.

The function $v$ is a nontrivial conserved density for .

Let ${\varphi}$ be a conserved density for , then its [*characteristic*]{} [@olver] (also called the [*generating function*]{} [@rb; @vin]) is the variational derivative of ${\varphi}$ $$\label{ch}
  \xi=\frac{\delta{\varphi}}{\delta u}=\sum_{i\ge 0}(-1)^iD_x^i(\frac{{\partial}{\varphi}}{{\partial}u_i}).$$ The characteristic satisfies the equation $$\label{che_g}
  -D_t(\xi)=\sum_{i=0}^m (-1)^iD_x^i(\frac{{\partial}F}{{\partial}u_i} \xi)$$ and is nonzero if and only if the density ${\varphi}$ is nontrivial, i.e., equivalent conserved densities have the same characteristic. The homological interpretation of these concepts can be found in [@rb; @vin].

For a function $h(x,t,u,u_1,\dots)$ the maximal integer $k$ such that ${{\partial}h}/{{\partial}u_k}\neq 0$ will be called the [*order*]{} of $h$ and denoted by ${\mathrm{o}}(h)$. If ${{\partial}h}/{{\partial}u_k}=0$ for all $k>0$, we set ${\mathrm{o}}(h)=0$.

\[xi\] The characteristic $\xi$ of any conserved density for has the form $$\label{ch_lin}
  \xi=a(x,t)+\sum_{i}b^i(x,t)u_i$$ such that the operator $\sum_i b^i(x,t)D_x^i$ is self-adjoint, i.e., $$\label{sa}
  \sum_i b^i(x,t)D_x^i(\psi)=\sum_i (-1)^iD_x^i(b^i(x,t)\psi).$$ for any smooth function $\psi=\psi(x,t,u,u_1,\dots)$.

We must prove $$\label{sol}
  \frac{{\partial}^2\xi}{{\partial}u_i{\partial}u_j}=0,\ \ \forall\,i,\,j\ge 0,$$ then follows from the properties of variational derivatives [@rb; @olver]. If is not true then there exist $i_0\ge j_0\ge 0$ such that $$\label{as}
  \frac{{\partial}^2\xi}{{\partial}u_{i_0}{\partial}u_{j_0}}\neq 0$$ and $$\label{sol1}
  \frac{{\partial}^2\xi}{{\partial}u_i{\partial}u_j}=0,\ \ \forall\,i,j: i>i_0,\,i+j\ge i_0+j_0.$$ For equation reads $$\label{che}
  -D_t(\xi)=\sum_{i=0}^m (-1)^iD_x^i(a^i(x,t)\xi).$$ Differentiating with respect to $u_{{\mathrm{o}}(\xi)+m}$, we get $$-a^m(x,t)\frac{{\partial}\xi}{{\partial}u_{{\mathrm{o}}(\xi)}}=
(-1)^m \cdot a^m(x,t)\frac{{\partial}\xi}{{\partial}u_{{\mathrm{o}}(\xi)}}.$$ Combining this identity with , we see that $m$ is odd. Now differentiating with respect to $u_{i_0+m-1},\,u_{j_0+1}$ and taking into account , one obtains $$-a^m(x,t)\frac{{\partial}^2\xi}{{\partial}u_{i_0-1}{\partial}u_{j_0+1}}
=-a^m(x,t)\frac{{\partial}^2\xi}{{\partial}u_{i_0-1}{\partial}u_{j_0+1}}
-m\cdot a^m(x,t)\frac{{\partial}^2\xi}{{\partial}u_{i_0}{\partial}u_{j_0}},$$ which contradicts to our assumption ; hence is true.

\[clB\] Any conserved density of a generalized Burgers equation is equivalent to $Cv$ for some $C\in{\mathbb{R}}$, i.e., the space of conservation laws is one-dimensional.

For the classical Burgers equation this fact is well known and follows from the general observation [@rb; @olver] that for any even order evolution equation there is an upper bound on the order of the characteristics of conservation laws. However, these arguments do not work for generalized Burgers equations of arbitrary order considered in this theorem.

For a conservation law $\Omega$ of consider a conserved density ${\varphi}\in\Omega$ of minimal order $k$. Replacing $v_k$ by $D_x^k(u_1/u)$ in ${\varphi}$, we obtain a conserved density $\tilde {\varphi}$ for . By the construction, $\tilde {\varphi}$ is invariant under the symmetry $u_k\mapsto\lambda u_k$ of , $k\ge 0,\,\lambda\in{\mathbb{R}}$. Then from for the characteristic $\xi$ of $\tilde {\varphi}$ one has $$\label{sym}
  \xi(x,t,\lambda u,\dots,\lambda u_{2k+2})=
  \lambda^{-1}\xi(x,t,u,\dots,u_{2k+2}),\ \ \forall\,\lambda\neq 0.$$ Combining with , we obtain $\xi=0$, that is, $$\label{triv}
  \tilde{\varphi}=D_x\psi(x,t,u,\dots,u_{k})$$ for some function $\psi$.

From we have ${{\partial}^2{\varphi}}/{{\partial}u_{k+1}^2}=0$, therefore, ${{\partial}^2{\varphi}}/{{\partial}v_k^2}=0$. This implies $k=0$, because if $k>0$, one easily constructs an equivalent conserved density ${\varphi}'$ with ${\mathrm{o}}({\varphi}')<k={\mathrm{o}}({\varphi})$, which contradicts to our assumption that ${\varphi}\in\Omega$ is of minimal order. Thus ${\varphi}=l(x,t)v+n(x,t)$, $\tilde{\varphi}=l(x,t)u_1/u+n(x,t)$, and from we obtain that $l(x,t)=l(t)$ does not depend on $x$. Finally, from the condition that ${\varphi}$ is a conserved density it follows that $l(t)$ is actually a constant.

In contrast to generalized Burgers equations, for a linear equation the space of conservation laws may be infinite-dimensional.

For a linear evolution equation with constant coefficients the space $\mathcal{C}\!\mathcal{L}$ of $(x,t)$-independent conservation laws is described as follows.

1.  If $a^0\neq 0$ then $\mathcal{C}\!\mathcal{L}=0$.

2.  If $a^0=0$, but $a^{2j}\neq 0$ for some $j\in\mathbb{N}$ then $\mathcal{C}\!\mathcal{L}$ is one-dimensional and generated by the conserved density $u$.

3.  If all the even coefficients $a^{2j}$ vanish, $\mathcal{C}\!\mathcal{L}$ is infinite-dimensional and generated by the conserved densities $$\label{cl}
      u,\ u_k^2,\ \ k\ge 0.$$

Let ${\varphi}={\varphi}(u,\dots,u_k)$ be an $(x,t)$-independent conserved density. From and it follows that the characteristic $\xi$ of any $(x,t)$-independent conserved density has the form $$\label{ch_const}
  \xi=c+\sum_{j} c^j u_{2j},\ \  c,\,c^j\in{\mathbb{R}}.$$ Equation reads $$\label{che1}
  -\sum_{j}\sum_{i=0}^m a^ic^ju_{i+2j}=
  \sum_{j}\sum_{i=0}^m (-1)^{i}a^ic^ju_{i+2j}+a^0c.$$ Clearly, if there is a nonzero even coefficient $a^{2j}\neq
0,\,j\in\mathbb{N},$ then implies $\xi=c\in{\mathbb{R}}$, while if $a^0\neq 0$ then $\xi=0$. On the other hand, if $a^{2j}=0$ for all $j$ then is valid for any constants $c,\,c^j$. It is easily seen that if $a^0=0$ then $cu$ is a conserved density with the characteristic $\xi=c$, while if $a^{2j}=0$ for all $j$ then the function ${\varphi}=u_j^2$ is a conserved density with characteristic $\xi=(-1)^j 2u_{2j}$. Hence every function satisfying is indeed a characteristic of a linear combination of the above described conserved densities, which thus span the whole space of conservation laws.

Conservation laws of a linear equation $$\label{lin3}
  u_t=\sum_{j\ge 0}a^{2j+1}u_{2j+1},\ a^{2j+1}\in\mathbb{R},$$ can be derived as follows. Performing the Galilean transformation $$x\mapsto x+a^1t,\ t\mapsto t,\ u\mapsto u,$$ we obtain an equation of the form with $a^1=0$. Such an equation possesses a bi-Hamiltonian structure [@H] $$\label{bH}
  \begin{aligned}
  u_t & = D_x\Bigl(\frac{\delta}{\delta u}\sum_j(-1)^j\frac12 a^{2j+1}u_{j}^2\Bigl),\\   
  u_t & =
   D_x^3\Bigl(\frac{\delta}{\delta u}\sum_{j>0}(-1)^{j-1}\frac12 a^{2j+1}u_{j-1}^2\Bigl).    
  \end{aligned}$$ It is easily seen that conservation laws are obtained from this by the Lenard scheme [@L; @olver] and, therefore, are in involution with respect to the both corresponding Poisson brackets. According to [@H], using and the Cole-Hopf transfomation, one can also construct a nonlocal bi-Hamiltonian structure for the generalized Burgers equation associated with .

Acknowledgments {#acknowledgments .unnumbered}
===============

The author would like to thank Prof. I. S. Krasil[[[$'$]{}]{}]{}shchik and Prof. R. Martini for useful remarks.

[99]{}

A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor[[[$'$]{}]{}]{}kova, I. S. Krasil[[[$'$]{}]{}]{}shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov. *Symmetries and Conservation Laws for Differential Equations of Mathematical Physics*. American Mathematical Society, Providence, RI, 1999.

F. Calogero. Why are certain nonlinear [P]{}[D]{}[E]{}s both widely applicable and integrable?, *What Is Integrability?*, 1–62. Springer-Verlag, Berlin, 1991.

I. Dorfman. *Dirac Structures and Inegrability of Nonlinear Evolution Equations*. John Wiley & Sons, Chichester, 1993.

D. Levi, O. Ragnisco, and M. Bruschi. Continuous and discrete matrix Burgers’ hierarchies, *Nuovo Cimento B (11)* **74** (1983), 33–51.

P. J. Olver, *Applications of Lie Groups to Differential Equations*. Springer-Verlag, New York, 1993.

A. Pickering. The Weiss-Tabor-Carnevale Painlevé test and Burgers’ hierarchy, *J. Math. Phys.* **35** (1994), 821–833.

H. Tasso. Hamiltonian formulation of odd [B]{}urgers hierarchy, *J. Phys. A: Math. Gen.* **29** (1996), 7779–7784.

A. M. Vinogradov. The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The nonlinear theory. *J. Math. Anal. Appl.* **100** (1984), 1–129.
