Complete Description of Local Conservation Laws for Generalized Dissipative Westervelt Equation

Abstract

We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables.

1 Introduction

Consider a partial differential equation in $n+1$ independent variables $t,x_1,\ldots ,x_n$ and dependent variable u,

$$\begin{aligned} \left( f(u)\right) _{tt}+a u_{ttt}+\Delta u=0, \end{aligned}$$

(1)

where n is a positive integer, a is a nonzero constant, f is an arbitrary nonlinear smooth function of u, and $\Delta =\sum _{i=1}^n\partial ^2/\partial x_i^2$; the subscripts indicate partial derivatives in the usual manner.

We shall hereinafter refer to this equation as to the generalized dissipative Westervelt equation (GDW equation for short), as it is a natural generalization of the well-known dissipative Westervelt equation (DWE), see e.g. [2, 12, 27]. Indeed, DWE is recovered from (1) if $f=k_2 u^2+k_1 u$ for constant $k_1$ and $k_2$.

Note that DWE has numerous applications in nonlinear acoustics etc. with u being interpreted as a (non-dimensional) pressure fluctuation, cf. e.g. [2, 12, 27] and references therein.

To the best of our knowledge, the only earlier work on conservation laws of (G)DW equation is [2], where for $n=1$ a complete list of inequivalent nontrivial local conservation laws with second-order characteristics for DWE was found but the matter of possible existence of nontrivial local conservation laws with higher-order characteristics was not settled.

Our goal is to provide, in Theorem 1 below, a complete description, modulo trivial conservation laws, of all nontrivial local conservation laws for (1) with $a\ne 0$ and nonlinear smooth f(u). To achieve this, we use, inter alia, a certain result from [14] (see Theorem 2 in Appendix for details) combined with the direct method for search of conservation laws [1]; cf. also Chap. 5 of [21].

A complete description of local conservation laws for a given PDE as outlined above is of interest as there are many important applications for conservation laws and associated integrals of motion, cf. e.g. Chap. 4 in [21], as well as [2, 4,5,6,7,8, 10, 11, 18,19,20, 22, 24, 25], and references therein. For instance, certain existence results for PDEs use conservation laws in their proofs, cf. e.g. [7, 8], and it is known that discretizations of a PDE tend to behave better when taking into account the (nontrivial) conservation laws admitted by the PDE in question, see e.g. [4, 10] and cf. related works [9, 15,16,17].

Finding all nontrivial local conservation laws for a given PDE is a difficult problem that was successfully addressed in the past only for a rather small number of equations, see e.g. [13, 14, 23, 26] and references therein, and to the best of our knowledge the GDW equation is not among those examples.

What is more, the overwhelming majority of PDEs with all nontrivial local conservation laws known is strictly evolutionary, i.e., the equations under study have the form of the time derivative of the dependent variable being equal to a function of independent variables, dependent variable and its partial derivatives with respect to independent variables not involving time. The GDW equation is not strictly evolutionary which makes the study of its conservation laws even more nontrivial.

The rest of the paper is organized as follows: in Sect. 2 we present our main result, Theorem 1, and discuss it, while Sect. 3 presents the proof thereof. In order to make the paper more self-contained, we have added an Appendix presenting the result from [14] that we use in the proof of Theorem 1 and a number of related definitions and formulas.

2 Main Result and Discussion

We are now ready to state our main theorem:

Theorem 1

For Eq. (1) with any nonlinear smooth function f(u) and with any nonzero constant a all nontrivial local conservation laws of all orders are, modulo the addition of trivial ones, of the following form:

$$\begin{aligned} \bigl ((f'(u)u_t+a u_{tt})\chi -(au_t+f(u))\chi _t\bigr )_{t} +\sum \limits _{j=1}^n \left( u_{x_j}\chi - u\chi _{x_j}\right) _{x_j}=0, \end{aligned}$$

(2)

where $\chi =\varphi _0+t\varphi _1$ and $\varphi _\alpha =\varphi _\alpha (x_1,\ldots ,x_n)$ are arbitrary smooth functions of their arguments satisfying

$$\begin{aligned} \Delta \varphi _\alpha =0,\quad \alpha =0,1. \end{aligned}$$

(3)

As usual, cf. e.g. [21], the conservation laws in (2) are written as differential identities holding modulo (1) and its differential consequences.

Before proceeding to the proof note that the above result shows, in particular, that the case of the original DWE for which $f=k_2 u^2+k_1 u$ with a nonzero constant $k_2$ is not distinguished by presence of additional nontrivial local conservation laws.

Also observe that for $n=1$ a general smooth solution of (3) takes a particularly simple form

$$\begin{aligned} \begin{array}{l} \varphi _\alpha =\eta ^{(0)}_\alpha +\eta ^{(1)}_\alpha x_1 \end{array} \end{aligned}$$

where $\eta ^{(\beta )}_\alpha$, $\alpha ,\beta =0,1$, are arbitrary constants.

If f is quadratic in u and $n=1$ then we recover, up to slight differences in notation, the local conservation laws for DWE found in [2] (our $a\ne 0$ corresponds to $\alpha \ne 0$ in the notation of [2]). Moreover, our result goes beyond that of [2] in that we show that the four conservation laws in question ((20)–(23) in [2]) exhaust all inequivalent nontrivial local conservation laws with characteristics of all orders for DWE with $n=1$ and quadratic f with nonzero quadratic term (recall that we have $a\ne 0$ in (1) from the outset).

Note also that using integral representations for solutions of homogeneous Laplace equation it is possible to write down a general smooth solution for (3) with $n=2,3,\ldots$, involving a total of four arbitrary functions of $n-1$ independent variables, but the resulting formulas are rather cumbersome and thus are left beyond the scope of the present paper.

It is now clear that for any integer $n>1$ Eq. (1) with nonzero a and nonlinear smooth f has infinitely many inequivalent nontrivial local conservation laws, while for $n=1$ there are just four inequivalent nontrivial local conservation laws for the equation under study.

3 Proof of the Main Result

Since Eq. (1) is normal in the sense of [21], any nontrivial local conservation law for (1) should have, see e.g. [21], a nonzero characteristic $\chi$ which is a local function, see Appendix for further details on the latter.

Now observe that Eq. (1) with arbitrary smooth function f of u and nonzero constant a can be written in the form (8) with $h=3$, $F=-(1/a)(\left( f(u)\right) _{tt}+\Delta u)$ and $x\text {-}{\text {ord}}F=2$. Since for (1) we have $h+x\text {-}{\text {ord}}F=5$, it is readily checked that (1) satisfies the conditions of Theorem 2 from Appendix with $O_F=-1$, so characteristics of local conservation laws of the equation under study may depend at most on the independent variables $t,x_1,\ldots ,x_n$.

Thus, any nontrivial local conservation law for (1) is equivalent (modulo the addition of a trivial local conservation law) to a nontrivial local conservation law for which we have

$$\begin{aligned} \rho _{t} +\sum \limits _{j=1}^n (\sigma _j)_{x_j}=\chi \cdot (f_{tt}+a u_{ttt}+\Delta u), \end{aligned}$$

(4)

where the characteristic $\chi$ may depend at most on independent variables $t,x_1,\ldots ,x_n$ and $\rho$ and $\sigma _j$, $j=1,\ldots ,n$, are local functions.

As per the discussion in Appendix, the above characteristic $\chi =\chi (t,x_1,\ldots ,x_n)$ must satisfy

$$\begin{aligned} f_u\chi _{tt}-a\chi _{ttt}+\Delta \chi =0 \end{aligned}$$

(5)

To proceed, observe that as $f_{uu}$ is nonzero by virtue of our assumption of f being nonlinear, 1 and $f_u$ are linearly independent as functions of u.

Equating to zero the coefficients at linearly independent functions $f_u$ and 1 in (5) yields

$$\begin{aligned} \chi _{tt}=0 \end{aligned}$$

(6)

and

$$\begin{aligned} -a\chi _{ttt} +\Delta \chi =0. \end{aligned}$$

(7)

From (6) we get

$$\begin{aligned} \chi =\varphi _0+t\varphi _1 \end{aligned}$$

where $\varphi _\alpha =\varphi _\alpha (x_1,\ldots ,x_n)$, $\alpha =0,1$, are arbitrary smooth functions of their arguments.

Substituting this into (7) and equating to zero the coefficients at the powers of t yields

$$\begin{aligned} \Delta \varphi _\alpha =0,\quad \alpha =0,1 \end{aligned}$$

and thus we arrive at (3).

It is now a straightforward matter to verify that any smooth $\chi =\varphi _0+t\varphi _1$ with smooth functions $\varphi _\alpha =\varphi _\alpha (x_1,\ldots ,x_n)$, $\alpha =0,1$, satisfying (3) is a characteristic for a local conservation law of the form (2) for (1), and the result follows.

Appendix: Conservation Laws for PDEs of the Form (8)

Given a multi-index $I=(i_1,\ldots ,i_n)\in \mathbb {Z}_+^n$, where $\mathbb {Z}_+$ denotes the set of nonnegative integers, and an integer $s\ge 0$, let $u_{s,I}=\partial ^{|I|+s} u/\partial t^s\partial x_1^{i_1}\cdots \partial x_n^{i_n}$, where $|I|=i_1+\cdots +i_n$, cf. [14].

Now, following [14], consider a PDE of the form

$$\begin{aligned} \partial ^h u/\partial t^h=F(x_1,\ldots ,x_n,t,u,\partial u/\partial t, \ldots \partial ^{h-1}u/\partial t^{h-1}, u_{0,I_1},\ldots ,u_{0,I_m}) \end{aligned}$$

(8)

where h and m are positive integers, and $I_k\in \mathbb {Z}_+^n$ are pairwise distinct multi-indices with $I_k\ne (0,\ldots ,0)$ for all $k=1,\ldots ,m$.

A smooth function depending at most on $t,x_1,\ldots ,x_n$ and finitely many $u_{s,I}$ with $s<h$ is called in this setting a local function.

For a local function f define¹ [14] its x-order $x\text {-}{\text {ord}}f$ as the greatest integer k such that there exist $s\in \mathbb {Z}_+$ and $I\in \mathbb {Z}_+^n$ for which $|I|=k$ and $\partial f/\partial u_{s,I}\ne 0$. If $\partial f/\partial u_{s,I}=0$ for all s and I then we set, following [14], $x\text {-}{\text {ord}}f=-1$.

Further following [14], consider the operators of total derivatives adapted to (8):

$$\begin{aligned} D_{x_i}= & {} \displaystyle \frac{\partial }{\partial x_i}+\sum _{I\in \mathbb {Z}_+^n}\sum _{s=0}^{h-1}u_{s,I+1_i}\frac{\partial }{\partial u_{s,I}},\quad D_t=\displaystyle \frac{\partial }{\partial t}+\sum _{s=0}^{h-2}\sum _{I\in \mathbb {Z}_+^n} u_{s+1,I}\frac{\partial }{\partial u_{s,I}}\\{} & {} \quad + \sum _{I\in \mathbb {Z}_+^n} D_x^I(F)\frac{\partial }{\partial u_{h-1,I}}, \end{aligned}$$

where $1_i$ is a multi-index with the only nonzero entry being 1 at the ith place, and for $I=(i_1,\ldots ,i_n)$ we have $D_x^I=D_{x_1}^{i_1}\circ \cdots \circ D_{x_n}^{i_n}$; consider [14] also an operator

$$\begin{aligned} \mathcal {M}_F=\sum _{I\in \mathbb {Z}_+^n} (-1)^{|I|}D_x^I\circ \frac{\partial F}{\partial u_{0,I}}+ \sum _{s=0}^{h-1}(-1)^s D_t^s\circ \frac{\partial F}{\partial u_{s,0}}\equiv \sum _{I\in \mathbb {Z}_+^n} f_I D_x^I+ \sum _{s=0}^{h-1}g_s D_t^s, \end{aligned}$$

and let

$$\begin{aligned} O_F=\max _{I,s}\{-1,\,x\text {-}{\text {ord}}f_I- x\text {-}{\text {ord}}F,\,x\text {-}{\text {ord}}g_s- x\text {-}{\text {ord}}F\}. \end{aligned}$$

A local conservation law for (8) is in this setting, cf. e.g. [14, 21], an identity of the form²

$$\begin{aligned} D_t J_0+\sum _{i=1}^n D_{x_i}J_i=0, \end{aligned}$$

(9)

where $J_0$ and $J_i$ are local functions; it can be shown, see e.g. [14], that the characteristic of this conservation law has the form $\sum _{I\in \mathbb {Z}_+^n} \partial J_0/\partial u_{h-1,I}$.

A local conservation law for (8) is nontrivial if its characteristic is nonzero and trivial otherwise (see e.g. [1, 21] and references therein for further details on these concepts).

Two local conservation laws for (8), say, (9) and

$$\begin{aligned} D_t \tilde{J}_0+\sum _{i=1}^n D_{x_i}\tilde{J}_i=0, \end{aligned}$$

are called equivalent if their difference, i.e.,

$$\begin{aligned} D_t (J_0-\tilde{J}_0)+\sum _{i=1}^n D_{x_i}(J_i-\tilde{J}_i)=0, \end{aligned}$$

is a trivial local conservation law, see e.g. [14, 21] for details.

In this connection recall [14] that if a local function $\chi$ is a characteristic for some local conservation law for (8) then $\chi$ must satisfy the equation

$$\begin{aligned} (-1)^h D_t^h(\chi )= \mathcal {M}_F(\chi ). \end{aligned}$$

(10)

For example, for a local characteristic $\chi$ of a local conservation law for (1), which can be rewritten in the form (8) as $u_{3,0}=-(1/a)(D_t^2(f(u))+\sum _{i=1}^n D_{x_{i}}^2(u))$, (10) takes the form

$$\begin{aligned} -D_t^3(\chi )=-\frac{1}{a}\left( \frac{\partial f}{\partial u} D_t^2(\chi )+\sum \limits _{i=1}^n D_{x_{i}}^2(\chi )\right) \end{aligned}$$

and if $\chi =\chi (t,x_1,\ldots ,x_n)$ then, upon multiplication by a which is nonzero by assumption, and bringing all terms to the left-hand side, the above equation boils down to (5), that is,

$$\begin{aligned} \frac{\partial f}{\partial u} \frac{\partial ^2\chi }{\partial t^2}-a\frac{\partial ^3\chi }{\partial t^3}+\sum \limits _{i=1}^n\frac{\partial ^2\chi }{\partial x_i^2}=0. \end{aligned}$$

For proving Theorem 1 we shall also need the following general result (Theorem 6 of [14]):

Theorem 2

[14] Consider a fixed PDE of the form (8) such that $h+x\text {-}{\text {ord}}F$ is an odd integer greater than one. Then the PDE in question admits no nontrivial local conservation laws with characteristics of x-order higher than $O_F$.

The power of this theorem rests in the fact that under rather mild assumptions it tells us on which (finite) set of derivatives of u can at most depend the characteristics of nontrivial local conservation laws for (8), i.e., an extremely difficult problem of search for conservation law characteristics of all orders boils down to that for characteristics whose x-order is not higher than $O_F$ (and such characteristics hence depend only on a finite number of $u_{s,I}$). This latter problem often can be readily handled in full generality using, if need be, available computer algebra software packages like e.g. Jets [3] for Maple^®. Some examples of application of the result in question, other than Theorem 1 of the present paper or the examples in [14], can be found e.g. in [13].