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.