Conservation Laws of The Generalized Riemann Equations
- 10.1080/14029251.2018.1440746How to use a DOI?
- conserved densities; generalized Riemann equation; Gurevich-Zybin equation; Monge-Ampere equation; reciprocal transformation
Two special classes of conserved densities involving arbitrary smooth functions are explicitly formulated for the generalized Riemann equation at arbitrary N. The particular case when N = 2 covers most of the known conserved densities of the equation, and the result is also extended to the famous Gurevich-Zybin, Monge-Ampere and two-component Hunter-Saxton equations by considering certain reductions.
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The Riemann equation
Many efforts have been made to construct conserved densities for the generalized Riemann equations (in general or special cases) [2,10–12]. It turns out that these equations have extraordinary features, namely abundant conserved densities of non-polynomial type, which are mainly generated by iterative procedures [11,12], such as (F, G. and H are smooth functions of u1, u2,⋯,uN and their derivatives)
If F solves a linear equation ∂tF = ku1, xF + u1Fx, then F1 / k is a conserved density such that 
If there is a conservation law ∂tH = λ∂x(u1H) and F satisfies ∂tF = 2λu1, xF + λu1Fx, then F / H is a new conserved density such that 
Let Rk be a first order differential operator defined as(1.4)and G and F respectively satisfy then for non-negative integers m and n both and are conserved densities such that 
Such diversity and richness of conserved densities distinguish the generalized Riemann equations from a typical integrable system such as the Korteweg-de Vries equation. Indeed, typical integrable systems normally possess infinitely many conserved densities of differential polynomial type, which may be generated by their linear spectral problems  or the Lenard scheme with bi-Hamiltonian structures .
It is interesting to observe that most conserved densities of the generalized Riemann equations in the literature surprisingly obey a common conservation law, namely
Other than those listed above, are there new conserved densities which also fulfill Eq. (1.5)?
Could the general solution of Eq. (1.5) be constructed under certain condition?
Is it possible to find conservation laws of new types for the generalized Riemann equations?
Indeed, available results for those systems, which are closely related to the generalized Riemann equations, motivate one to expect more general conserved densities for the latter. On one hand, the Riemann equation (1.1) was shown by Olver and Nutku to have a class of conservation laws , given bywhere , and G is an arbitrary smooth function of its arguments. On the other hand, similar results have recently been achieved for the Hunter-Saxton equation  which also admits conserved densities involving arbitrary smooth functions.
The aim of this paper is to provide answers to above questions and offer better understanding on richness of conserved densities of the generalized Riemann equations. A method will be proposed to solve Eq. (1.5) so that conserved densities of the generalized Riemann equations are constructed. For the general N, a set of appropriate dynamical variables will be introduced and with the help of those variables the assumed conservation law will be brought to a first order linear Partial Differential Equation (PDE). Conserved densities will be obtained as the general solution to the linear PDE by the standard method of characteristics, and hence would involve arbitrary smooth functions which cast light on the diversity and richness of conserved densities presented by various authors [2,10–12]. Furthermore, conservation laws, which are different from Eq. (1.5), will be directly constructed from a simple ansatz on their densities. The generalized Riemann equation at N = 2, which appears to be the most important case and has close relation with several known systems, will be taken as an enlightening example to explain our methodology in detail, while results in the generic case are presented without proofs.
This paper is organized as follows. In section 2, the generalized Riemann equation at N = 2 is shown to have two kinds of conserved densities involving arbitrary functions by solving some assumed conservation laws (like Eq. (1.5)) with suitable dynamical variables. Three well known systems related to it, namely the Gurevich-Zybin system, the Monge-Ampere equation and the twocomponent Hunter-Saxton equation, are considered in section 3, and their conserved densities are deduced from those of the generalized Riemann equation at N = 2. In section 4, new dynamical variables are introduced for the generalized Riemann equation at arbitrary N, and they enable us to explicitly figure out conserved densities involving arbitrary smooth functions. As illustrations, general formulas are applied to the cases when N = 3,4. Conclusions are given in the last section.
2. The N = 2 case
By rescaling dependent variables as u1 = − 2u and u2 = 4v, the generalized Riemann equation at N = 2 is rewritten as
Differentiating the system (2.1) with respect to x yields
With the aid of 𝒯1 and 𝒯2, one can reduce the assumed conservation law (2.2) of the system (2.1) to a solvable PDE, and figure out conserved densities. According to (2.4), Eq. (2.2) is changed by 𝒯1 to (We temporarily omit the arguments of F for simplicity.)By means of 𝒯2 or in terms of p and q, above equation becomes
Based on the discussions so far, we conclude that E q. (2.2) is a conservation law of the system (2.1) if and only if the density F(ux, vx, u2x, v2 x,⋯,un x, vnx) is of the form
It is remarked that, the quantities α(j)’s and β(j)’s, which are simply deduced by applying to p, q and their derivatives with respect to y, play the important roles in formulating conserved densities. Next, we demonstrate that they have interesting properties, which lead us to more general results and bring us new ideas to study conservation laws in the general case as well.
When the system (2.1) holds,
Proof. For and direct calculations give us
It is noticed that both α(j) and β(j)(j ≥ 2) are defined by the same recursive relation
For any n ∈ ℤ + , the change of dynamical variablesis invertible.
Proof. To prove the invertibility of Γ, it would be enough to formulate (ux, vx,⋯,unx, vnx) in terms of (α(1), β(1), α(2), β(2), ⋯, α(n), β(n)). Solving ux and vx from explicit expressions of α(1) and β(1) (cf. Eqs. (2.10)) yieldsInferred from Eqs. (2.10), we get where 𝔾1 is a 2-dimensional vector function. The 2 × 2 matrix in the right hand side is non-singular, so we could solve ujx and vjx step by step, and formulate them in terms of (α(1), β(1), α(2), β(2), ⋯, α(j), β(j)). □
Via the invertible transformation Γ, any smooth function F(u, v, ux, vx,⋯,unx, vnx) could be reformulated as and vice versa, then its time-evolution is easily determined according to Lemma 2.1. Taking account of both lemmas, we will determine all conserved densities, which may explicitly depend on u and v, by solving the assumed conservation law
Indeed, in terms of α(j)’s and β(j)’s appears aswhich, after expanding all derivatives, yields Now taking into consideration the system (2.1) and Lemma 2.1, we find that above equation is simplified as which is a first-order non-homogeneous linear PDE for . Solving it by the standard method of characteristics, we obtain where G is an arbitrary smooth function of its arguments. Taking account of Eqs. (2.10) we find, in terms of the original variables, the conserved density namely . The result is summarized as
The system (2.1) admits the conservation law (2.12) if and only if
In particular, if the conserved densities of the type (2.13) do not explicitly depend on u and v, we recover the conserved densities of the type (2.9).
In the rest part of the section, we manage to find conservation laws other than in the form of Eq. (2.2) for the system (2.1). Since the conserved densities of the type (2.9) do not explicitly depend on β(1), it is interesting to construct the new conservation laws explicitly depending on β(1). Let us consider a smooth functionOn solutions to the system (2.1), by means of Lemma 2.1 we straightforwardly get A simple but interesting observation on the above expression is that if there exists a function Q(α(1), β(1), α(2), β(2), ⋯, α(n − 1), β(n −1)) such that
We summarize above discussions as
When the system (2.1) holds, given an arbitrary smooth function Q(α(1), β(1), α(2), β(2), ⋯, α(n − 1), β(n − 1)), let
As an implementation of Proposition 2.2, we construct a conserved density from the smooth function Q(α(1)) β(1). Now the formula (2.15) leads towhere Q′(α(1)) = ∂ Q / ∂α(1) and T(α(1), α(2), β(2)) is the “constant” of integration. Then, we obtain a conserved density such that
To conclude this section, we point out that while above results show that the generalized Riemann equation at N = 2 does possess abundant conserved densities, they by no means exhaust all possibilities since there are extra conservation laws explicitly depending on time , such as
3. The systems related to the N = 2 case
The generalized Riemann equation at N = 2 is interesting since it is associated with some important systems. In fact, through the appropriate changes of variables and/or reductions, the system (2.1) is connected/reduced to the Gurevich-Zybin, Monge-Ampere, two-component Hunter-Saxton and Hunter-Saxton equations. In this section, we demonstrate that the results obtained in last section allow us to construct the conserved densities for those systems, and we concentrate on the reductions of the conserved densities ( 2.13 ) in particular.
3.1. The Gurevich-Zybin system and the Monge-Ampere equation
For the system (2.1), let and v = −Φx / 4, then the first equation is changed to
Taking above relations into consideration, one could deduce their conserved densities from those of the system (2.1). For instance, according to Proposition 2.1, the Gurevich-Zybin system (3.1) and (3.3) has the conserved densities of the formsuch that , where denotes the inverse of ∂x such that , while Similarly, replacing u by Φxt /(2 Φx x), while v by −Φx / 4 in (2.13) yields the conserved density of the Monge-Ampere equation (3.4).
3.2. The two-component Hunter-Saxton equation
Let , then the system (2.3a)–(2.3b) is converted to
Replacing vx by in (2.13) immediately gives us the conserved density
When η = 0, all ’s vanish and the density (3.6) reduces towhich is nothing but the conserved density with arbitrary function of the Hunter-Saxton equation recently reported by two of the authors .
4. The general case
In this section, we extend the ideas proposed in section 2 to the general case. We shall introduce a set of dynamical variables for the generalized Riemann equation at arbitrary N, i.e. the system (1.3). With the help of these variables, we shall explicitly present conserved densities involving arbitrary smooth functions for the general case. As the validity of these results can be checked by direct calculations, we omit the details in the subsequent discussions.
Let Q0 = 1 / uNx and Qi = ui, x / uN, x(i = 1,2,⋯,N), then it is shown by direct calculations that any monomial of degree m in Qi’s, like satisfies
The introduction of those fundamental variables is motivated by the considerations of the N = 2 case. Indeed, the dynamical variables α(1) and β(1) defined by (2.10) for the N = 2 case are and up to the scaling u1 = − 2u, u2 = 4v.
Other dynamical variables are iteratively defined as
The first order differential operator in iterative relations (4.3) of dynamical variables is different from the operator (1.4), which is used to generate non-polynomial conserved densities .
On the basis of Eqs. (4.2) and (4.4), it is straightforward to show that the system (1.3), namely the generalized Riemann equation at arbitrary N, has the following conservation laws
∂t(F) = ∂x( − u1F), where , and G is an arbitrary smooth function of its arguments.
∂t(H) = ∂x(−u1H + P), where stands for an arbitrary smooth function of its arguments, and
As implementations, the conserved densities in fundamental dynamical variables are presented for the generalized Riemann equations at N = 3, 4, and the corresponding conservation laws could be checked by direct calculations.
When N = 3, the fundamental dynamical variables areSubstituting Q0 = 1 / u3, x and Qi = ui, x / u3, x(i = 1,2,3) into them, we get Then the generalized Riemann equation at N = 3 has the conserved density such that ∂t(F) = ∂x( − u1F).
For the generalized Riemann equation at N = 4, the fundamental dynamical variables areTaking Q0 = 1 / u4, x and Qi = ui, x / u4, x(i = 1,2,3,4) into account, we obtain Hence, the quantity is conserved, and satisfies ∂t(F) = ∂x( − u1F) when the generalized Riemann equation at N = 4 holds.
Two comments are in order. The conservation laws found by Olver and Nutku  for the Riemann equation (1.1) could be recovered from above general formulas by taking N = 1. Also, we remark that, while the generalized Riemann equation at N − 1 may be obtained from the equation at N, it does not mean that we may construct the conserved densities of the former from those of the latter. The obstacle is indeed the iterative relations (4.3), where uN x appears as a denominator.
The conserved densities, involving arbitrary smooth functions, have been worked out explicitly for the generalized Riemann equations, and in this way the previous results about this topic have been extended. As illustrated by the N = 2 case, the proper changes of variables 𝒯1 and 𝒯2, which linearize Eqs. (2.3a)–(2.3b), convert the assumed conservation law (2.2) to a first order PDE with separable variables and result in the conserved densities of the type (2.9). It is noticed that for the Gurevich-Zybin, Monge-Ampere and two-component Hunter-Saxton equations, the conservation laws involving arbitrary functions are established by reducing those in Proposition 2.1. However, rather than concrete expressions, a formula is given by Proposition 2.2 for the conserved quantities. Thus, the conservation laws for these three systems may not be recovered from this Proposition. It is also believed that the existence of the conserved densities depending on arbitrary functions have their roots in the linearizable property of the generalized Riemann equations. However, further investigations are needed to reveal this deep connections for the generic case. It will be interesting to consider potential applications of the conserved densities presented in this paper, such as constructing new solutions, exploiting new nonlinear phenomenon and so on.
We greatly appreciate the anonymous referees' constructive comments, which are very helpful to improve the paper. This work is supported by the National Natural Science Foundation of China (grant numbers: 11271366,11331008 and 11505284) and the Fundamental Research Funds for Central Universities.
Cite this article
TY - JOUR AU - Binfang Gao AU - Kai Tian AU - Q. P. Liu AU - Lujuan Feng PY - 2021 DA - 2021/01/06 TI - Conservation Laws of The Generalized Riemann Equations JO - Journal of Nonlinear Mathematical Physics SP - 122 EP - 135 VL - 25 IS - 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2018.1440746 DO - 10.1080/14029251.2018.1440746 ID - Gao2021 ER -