Nonlocal Hydrodynamic Type of Equations

TL;DR

This paper demonstrates that integrable hydrodynamic equations can be reduced to nonlocal forms, which remain integrable and possess Lax representations, expanding the understanding of their structure and conservation laws.

Contribution

It introduces a method to construct nonlocal reductions of integrable hydrodynamic equations and provides explicit examples demonstrating their integrability and conserved quantities.

Findings

Nonlocal reductions of hydrodynamic equations are integrable.

Reduced equations admit Lax representations.

They possess infinitely many conserved quantities.

Abstract

We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. They admit Lax representations and hence possess infinitely many conserved quantities.

Equations

$$\begin{array}[]{l}q^{i}_{t}=F(q^{j},q^{j}_{x},r^{j},r^{j}_{x}),\qquad i,j=1,\cdots,L,\\ r^{i}_{t}=G(q^{j},q^{j}_{x},r^{j},r^{j}_{x}),\qquad i,j=1,\cdots,L,\\ \end{array}$$

$$r^i(x,t)=\rho q^i({\epsilon }_{1}x,{\epsilon }_{2}t),i=1,\cdots ,L,$$

$$r^i(x,t)=\rho {\bar{q}}^i({\epsilon }_{1}x,{\epsilon }_{2}t),i=1,\cdots ,L.$$

$$\mathcal{A}=\left\{\alpha =\sum _{-\infty }^{\infty }{\alpha }_n(x,t)p^n:{\alpha }_n(x,t)\in Y\right\},$$

$$\{f,g{\}}_k=p^k\left(\frac{\partial f}{\partial p}\frac{\partial g}{\partial x}-\frac{\partial f}{\partial x}\frac{\partial g}{\partial p}\right)f,g\in \mathcal{A},$$

$$\text{mbox}{tr}_k\alpha ={\int }_I\text{mbox}{Res}_k\alpha dx\alpha \in \mathcal{A},$$

$$(\alpha ,\beta )=\text{mbox}{tr}_k\alpha \cdot \beta ,\alpha ,\beta \in \mathcal{A}.$$

$$\mathcal{A}={\mathcal{A}}_{\ge -k+1}\oplus {\mathcal{A}}_{<-k+1},$$

$${\mathcal{A}}_{\ge -k+1}=\left\{{\alpha }_{\ge -k+1}=\sum _{-k+1}^{\infty }{\alpha }_n(x,t)p^n:{\alpha }_n(x,t)\in Y\right\}$$

$${\mathcal{A}}_{<-k+1}=\left\{{\alpha }_{<-k+1}=\sum _{-\infty }^{-k}{\alpha }_n(x,t)p^n:{\alpha }_n(x,t)\in Y\right\}.$$

$$R_k=\frac{1}{2}({\Pi }_{\ge -k+1}-{\Pi }_{<-k+1})$$

$$L=p^{N-1}+\sum _{i=-1}^{N-2}{p}^i{S}_i(x,t),N\in \mathbb{N},$$

$$\frac{\partial L}{\partial t_n}={\left\{R_k(L^{\frac{m}{N-1}+n}),L\right\}}_k={\left\{L_{\ge -k+1}^{\frac{m}{N-1}+n},L\right\}}_k,n=1,2,\cdots ,$$

$${\mathcal{Q}}_n=\text{mbox}{tr}_k{L}^{\frac{m}{N-1}+n},n=1,2,\cdots .$$

$$L=\frac{1}{p}\prod _{j=1}^N(p-u_j).$$

$$L=\frac{1}{p}(p-u)(p-v).$$

$$\begin{array}[]{l}\frac{1}{2}u_{t_{2}}=(u+v)u_{x}+uv_{x},\\ \\ \frac{1}{2}v_{t_{2}}=(u+v)v_{x}+vu_{x}.\end{array}$$

$$\begin{array}[]{l}\frac{1}{3}u_{t_{3}}=(u^{2}+4uv+v^{2})u_{x}+(2u^{2}+2uv)v_{x},\\ \\ \frac{1}{3}v_{t_{3}}=(2v^{2}+2uv)u_{x}+(v^{2}+4uv+u^{2})v_{x},\end{array}$$

$$\begin{array}[]{l}\frac{1}{4}u_{t_{4}}=(u^{3}+9u^{2}v+9uv^{2}+v^{3})u_{x}+(3u^{3}+9u^{2}v+3uv^{2})v_{x},\\ \\ \frac{1}{4}v_{t_{4}}=(3v^{3}+9v^{2}u+3vu^{2})u_{x}+(v^{3}+9v^{2}u+9vvu^{2}+u^{3})v_{x}.\end{array}$$

$${\mathcal{Q}}_2={\int }_I(u^{2}v+u{v}^2)dx,$$

$${\mathcal{Q}}_3={\int }_I(u^{3}v+3u^2{v}^2+u{v}^3)dx,$$

$${\mathcal{Q}}_4={\int }_I(u^{4}v+6u^3{v}^2+6u^2{v}^3+6u{v}^4)dx.$$

$$\begin{array}[]{l}u_{t_{1}}=uv_{x},\\ v_{t_{1}}=vu_{x}.\end{array}$$

$${\mathcal{Q}}_1={\int }_I(u+v)dx,$$

$${\mathcal{Q}}_2={\int }_I(u^2+4uv+v^2)dx,$$

$${\mathcal{Q}}_3={\int }_I(u^3+9u^{2}v+9u{v}^2+v^3)dx.$$

$$L=\frac{1}{p}(p-u)(p-v)(p-w),$$

$$\begin{array}[]{l}\displaystyle\frac{8}{3}u_{t_{1}}=(-u^{2}+v^{2}+w^{2}+4uv++4uw+6vw)u_{x}+(2u^{2}+2uv+6uw)v_{x}\\ \hskip 56.9055pt+(2u^{2}+6uv+2uw)w_{x},\\ \displaystyle\frac{8}{3}v_{t_{1}}=(2v^{2}+2uv+6vw)u_{x}+(-v^{2}+u^{2}+w^{2}+4uv++4vw+6vw)v_{x}\\ \hskip 56.9055pt+(2v^{2}+6uv+2vw)w_{x},\\ \displaystyle\frac{8}{3}w_{t_{1}}=(2w^{2}+6wv+2uw)u_{x})+(2w^{2}+2wv+6uw)v_{x}\\ \hskip 56.9055pt+(-w^{2}+v^{2}+u^{2}+4wv+4uw+6uv)w_{x}.\end{array}$$

$${\mathcal{Q}}_1={\int }_I(u^2+v^2+w^2-2uv-2uw-2vw)dx,$$

$${\mathcal{Q}}_2={\int }_I(u^4+v^4+w^4-4(u^{3}v-u^{3}w-u{v}^3-v^{3}w-u{w}^3-v{w}^3)+6(u^2{v}^2+u^2{w}^2+6v^2{w}^2)$$

$$+36(u^{2}vw+u{v}^{2}w+uv{w}^2))dx,$$

$$L=\frac{1}{p}(p-u)(p-u_1)(p-v)(p-v_1)$$

$$\begin{array}[]{l}u_{t_{1}}=(u_{1}v+u_{1}v_{1}+vv_{1})u_{x}+(uv+uv_{1})u_{1x}+(uu_{1}+uv_{1})v_{x}+(uu_{1}+uv)v_{1x},\\ u_{1t_{1}}=(uv+uv_{1}+vv_{1})u_{1x}+(u_{1}v+u_{1}v_{1})u_{x}+(uu_{1}+u_{1}v_{1})v_{x}+(u_{1}u+u_{1}v)v_{1x},\\ v_{t_{1}}=(uu_{1}+uv_{1}+u_{1}v_{1})v_{x}+(u_{1}v+vv_{1})u_{x}+(uv+vv_{1})u_{1x}+(uv+u_{1}v)v_{1x},\\ v_{1t_{1}}=(uu_{1}+uv+u_{1}v)v_{1x}+(u_{1}v_{1}+vv_{1})u_{x}+(uv_{1}+vv_{1})u_{1x}+(uv_{1}+u_{1}v_{1})v_{x}.\\ \end{array}$$

Full text

Keywords: Hydrodynamic equations, Lax Representations, Conserved Quantities, Nonlocal Reductions.

1 Introduction

One way of obtaining a system of integrable equations, in Gelfand-Dikii formalism [1], is to construct a Lax operator on some Lie algebra. The examples of such algebras are the matrix algebra, algebra of differential operators, and algebra of Laurent series (see [2]-[4] and references there in). In most cases the Lax operator is polynomial. It is a polynomial of the spectral parameter in the matrix algebra, the operator $D_{x}$ in the algebra of differential operators, and an auxiliary variable $p$ (momentum) in the case of the algebra of Laurent series.

Writing the Lax equation on the algebra of Laurent series we obtain equations of the hydrodynamic type. These equations are integrable. This means that they admit recursion operators, multi-Hamiltonian representation (see [2], [5]-[7]). They also have many types of reductions.

For a hydrodynamic type system let us give an example

[TABLE]

where $F,G$ are functions of the dynamical variables $q^{i}(t,x)$, $r^{i}(t,x)$, and their first partial derivatives with respect to $x$. We define a reduction as the following relations

[TABLE]

where $\varepsilon_{1},\,\varepsilon_{2},\,\rho$ are real constants such that $\rho^{2}=\varepsilon_{1}^{2}=\varepsilon_{2}^{2}=1$. In the case of complex valued dependent variables we also consider the following relations

[TABLE]

The system (1) must remain consistent with respect to the reductions (2) and (3).

If $\varepsilon_{1}=\varepsilon_{2}=1$ we have a local reduction. Such reductions were considered in [5]. If $\varepsilon_{1}=-1$ or $\varepsilon_{2}=-1$ we have nonlocal reductions. In this work we address to the problem of nonlocal reductions for the Lax equations on the algebra of Laurent series.

The nonlocal reductions of systems of integrable equations were first consistently applied to the nonlinear Schrödinger system, which led to the nonlocal nonlinear Schrödinger equation (nNLS) [8]-[10]. It has been later shown that there are many other systems where the consistent nonlocal reductions are possible [11]-[37]. For a recent review of this subject see [35]. See also [38] for the discussion of superposition of nonlocal integrable equations and [39] for the origin of nonlocal reductions.

The layout of the paper is as follows. In Section 2 we give a short review on the hydrodynamic type of (dispersionless) equations and their Lax representations. In Section 3 we give nonlocal reductions for real and complex valued fields for $N=2$. We obtain some explicit examples of nonlocal hydrodynamic type of equations with some conserved quantities. In Section 4 we discuss the nonlocal reductions in general and give examples for higher values of $N$.

2 Lax equations

In this section we describe the algebra of Laurent series, introduce necessary definitions to write a Lax equation and give some examples. In later sections we give nonlocal reductions of these equations.

We consider an algebra $\cal A$, given by

[TABLE]

where $Y$ is either $C^{\infty}(S^{1})$-the space of periodic functions or $S({\mathbb{R}})$-the spaces of smooth asymptotically decreasing functions on $\mathbb{R}$. A Poisson bracket on $\cal A$, is given by

[TABLE]

where $k\in\mathbb{Z}$.

On the algebra $\cal A$ we can define a trace functional

[TABLE]

where $\mbox{Res}_{k}\,\alpha=\alpha_{k-1}(x,t)$. The set $I$ is either $S^{1}$ for the space of periodic functions or $\mathbb{R}$ for the space of asymptotically decaying functions. Using the trace functional we define a non-degenerate, ad-invariant, symmetric pairing

[TABLE]

The algebra $\cal A$ admits a decomposition into sub-algebras, closed with respect to the Poisson bracket,

[TABLE]

where

[TABLE]

and

[TABLE]

So, we can define $r$-matrix mapping on $\cal A$ (see [4] and references therein)

[TABLE]

where $\Pi_{\geq-k+1}$ and $\Pi_{<-k+1}$ are projections on the sub-algebras ${\cal A}_{\geq-k+1}$ and ${\cal A}_{<-k+1}$ respectively.

Now for a Lax operator

[TABLE]

we can consider a hierarchy of the Lax equations (see [5])

[TABLE]

where $m=0,1,2,\cdots,(N-2)$ is fixed. This is an integrable hierarchy that is the Lax equations admit multi-Hamiltonian representation (see [2]). In particular the conserved quantities for the hierarchy are given by

[TABLE]

We note that to consider nonlocal reductions it is more convenient to write the equation (12) in terms of zeros of the Lax operator. That is we set

[TABLE]

Let us give some examples of the Lax equations and their conserved quantities. First we consider examples of Lax equations corresponding to the Lax operator

[TABLE]

Example 1**.**

We take the algebra $\cal A$ with $k=0$ and consider the Lax equation (12) with a Lax operator (15). For $n=2$, we obtain the shallow water waves system

[TABLE]

Taking $n=3,4,\cdots$, we can have generalized symmetries of the shallow water waves system. For instance, the first two symmetries are

[TABLE]

and

[TABLE]

The above systems admit infinitely many conserved quantities. The first three of the conserved quantities are

[TABLE]

Example 2**.**

We take the algebra $\cal A$ with $k=1$ and consider the Lax equation (12) with a Lax operator (15). For $n=1$ we obtain the Toda system

[TABLE]

Taking $n=2,3,\cdots$, we can have generalized symmetries of the Toda system. The Toda system admits infinitely many conserved quantities. The first three of the conserved quantities are

[TABLE]

Let us give examples of the Lax equation with more general Lax operator.

Example 3**.**

We take the algebra $\cal A$ with $k=0$ and consider the Lax equation (12) with a Lax operator

[TABLE]

and $m=1$. For $n=1$ we obtain the following system

[TABLE]

Taking $n=2,3,\cdots$, we can have generalized symmetries of the above system. The above system admits infinitely many conserved quantities. The first two of the conserved quantities are

[TABLE]

Example 4**.**

For simplicity, we take the algebra $\cal A$ with $k=0$ and consider the Lax equation (12) with a Lax operator

[TABLE]

and $m=0$. For $n=1$ we obtain the following system

[TABLE]

Taking $n=2,3,\cdots$, we can have generalized symmetries of the above system. The above system admits infinitely many conserved quantities. The first two of the conserved quantities are

[TABLE]

3 Nonlocal reductions for two component Lax operator

To study nonlocal reductions of the Lax equation (12), we start with the case of two component systems. That is we study the hierarchy

[TABLE]

with the Lax operator (15).

3.1 Nonlocal reductions for real valued dependent variables

In this section we assume that our dependent variables are real valued and consider reduction of the form

[TABLE]

The reduced equation is the Lax equation (34) with the Lax operator

[TABLE]

To find admissible reductions we introduce the following change of variables

[TABLE]

In new variables the Lax operator becomes

[TABLE]

where $\displaystyle{\tilde{L}=\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho u^{\varepsilon})}$. The Lax equation (34) takes the form

[TABLE]

or

[TABLE]

It is easy to see that if $\rho^{k+n+1}\varepsilon_{2}\varepsilon_{1}=1$ then the hierarchy (34) with Lax operator (36) is invariant under the change of variables and we have a consistent nonlocal reduction.

Since $\rho^{2}=1$, we have one condition $\rho^{k}\varepsilon_{2}\varepsilon_{1}=1$ for all odd $n$ and another condition $\rho^{k+1}\varepsilon_{2}\varepsilon_{1}=1$ for all even $n$. Thus, to consider nonlocal reduction of the whole hierarchy (34) we have to split it into two hierarchies

[TABLE]

and

[TABLE]

Taking $\rho,\,\varepsilon_{1}$, and $\varepsilon_{2}$ satisfying

[TABLE]

we obtain nonlocal reductions for the hierarchy (41). Note that these equations admit the following conserved quantities

[TABLE]

where $L$ is given by (36).

Taking $\rho,\,\varepsilon_{1}$, and $\varepsilon_{2}$ satisfying

[TABLE]

we obtain nonlocal reductions for the hierarchy (42). Note that these equations admit the following conserved quantities

[TABLE]

where $L$ is given by (36).

Let us give some examples. As first example we consider the shallow water waves system.

Example 5**.**

To have nonlocal reductions of the shallow water waves system (16), we need to work with the hierarchy (41). Taking $\rho,\,\varepsilon_{1}$, and $\varepsilon_{2}$ satisfying (43) we obtain possible reductions.

a.* S-symmetric reduction, $\rho=-1,\varepsilon_{1}=-1,\varepsilon_{2}=1$, is*

[TABLE]

It has infinitely many symmetries, with the first symmetry, the reduction of (18), being

[TABLE]

The above equations have infinitely many conserved quantities. The first two conserved quantities are

[TABLE]

b.* T-symmetric reduction, $\rho=-1,\varepsilon_{1}=1,\varepsilon_{2}=-1$, is*

[TABLE]

It has infinitely many symmetries, with the first symmetry, the reduction of (18), being

[TABLE]

The above equations have infinitely many conserved quantities. The first two conserved quantities are

[TABLE]

c.* ST-symmetric reduction, $\rho=1,\varepsilon_{1}=-1,\varepsilon_{2}=-1$, is*

[TABLE]

It has infinitely many symmetries with the first symmetry, the reduction of (18), being

[TABLE]

The above equations have infinitely many conserved quantities. The first two conserved quantities are

[TABLE]

We can also consider the reductions for the odd symmetries of the shallow water waves equations.

Example 6**.**

To have nonlocal reductions of the system (17), we need to work with the hierarchy (42). Taking $\rho,\,\varepsilon_{1}$, and $\varepsilon_{2},$ satisfying (45) we obtain one possible reduction.

ST-symmetric reduction, $\rho=1,\varepsilon_{1}=-1,\varepsilon_{2}=-1$, is

[TABLE]

The above system has infinitely many symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

Let us also consider the reductions of the Toda system.

Example 7**.**

To have nonlocal reductions of the Toda system (22), we need to work with the hierarchy (42). Taking $\rho,\,\varepsilon_{1},$ and $\varepsilon_{2}$ satisfying (45) we can obtain the following reductions: S-symmetric reduction, $\rho=-1,\varepsilon_{1}=-1,\varepsilon_{2}=1$; T-symmetric reduction, $\rho=-1,\varepsilon_{1}=1,\varepsilon_{2}=-1$; ST-symmetric reduction, $\rho=1,\varepsilon_{1}=-1,\varepsilon_{2}=-1$. For instance, the ST-symmetric reduction of the Toda system is

[TABLE]

It has infinitely many symmetries, with the first symmetry

[TABLE]

which is the ST-symmetric reduction of the second equation in the hierarchy (34). It also has infinitely many conserved quantities. The first two conserved quantities are

[TABLE]

Remark 3.1**.**

We considered the reductions for the whole hierarchy (34). We can also consider nonlocal reductions just for one equation from the hierarchy. For instance, taking $k=0$ and $n=2$ in (34) we get the following equations:

[TABLE]

Let us consider nonlocal reductions of these equations only. Assume that $v(x,t)=\rho u(\varepsilon_{1}x,\varepsilon_{2}t)=\rho u^{\varepsilon}$, where $\varepsilon_{1}^{2}=\varepsilon_{2}^{2}=1$, and $\rho$ is a real constant, then the above system takes form

[TABLE]

For consistency, we must have $\rho=\varepsilon_{1}\varepsilon_{2}$ which is the same condition as equality (43). Therefore working with the whole hierarchy we obtain all possible cases.

3.2 Nonlocal reductions for complex valued dependent variables

In this section we assume that all dependent variables are complex valued functions. To study reductions of such equations it is convenient to introduce a constant in the Lax equation (34) by setting new time $t^{\prime}=at$, $a\in{\mathbb{C}}$. So we consider a hierarchy

[TABLE]

Note that after the change of the time variable we shall use again $t$ to denote new time.

The reduction is of the form

[TABLE]

To find admissible reductions we use the following change of variables

[TABLE]

In new variables the Lax operator becomes

[TABLE]

where $\displaystyle{L^{\prime}=\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho\bar{u}^{\varepsilon})}$ and the Lax equation (34) takes the form

[TABLE]

Taking complex conjugates of both sides in the above equality we can get

[TABLE]

If $\overline{a}\rho^{k+n+1}\varepsilon_{2}\varepsilon_{1}=a$ then the hierarchy (70) is invariant under the change of variables and we have a consistent nonlocal reduction.

Since $\rho^{2}=1$ we have one condition $\overline{a}\rho^{k}\varepsilon_{2}\varepsilon_{1}=a$ for all odd $n$ and another condition $\overline{a}\rho^{k+1}\varepsilon_{2}\varepsilon_{1}=a$ for all even $n$. Thus, to consider local reduction of the hierarchy (70) we have to split it into two hierarchies (41) and (42) as in the real case.

Let us give one example of the reduction for complex valued dependent variables.

Example 8**.**

We take $k=0$ and consider the hierarchy (70) with $n$ even. The first system corresponding to $n=2$ is

[TABLE]

*Taking $\rho,\,\varepsilon_{1},$ and $\varepsilon_{2}$ satisfying $\overline{a}\rho\varepsilon_{2}\varepsilon_{1}=a$ we obtain possible reductions.

a.* S-symmetric reduction, $\varepsilon_{1}=-1,\,\varepsilon_{2}=1$, is*

[TABLE]

where we can take $\rho=1$ and the constant $a$ equal to a pure imaginary number or $\rho=-1$ and the constant $a$ equal to a real number. The above equation has infinitely many symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

b.* T-symmetric reduction, $\varepsilon_{1}=1,\,\varepsilon_{2}=-1$, is*

[TABLE]

where we can take $\rho=1$ and the constant $a$ equal to a pure imaginary number or $\rho=-1$ and the constant $a$ equal to a real number. The above equation has infinitely many symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

c.* ST-symmetric reduction, $\varepsilon_{1}=-1,\,\varepsilon_{2}=-1$, is*

[TABLE]

where we can take $\rho=1$ and the constant $a$ equal to a real number or $\rho=-1$ and the constant $a$ equal to a pure imaginary number. The above equation has infinitely many symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

4 Reductions with a general Lax operator

To study nonlocal reductions of the Lax equation (12) with the general Lax operator (14), we can divide the variables $u_{i}$ into pairs and set variables that do not have a pair to zero. For each pair of variables we consider a relation similar to (35) and proceed as before. As an example let us consider two special cases.

4.1 Three component Lax operator

In this section we take the Lax operator

[TABLE]

and consider the hierarchy (12) with $m=1$. Thus we have

[TABLE]

4.1.1 Real valued dependent variables

First we study reductions for real valued dependent variables. A nonlocal reduction has a form

[TABLE]

To find admissible reductions we use the following change of variables

[TABLE]

In new variables the Lax operator becomes

[TABLE]

where $\tilde{L}=\displaystyle{\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho u^{\varepsilon})(\tilde{p}-\rho w^{\varepsilon})}$. Also in new variables we have $L^{\frac{1}{2}}=\rho\tilde{L}^{\frac{1}{2}}$.

The Lax equation (87) takes the form

[TABLE]

or

[TABLE]

Now we set $w=0$ in the equation (87) and $w^{\varepsilon}=0$ in the equation (92). It is easy to see that under these conditions the hierarchy (87) is invariant under the change of variables if

[TABLE]

So, we have a consistent nonlocal reduction if the above condition is satisfied.

Example 9**.**

Let us consider reductions for the system (27). The condition (93) takes form $\varepsilon_{2}\varepsilon_{1}=1$. Thus we can have only $ST$-symmetric reduction. Taking $\varepsilon_{1}=-1$ and $\varepsilon_{2}=-1$ we get $ST$-symmetric nonlocal reduction. For $\rho=1$, the reduced equation is

[TABLE]

The above system has infinitely many generalized symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

4.1.2 Complex valued dependent variables

In this section we assume that all variables in the Lax equation (87) are complex valued functions. As in the case $N=2$, we transform the Lax equation to the form

[TABLE]

The reduction is of the form

[TABLE]

To find admissible reductions we use the following change of variables

[TABLE]

for the Lax equation (97).

In new variables the Lax operator becomes

[TABLE]

where $L^{\prime}=\displaystyle{\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho\bar{u}^{\varepsilon})(\tilde{p}-\rho\bar{w}^{\varepsilon})}$. Also in new variables we have $L^{\frac{1}{2}}=\rho\bar{L^{\prime}}^{\frac{1}{2}}$.

The Lax equation (97) takes the form

[TABLE]

Taking complex conjugates of both sides in the above equality we get

[TABLE]

Now we set $w=0$ in the equation (87) and $w^{\varepsilon}=0$ in the equation (92). It is easy to see that under these conditions the hierarchy (87) is invariant under the change of variables if

[TABLE]

So, we have a consistent nonlocal reduction if the above condition is satisfied.

Example 10**.**

Let us consider reductions for the modified system (27) (the time is redefined to introduce the multiple $a$). The condition (103) takes the form $\bar{a}\varepsilon_{2}\varepsilon_{1}=a$. Thus we can have $S$-symmetric, $T$-symmetric, and $ST$-symmetric reductions. For instance, taking $a$ purely imaginary, $\varepsilon_{1}=-1$, and $\varepsilon_{2}=1$ we get $S$-symmetric nonlocal reduction. For $\rho=1$, the reduced equation is

[TABLE]

The above system has infinitely many generalized symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

4.2 Four component Lax operator

In this section we take the Lax operator

[TABLE]

and consider the hierarchy (12) with $m=0$, for simplicity. Thus we have

[TABLE]

4.2.1 Real valued dependent variables

First we study reductions for real valued dependent variables. A nonlocal reduction has a form

[TABLE]

To find admissible reductions we use the following change of variables

[TABLE]

In new variables the Lax operator becomes

[TABLE]

where $\displaystyle{\tilde{L}=\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho u^{\varepsilon})(\rho\tilde{p}-v^{\varepsilon})(\rho\tilde{p}-\rho v)}$. The Lax equation (108) takes the form

[TABLE]

or

[TABLE]

If

[TABLE]

then the hierarchy (108) is invariant under change of variables and we have a consistent nonlocal reduction. As in the case $N=2$, to have integrable hierarchy of nonlocal systems we have to split our hierarchy into two hierarchies: one with even $n$ and another with odd $n$.

Example 11**.**

Let us consider reductions for the system (31). It follows that for nonlocal reductions we have to consider the systems of hierarchy (107), corresponding to odd $n$. The condition (114) takes the form $\varepsilon_{2}\varepsilon_{1}=1$. Thus we can have only $ST$-symmetric reduction, $\varepsilon_{1}=-1,\varepsilon_{2}=-1$. Taking $\rho=1$ and setting $u_{1}(x,t)=u(-x,-t)=u^{\varepsilon}$ and $v_{1}(x,t)=v(-x,-t)=v^{\varepsilon}$ we obtain ST-symmetric nonlocal reduction.

[TABLE]

The above system has infinitely many generalized symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

4.2.2 Complex valued dependent variables

In this section we assume that all variables in the Lax equation (108) are complex valued functions. As in the case $N=2$, we transform the Lax equation to the form

[TABLE]

The reduction is of the form

[TABLE]

To find admissible reductions we use the following change of variables

[TABLE]

for the Lax equation (118).

In new variables the Lax operator becomes

[TABLE]

where $\displaystyle{L^{\prime}=\frac{1}{\tilde{p}}(\tilde{p}-u)(\tilde{p}-\rho\bar{u}^{\varepsilon})(\tilde{p}-v)(\tilde{p}-\rho\bar{v}^{\varepsilon})}$. The Lax equation (118) takes the form

[TABLE]

Taking complex conjugates of both sides in the above equality we get

[TABLE]

If

[TABLE]

then the hierarchy (118) is invariant under the change of variables and we have a consistent nonlocal reduction.

Since $\rho^{2}=1$ we have one condition $\overline{a}\rho^{k}\varepsilon_{2}\varepsilon_{1}=a$ for all odd $n$ and another condition $\overline{a}\rho^{k+1}\varepsilon_{2}\varepsilon_{1}=a$ for all even $n$. Thus, to consider nonlocal reduction of the hierarchy (70) we have to split it into two hierarchies as in the real case.

Example 12**.**

Let us consider reductions for the modified system (31) (the time is redefined to introduce the multiple $a$). It follows that for nonlocal reductions we have to consider the systems of hierarchy (118) corresponding to odd $n$. The condition (124) takes form $\bar{a}\varepsilon_{2}\varepsilon_{1}=a$. Thus we can have $S$-symmetric, $T$-symmetric and $ST$-symmetric reduction. For instance, taking $a$ purely imaginary and $\varepsilon_{1}=-1$ ,$\varepsilon_{2}=1$ we get $S$-symmetric nonlocal reduction. Taking $\rho=1$ and setting $u_{1}(x,t)=\bar{u}(-x,t)=\bar{u}^{\varepsilon}$ and $v_{1}(x,t)=\bar{v}(-x,t)=\bar{v}^{\varepsilon}$ we can write the reduced equation as

[TABLE]

The above system has infinitely many generalized symmetries and conserved quantities. The first two conserved quantities are

[TABLE]

5 Conclusion

We have found all possible nonlocal reductions of the hydrodynamic type of equations in $1+1$ dimensions. Such reductions are possible only for all even numbered systems. We have given examples of nonlocal reductions for $N=2$ and $N=4$ when the dynamical variables are real and complex valued functions. For the complex valued dynamical variables we have all kinds of nonlocal reductions, namely space reflection, time reflection, and space-time reflections. The new nonlocal systems admit Lax representations and possess infinitely many conserved quantities. We gave some examples of these conserved quantities of the new nonlocal reduced systems.

6 Acknowledgment

This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).