# Optimal linear approximation and isometric extensions

Alexander Kushpel  Çankaya University, Department of Mathematics
Ankara, Turkey<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a Banach space with the unit ball $B(X)$ and $A\subset X$ be a
convex origin-symmetric compact in $X$. Let
$\mathrm{j}:X\rightarrow\widetilde{X}$ be an isometric extension of $X$. It is
well-known that linear widths
$\lambda_{n}\left(\mathrm{j}\left(A\right)\text{,}\widetilde{X}\right)$ may
decrease in order when compared with $\lambda_{n}\left(A\text{,}X\right)$ and
absolute widths
$\Lambda\left(A,\widehat{X}\right)=\inf_{\mathrm{j}}\left(\mathrm{j}\left(A\right),\widetilde{X}\right)$
are realized in the space $\widehat{X}$ which is the Banach space of bounded
functions $f:B\left(X^{\ast}\right)\rightarrow\mathbb{R}$ on the unit ball
$B\left(X^{\ast}\right)$ of the conjugate space $X^{\ast}$. We show that it is
sufficient to use just $n$-dimensional extensions of $X$ to attain absolute
linear widths. This unexpected fact significantly reduces the space $\
\widehat{X}$. This allows us to introduce the notion of preabsolute widths. We
give the respective optimal extensions explicitly and establish order
estimates for preabsolute widths of a wide range of sets of smooth functions
considered in [3]. In particular, in the case of super-small and super-high
smoothness considered in [3] the orders of preabsolute linear widths coincide
with the orders of absolute linear widths. In the intermediate cases of finite
and infinite smoothness the respective orders are different.

###### Key words and phrases:

Optimal linear approximation, absolute widths, multiplier

###### 2010 Mathematics Subject Classification:

41A46, 42A45

##  1\. Introduction

Optimal linear approximation and recovery play an important role in
Approximation Theory and they are connected via absolute linear widths and
duality with nonlinear approximation. $n$-Widths were introduced in 1936 by
Kolmogorov to compare the efficiency of numerical algorithms [4]. Let
$\left(X,\left\|\cdot\right\|_{X}\right)$ be a Banach space with the unit ball
$B\left(X\right)$ and $A\subset X$ be a compact, convex and origin symmetric
set in $X$. The Kolmogorov $n$-width of $A$ in $X$ is defined as

$d_{n}\left(A,X\right)=\inf_{L_{n}\subset X}\sup_{x\in A}\inf_{y\in
L_{n}}\left\|x-y\right\|_{X}$

Let

$d^{n}\left(A,X\right)=\inf_{L^{n}}\sup_{x\in A\cap
L^{n}}\left\|x\right\|_{X},$

be the Gelfand $n$-width [2]. Here $L^{n}$ runs over all subspaces of
codimension at most $n$. We shall concentrate here on linear widths introduced
in [11]. The linear $n$-width of $A$ in $X$ is defined by

$\lambda_{n}\left(A,X\right)=\inf_{\mathrm{P}_{n}}\sup_{x\in
A}\left\|x-\mathrm{P}_{n}x\right\|_{X},$

where $\mathrm{P}_{n}:X\rightarrow X$ varies over all linear operators of rank
at most $n$. Let $X$ and $Z$ be Banach spaces, $u:X\rightarrow Z$,
$u\in\mathcal{L}\left(X,Z\right)$ be a bounded linear operator and $u^{\ast}$
be its adjoint. It is well-known if $u$ is compact or $Z$ is reflexive (see
e.g. [10], [9]) then

(1.1) $d^{n}\left(u^{\ast}\right)=d_{n}\left(u\right)$

and

$\lambda_{n}\left(u\right)=\lambda_{n}\left(u^{\ast}\right),$

where

$d_{n}\left(u\right)=d_{n}\left(uB\left(X\right),Z\right)=\inf_{L_{n}\subset
X}\sup_{x\in B\left(X\right)}\inf_{y\in L_{n}}\left\|ux-y\right\|_{X},$
$\lambda_{n}\left(u\right)=\lambda_{n}\left(uB\left(X\right),Z\right)=\inf_{\mathrm{P}_{n}}\sup_{x\in
B\left(X\right)}\left\|ux-\mathrm{P}_{n}ux\right\|_{X}.$

Let $\left(\widetilde{X},\mathrm{j}\right)$ be an extension of
$X\subset\widetilde{X}$, where $\mathrm{j}:X\rightarrow\widetilde{X}$ is a
linear isometry. It was noticed by Kolmogorov and demonstrated on a concrete
example by Tikhomirov [11] that the linear $n$-width of $A$ in $X$ may
decrease in an isometric extension $\widetilde{X}$ of $X\subset$
$\widetilde{X}$, since $\widetilde{X}$ contains more subspaces to approximate
$A$. Hence it is natural to consider

$\Lambda_{n}\left(A,X\right)=\inf\lambda_{n}\left(\mathrm{j}\left(A\right),\widetilde{X}\right),$

where $\inf$ is taken over all isometric extensions
$\mathrm{j}:X\rightarrow\widetilde{X}$. The width
$\Lambda_{n}\left(A,X\right)$ is the absolute linear $n$-width introduced by
Ismagilov [2]. It is known that
$\Lambda_{n}\left(A,X\right)=d^{n}\left(A,X\right)$. Moreover, the absolute
linear width is realized in the so-called universal isometric extension
$\widehat{X}$ which can be constructed as following. Let
$B\left(X^{\ast}\right)$ be the unit ball in the dual of $X$ and $\widehat{X}$
be the Banach space of bounded functions
$f:B\left(X^{\ast}\right)\rightarrow\mathbb{R}$ with the usual norm

$\left\|f\left(\phi\right)\right\|_{\widehat{X}}=\sup_{\phi\in
B\left(X^{\ast}\right)}\left|f\left(\phi\right)\right|.$

Clearly, $f\left(\phi\right)=\left\langle x,\phi\right\rangle\in\widehat{X}$
for any $x\in X$. By this way we get the linear isometric extension
$\mathrm{j}:X\rightarrow\widehat{X}$. Observe that Gelfand $n$-widths are
closely connected to the linear cowidths. Let $\mathbb{R}^{n}$ be the coding
set, i.e. the set which contains information on the elements of $A$ and
$\mathcal{L}\left(\mathrm{lin}\left(A\right),\mathbb{R}^{n}\right)$ be a
family of coding operators, $\phi:A\rightarrow\mathbb{R}^{n}$. Let $D\subset
X$,

$\mathrm{diam}\left(D,X\right)=\sup\left\\{\left\|x-y\right\|_{X}\left|x,y\in
D\right.\right\\}$

and

$\phi^{-1}\left(z\right)=\left\\{y\left|y\in X\text{,
}\phi\left(y\right)=z\right.\right\\}$

be the diameter of $D$ in $X$ and preimage of $z\in X$ respectively. The
linear cowidth is defined as

$\lambda^{n}\left(A,X\right)=\inf_{\phi\in\mathcal{L}\left(\mathrm{lin}\left(A\right),\mathbb{R}^{n}\right)}\sup_{x\in
A}\mathrm{diam}\left\\{\phi^{-1}\left(\phi\left(x\right)\right)\right\\}.$

Clearly,

$\lambda^{n}\left(A,X\right)=2d^{n}\left(A,X\right).$

In Section 2 we demonstrate an unexpected phenomenon. Namely, we show that
instead of a considerably big extension $\widehat{X}$ of $X$ it is sufficient
to use $n$-dimensional extensions constructed in Theorem 1 to attain absolute
$n$-widths $\Lambda_{n}\left(A,X\right)$. Hence it is natural to introduce a
new notion of preabsolute $n$-widths, $\Lambda_{n,m}\left(A,X\right)$ (see
Definition 1), where we allow $m$-dimensional isometric extensions of $X$,
$0\leq m\leq n$. In Section 3 we present two-side estimates for preabsolute
$n$-widths $\Lambda_{n,m}\left(A,X\right)$ on a wide range of sets of smooth
functions considered in [3]. More precisely, denote by $\mathcal{T}_{n}$,
$n\in\mathbb{N}$ the sequence of subspaces of trigonometric polynomials with
the usual order, i.e. $\mathcal{T}_{n}=\mathrm{lin}\left\\{1,\cos kx,\sin
kx\text{, }k\in\mathbb{N}\right\\}$. Consider usual spaces $L_{p}$, $1\leq
p\leq\infty$, of $p$-integrable functions $\phi$ on the unit circle
$\mathbb{T}$ with the Lebesgue measure $dx$,

$\left\|\phi\right\|_{p}=\left(\int_{\mathbb{T}}\left|\phi\right|^{p}dx\right)^{\frac{1}{p}}<\infty\text{.}$

Let $\phi\in L_{p}$ with the formal Fourier series

$\phi\sim\sum_{k=1}^{\infty}a_{k}\left(\phi\right)\cos
kx+b_{k}\left(\phi\right)\sin kx,$

where

$a_{k}\left(\phi\right)=\frac{1}{\pi}\int_{-\pi}^{\pi}\phi\left(t\right)\cos
ktdt,$
$b_{k}\left(\phi\right)=\frac{1}{\pi}\int_{-\pi}^{\pi}\phi\left(t\right)\sin
ktdt$

and

(1.2) $S_{n}\left(\phi,x\right)=\sum_{k=1}^{n}a_{k}\left(\phi\right)\cos
kx+b_{k}\left(\phi\right)\sin kx$

be its $n^{\frac{th}{{}}}$ Fourier sum. We introduce sets of smooth functions
using multipliers $\Lambda=\left\\{\lambda\left(k\right)\text{,
}k\in\mathbb{N}\right\\}$ [3]. We say that $f\in\Lambda_{\beta}U_{p}$ if

(1.3)
$f\sim\sum_{k=1}^{\infty}\lambda\left(k\right)\left(a_{k}\left(\phi\right)\cos\left(kx-\frac{\beta\pi}{2}\right)+b_{k}\left(\phi\right)\sin\left(kx-\frac{\beta\pi}{2}\right)\right),$

where $\phi\in U_{p}=\left\\{\phi\left|\left\|\phi\right\|_{p}\leq
1\right.\right\\}$ is the unit ball in $L_{p}$. If $\beta=0$ then we write
$\Lambda_{\beta}=\Lambda$. If there exists $K\in L_{1}$ such that

$K\sim\sum_{k=1}^{\infty}\lambda\left(k\right)\cos\left(kx-\frac{\beta\pi}{2}\right)$

then $\Lambda_{\beta}U_{p}$ is the set of functions $f$ representable in the
form

$f\left(x\right)=\int_{\mathbb{T}}K\left(x-y\right)\phi\left(y\right)dy,$

i.e. in this case $\Lambda_{\beta}U_{p}=K\ast U_{p}$. Observe that the
smoothness of the classes $\Lambda_{\beta}U_{p}$ is determined by the rate of
decay of the sequence $\Lambda$. In particular, if
$\lambda\left(k\right)=k^{-r}$, $\beta=r$, $r>0$ we get standard Sobolev
classes $W_{p}^{r}$. If $\lambda\left(k\right)=\exp\left(-\mu
k^{\gamma}\right)$, $\beta\in\mathbb{R}$, $\mu>0$, $0<\gamma<1$, then the set
$\Lambda_{\beta}U_{p}$ consists of infinitely differentiable functions. In the
case $\gamma=1$ we get classes of analytic functions. If $\gamma>1$, then we
obtain classes of entire functions. To simplify technical notations we present
our results just in the case $1<p,q<\infty$ and $\beta=0$.

We show here that isometric extensions may decrease the order of linear widths
of $\Lambda U_{p}$ in $L_{p}$ if $1<p<q\leq 2$ in the case of small and finite
smoothness, i.e. if

$\lambda\left(k\right)=k^{-r}\text{,
}r>\left(\frac{1}{p}-\frac{1}{q}\right)_{+},$

where $\left(a\right)_{+}=\max\left\\{a,0\right\\}$, $a\in\mathbb{R}$. From
the other side, it is shown that in the case of super-small smoothness, i.e.
if

$\lambda\left(k\right)=\phi\left(k\right)k^{-\left(\frac{1}{p}-\frac{1}{q}\right)_{+}},$

where $\phi\left(k\right)$ is a decreasing function,
$\lim_{k\rightarrow\infty}\phi\left(k\right)=0$ and
$\phi\left(k^{s}\right)\asymp\phi\left(k\right)$ for any fixed
$s\in\mathbb{N}$, isometric extensions can not decrease the order of
preabsolute linear widths $\Lambda_{n,m}\left(\Lambda U_{p},L_{q}\right)$,
$0\leq m\leq n$. A typical example of the sequence $\lambda\left(k\right)$ is
given by

$\lambda\left(k\right)=\left(\ln\left(k+1\right)\right)^{-\varrho}k^{-\left(\frac{1}{p}-\frac{1}{q}\right)_{+}}\text{,
}\varrho>0,k\in\mathbb{N}.$

Similarly, in the case of super-high smoothness, i.e. if

$\lambda\left(k\right)=\exp\left(-\mu n^{\gamma}\right)\text{, }\mu>0\text{,
}\gamma\geq 1$

the order of preabsolute linear widths $\Lambda_{n,m}\left(\Lambda
U_{p},L_{q}\right)$, $\left(p,q\right)\in I$, $0\leq m\leq n$ remains the same
as $\lambda_{n}\left(\Lambda U_{p},L_{q}\right)$ (see (3.4)). In this sense,
the results presented here complement the results obtained in [3].

For easy of notation we will put $a_{n}\gg b_{n}$ for two sequences, if
$a_{n}>Cb_{n}$ for some $C>0$ and any $n\in\mathbb{N}$ and $a_{n}\asymp b_{n}$
if $C_{1}b_{n}\leq a_{n}\leq C_{2}b_{n}$ for all $n\in\mathbb{N}$ and some
constants $C_{1}>0$ and $C_{2}>0$.

##  2\. Preabsolute linear widths

Our main result significantly reduces the space $\widehat{X}$ and gives an
explicit representation of the extension $\mathrm{j}$ which is important for
applications. Consider Banach space $\mathrm{lin}\left(A\right)$ with the unit
ball $A$ and $\left(\mathrm{lin}\left(A\right)\right)^{\ast}$ its conjugate
with the usual norm
$\left\|\cdot\right\|_{\left(\mathrm{lin}\left(A\right)\right)^{\ast}}$.

######  Theorem 1.

Let $A\subset X$ be a convex origin symmetric compact,
$\mathrm{diam}\left(A,X\right)<\infty$, $\phi_{k}\in X^{\ast}$ be such that

$\sup\left\\{\left\|x\right\|_{X}\left|x\in A\text{, }\left\langle
x,\phi_{k}\right\rangle=0\text{, }1\leq k\leq n\right.\right\\}\leq
d^{n}\left(A,X\right)+\epsilon\text{, }\forall\epsilon>0$

and $c_{k}:B\left(X^{\ast}\right)\rightarrow\mathbb{R}$, $1\leq k\leq n$ be
the functionals of the best approximation of $\phi\in B\left(X^{\ast}\right)$
by $\mathrm{lin}\left\\{\phi_{k}\text{, }1\leq k\leq n\right\\}$ in
$\left\|\cdot\right\|_{\left(\mathrm{lin}\left(A\right)\right)^{\ast}}$. Let

$\left[c_{k}\right]=\left\\{\begin{array}[]{cc}0,&c_{k}\in X^{\ast\ast},\\\
c_{k},&c_{k}\notin X^{\ast\ast}.\end{array}\right.$

Then

$\lambda_{n}\left(\mathrm{j}\left(A\right),\overline{X}\right)=\Lambda_{n}\left(A,X\right),$

where
$\mathrm{j}:X\rightarrow\overline{X}=\mathrm{lin}\left\\{X^{\ast\ast},\left[c_{1}\right],\cdots,\left[c_{n}\right]\right\\}\subset\widehat{X}$.

###### Proof.

By the definition of linear width, for any extension
$\mathrm{j}:X\rightarrow\widetilde{X}$ and $\epsilon>0$ there exist
$\phi_{k}\in$ $\widetilde{X}^{\ast}$and $x_{k}\in\widetilde{X}$, $1\leq k\leq
n$ such that

$\sup_{x\in
A}\left\|\mathrm{j}\left(x\right)-\sum_{k=1}^{n}\left\langle\mathrm{j}\left(x\right),\phi_{k}\right\rangle
x_{k}\right\|_{\widetilde{X}}\leq\lambda_{n}\left(\mathrm{j}\left(A\right),\widetilde{X}\right)+\epsilon.$

Consequently, by the definition of Gelfand widths (see [2]),

$d^{n}\left(A,X\right)\leq\sup\left\\{\left\|\mathrm{j}\left(x\right)\right\|_{\widetilde{X}}\left|x\in
A\text{, }\left\langle\mathrm{j}\left(x\right),\phi_{k}\right\rangle=0\text{,
}1\leq k\leq n\right.\right\\}$ (2.1)
$\leq\lambda_{n}\left(\mathrm{j}\left(A\right),\widetilde{X}\right)+\epsilon.$

Also, by the definition of Gelfand width there are such $\phi_{k}\in
X^{\ast}$, $1\leq k\leq n$ that

(2.2) $\sup\left\\{\left\|x\right\|_{X}\left|x\in A\text{, }\left\langle
x,\phi_{k}\right\rangle=0\text{, }1\leq k\leq n\right.\right\\}\leq
d^{n}\left(A,X\right)+\epsilon$

for any $\epsilon>0$. Let $\mathrm{lin}\left(A\right)$ be the Banach space
with the unit ball $A$ and $\left(\mathrm{lin}\left(A\right)\right)^{\ast}$ be
its conjugate with the usual norm

$\left\|\phi\right\|_{\left(\mathrm{lin}\left(A\right)\right)^{\ast}}=\sup\left\\{\left|\left\langle
x,\phi\right\rangle\right|\left|x\in A\right.\text{ }\right\\}.$

Then, $\forall\phi\in B\left(X^{\ast}\right)$, by duality and (2.2) we get

$\inf\left\\{\left\|\phi-\sum_{k=1}^{n}c_{k}\left(\phi\right)\phi_{k}\right\|_{\left(\mathrm{lin}\left(A\right)\right)^{\ast}}\left|c_{k}\text{,
}1\leq k\leq n\right.\right\\}$ $\leq\sup\left\\{\left|\left\langle
x,\phi\right\rangle\right|\left|x\in A\text{, }\left\langle
x,\phi_{k}\right\rangle=0\text{, }1\leq k\leq n\right.\text{ }\right\\}$ $\leq
d^{n}\left(A,X\right)+\epsilon.$

Since $\mathrm{diam}\left(A,X\right)<\infty$ then there are such bounded
functions $\phi\mapsto c_{k}\left(\phi\right)$, $1\leq k\leq n$, $\phi\in
B\left(X^{\ast}\right)$ that

$\sup\left\\{\left\|\phi-\sum_{k=1}^{n}c_{k}\left(\phi\right)\phi_{k}\right\|_{\left(\mathrm{lin}\left(A\right)\right)^{\ast}}\left|\phi\in
B\left(X^{\ast}\right)\right.\right\\}\leq d^{n}\left(A,X\right)+\epsilon,$

or

$\sup_{\phi\in B\left(X^{\ast}\right)}\sup_{x\in A}\left|\left\langle
x,\phi\right\rangle-\sum_{k=1}^{n}c_{k}\left(\phi\right)\left\langle
x,\phi_{k}\right\rangle\right|$ $=\sup_{x\in A}\sup_{\phi\in
B\left(X^{\ast}\right)}\left|\left\langle
x,\phi\right\rangle-\sum_{k=1}^{n}c_{k}\left(\phi\right)\left\langle
x,\phi_{k}\right\rangle\right|$ (2.3) $=\sup_{x\in
A}\left\|\mathrm{j}\left(x\right)-\sum_{k=1}^{n}\left\langle
x,\phi_{k}\right\rangle c_{k}\right\|_{\widehat{X}}\leq
d^{n}\left(A,X\right)+\epsilon.$

Since $\phi\in B\left(X^{\ast}\right)$ then $\left\langle
x,\phi\right\rangle\in X^{\ast\ast}$ and $c_{k}\in\widehat{X}$. Consequently,

$\left\langle x,\phi\right\rangle-\sum_{k=1}^{n}\left\langle
x,\phi_{k}\right\rangle c_{k}\in\mathrm{lin}\left\\{X^{\ast\ast}\text{,
}c_{k}\text{, }1\leq k\leq n\right\\}$
$=\mathrm{lin}\left\\{X^{\ast\ast}\text{, }\left[c_{k}\right]\text{, }1\leq
k\leq n\right\\},$

where $c_{k}$, $1\leq k\leq n$ are defined by (2.3). Observe that if among
$c_{k}$, $1\leq k\leq n$ there are linear functionals $c_{s}$ on $X^{\ast}$
then $c_{s}\in X^{\ast\ast}$, or $\left[c_{s}\right]=0$. Comparing (2.1) and
(2.3) we get the proof. ∎

Theorem 1 allows us to introduce the following notion.

######  Definition 1.

Let $\left(X,\left\|\cdot\right\|_{X}\right)$ be a Banach space and $A\subset
X$ be a compact, convex and origin symmetric set in $X$. The $m$-preabsolute
linear $n$-width of $A$ in $X$ is defined by

$\Lambda_{n,m}\left(A,X\right)=\inf\lambda_{n}\left(\mathrm{j}_{m}\left(A\right),\widetilde{X}\right)\text{,
}0\leq m\leq n,$

where $\inf$ is taken over all isometric extensions

$\mathrm{j}_{m}:X\rightarrow\widetilde{X}=\mathrm{lin}\left\\{X,c_{k}\text{,
}k\leq m\text{, }c_{0}=0\right\\}$

and

$c_{k}:B\left(X^{\ast}\right)\rightarrow\mathbb{R}\text{, }k\leq m\text{. }$

Observe that, by Theorem 1,

$\lambda_{n}\left(A,X\right)=\Lambda_{n,0}\left(A,X\right)\geq\Lambda_{n,1}\left(A,X\right)\geq\cdots\geq\Lambda_{n,n}\left(A,X\right)$
(2.4) $=\Lambda_{n}\left(A,X\right)=d^{n}\left(A,X\right).$

##  3\. Examples and application

In this section we consider several motivating examples in the case $1<p<q\leq
2$ to underline the dependence of preabsolute widths on smoothness. In
particular, it is shown that in the cases of super-small and super-high
smoothness isometric extensions can not decrease the order of linear widths.

######  Theorem 2.

1\. Let $1<p<q\leq 2$ and

$\lambda\left(k\right)=\varphi\left(k\right)k^{-\left(\frac{1}{p}-\frac{1}{q}\right)_{+}},k\in\mathbb{N}$

in (1.3), where $\left(a\right)_{+}=\max\left\\{a,0\right\\}$,
$\varphi\left(k\right)$ is a decreasing function,
$\lim_{k\rightarrow\infty}\varphi\left(k\right)=0$ and
$\varphi\left(k^{s}\right)\asymp\varphi\left(k\right)$ for any fixed $s>0$
(i.e. the case of super-small smoothness). Then

(3.1) $\Lambda_{n,m}\left(\Lambda
U_{p},L_{q}\right)\asymp\varphi\left(n\right)\text{, }1<p,q<\infty\text{,
}0\leq m\leq n.$

2\. If $\lambda\left(k\right)=k^{-r}$ where

$\frac{1}{p}-\frac{1}{q}<r<\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)/\left(\frac{1}{p}-\frac{1}{2}\right)\text{,
}1<p<q\leq 2$

(i.e. the case of small smoothness) then

(3.2)
$n^{\frac{p}{2\left(p-1\right)}\left(-r+\frac{1}{p}-\frac{1}{q}\right)}\ll\Lambda_{n,m}\left(W_{p}^{r},L_{q}\right)\ll
n^{-r+\frac{1}{p}-\frac{1}{q}}\text{, }0\leq m\leq n.$

3\. If $\lambda\left(k\right)=k^{-r}$ where

$r>\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)/\left(\frac{1}{p}-\frac{1}{2}\right)$

(i.e. the case of finite smoothness) then

(3.3) $n^{-r}\ll\Lambda_{n,m}\left(W_{p}^{r},L_{q}\right)\ll
n^{-r+\frac{1}{p}-\frac{1}{q}}\text{, }0\leq m\leq n\text{.}$

4\. Let $\lambda\left(k\right)=\exp\left(\mu k^{\gamma}\right)$, $\mu>0$,
$0<\gamma<1$ in (1.3) (i.e. the case of infinite smoothness) then

(3.4) $\exp\left(-\mu n^{\gamma}\right)\ll\Lambda_{n,m}\left(\Lambda
U_{p},L_{q}\right)\ll\exp\left(-\mu
n^{\gamma}\right)n^{\left(1-\gamma\right)\left(\frac{1}{p}-\frac{1}{q}\right)}\text{,
}0\leq m\leq n.$

5\. Let $\lambda\left(k\right)=\exp\left(\mu k^{\gamma}\right)$, $\mu>0$,
$\gamma\geq 1$ in (1.3) (i.e. the case of super-high smoothness) then

(3.5) $\Lambda_{n,m}\left(\Lambda U_{p},L_{q}\right)\asymp\exp\left(-\mu
n^{\gamma}\right)\text{, }\mu>0\text{, }\gamma\geq 1\text{, }0\leq m\leq n.$

###### Proof.

Let us consider the case of super-small smoothness (3.1). It was shown in [3]
that in this case

$d^{n}\left(\Lambda U_{p},L_{q}\right)\asymp\lambda_{n}\left(\Lambda
U_{p},L_{q}\right)$

$\asymp\sup_{f\in\Lambda
U_{p}}\left\|f-S_{n}\left(f\right)\right\|_{q}\asymp\varphi\left(n\right)\text{.}$

Consequently, in this case

(3.6) $\Lambda_{n,m}\left(\Lambda
U_{p},L_{q}\right)\asymp\varphi\left(n\right)\text{, }1<p,q<\infty$

for any $0\leq m\leq n$.

Let $\Lambda U_{p}=W_{p}^{r}$ be Sobolev class. In this case
$\lambda\left(k\right)=k^{-r}$. We show (3.3). It is known [3] that

(3.7) $\lambda_{n}\left(W_{p}^{r},L_{q}\right)\ll\sup_{f\in
W_{p}^{r}}\left\|f-S_{n}\left(f\right)\right\|_{q}\ll
n^{-r+\frac{1}{p}-\frac{1}{q}}\text{, }1<p<q\leq 2\text{,
}r>\frac{1}{p}-\frac{1}{q},$

where $S_{n}$ is defined by (1.2). By the Theorem 2, (2.4) and (1.1) for any
$0\leq m\leq n$ we have

$\Lambda_{n,m}\left(W_{p}^{r},L_{q}\right)\geq\Lambda_{n,n}\left(W_{p}^{r},L_{q}\right)=\Lambda_{n}\left(W_{p}^{r},L_{q}\right)$
$=d^{n}\left(W_{p}^{r},L_{q}\right)=d_{n}\left(W_{q^{{}^{\prime}}}^{r},L_{p^{{}^{\prime}}}\right),$

where

$p^{{}^{\prime}}=\left\\{\begin{array}[]{cc}\frac{p}{p-1},&1<p<\infty,\\\
1,&p=\infty,\\\ \infty,&p=1.\end{array}\right.$

Since $1<p<q\leq 2$ then $2\leq q^{{}^{\prime}}<p^{{}^{\prime}}<\infty$ and if

(3.8)
$r>\frac{1}{2}\left(\frac{1}{q^{{}^{\prime}}}-\frac{1}{p^{{}^{\prime}}}\right)/\left(\frac{1}{2}-\frac{1}{p^{{}^{\prime}}}\right)=\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)/\left(\frac{1}{p}-\frac{1}{2}\right)$

then

$d_{n}\left(W_{q^{{}^{\prime}}}^{r},L_{p^{{}^{\prime}}}\right)\asymp n^{-r}.$

Clearly,

$\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)/\left(\frac{1}{p}-\frac{1}{2}\right)>\frac{1}{p}-\frac{1}{q}.$

Hence

$n^{-r}\ll\Lambda_{n,m}\left(W_{p}^{r},L_{q}\right)\ll
n^{-r+\frac{1}{p}-\frac{1}{q}}\text{, }0\leq m\leq n$

if (3.8) is satisfied. This proves (3.3).

To show (3.2) we remark that if $2\leq p<q<\infty$ then [1], [12], [3]

$d_{n}\left(W_{p}^{r},L_{q}\right)\asymp
n^{\frac{q}{2}\left(-r+\frac{1}{p}-\frac{1}{q}\right)},$

where

$\frac{1}{p}-\frac{1}{q}<r<\frac{1}{2}\left(\frac{1}{p}-\frac{1}{q}\right)/\left(\frac{1}{2}-\frac{1}{q}\right).$

Consequently, by (2.4) and (1.1) we get

$\Lambda_{n,m}\left(W_{p}^{r},L_{q}\right)\geq
d^{n}\left(W_{p}^{r},L_{q}\right)=d_{n}\left(W_{q^{{}^{\prime}}}^{r},L_{p^{{}^{\prime}}}\right)$
$\asymp
n^{\frac{p^{{}^{\prime}}}{2}\left(-r+\frac{1}{q^{{}^{\prime}}}-\frac{1}{p^{{}^{\prime}}}\right)}=n^{\frac{p}{2\left(p-1\right)}\left(-r+\frac{1}{p}-\frac{1}{q}\right)}.$

The respective upper bounds follow from (3.7). This proves (3.2).

The case (3.4) can be treated similarly. Namely, since

$\lambda_{2n}\left(\Lambda U_{p},L_{q}\right)\ll\sup_{f\in
W_{p}^{r}}\left\|f-S_{n}\left(f\right)\right\|_{q}\ll\exp\left(-\mu
n^{\gamma}\right)n^{\left(1-\gamma\right)\left(\frac{1}{p}-\frac{1}{q}\right)_{+}}\text{,
}$

where

$1<p<q\leq 2\text{, }\mu>0\text{, }0<\gamma<1.$

[3], [5], [6], [7], [8] and by (1.1)

$d^{2n}\left(\Lambda U_{p},L_{q}\right)\asymp d_{2n}\left(\Lambda
U_{p^{{}^{\prime}}},L_{q^{{}^{\prime}}}\right)\asymp\exp\left(-\mu
n^{\gamma}\right),$ $2\leq p^{{}^{\prime}}<q^{{}^{\prime}}<\infty$

then

$\exp\left(-\mu n^{\gamma}\right)\ll\Lambda_{2n,2m}\left(\Lambda
U_{p},L_{q}\right)\ll\exp\left(-\mu
n^{\gamma}\right)n^{\left(1-\gamma\right)\left(\frac{1}{p}-\frac{1}{q}\right)}\text{,
}0\leq m\leq n.$

Finally, consider the case of super-high smoothness (3.5). Namely, if

$\lambda\left(k\right)=\exp\left(\mu k^{\gamma}\right)\text{, }\mu>0\text{,
}\gamma\geq 1.$

In this case [5]

$\lambda_{2n}\left(\Lambda U_{p},L_{q}\right)\ll\sup_{f\in
W_{p}^{r}}\left\|f-S_{n}\left(f\right)\right\|_{q}\ll\exp\left(-\mu
n^{\gamma}\right)\text{, }1<p,q<\infty$

and

$d^{2n}\left(\Lambda U_{p},L_{q}\right)\asymp d_{2n}\left(\Lambda
U_{p^{{}^{\prime}}},L_{q^{{}^{\prime}}}\right)\asymp\exp\left(-\mu
n^{\gamma}\right)\text{, }1<p,q<\infty.$

Consequently,

$\Lambda_{n,m}\left(\Lambda U_{p},L_{q}\right)\asymp\exp\left(-\mu
n^{\gamma}\right)\text{, }\mu>0\text{, }\gamma\geq 1$

for any $0\leq m\leq n$. ∎

## Acknowledgements

The Author wishes to thank the organisers of the $2^{\frac{nd}{{}}}$ Gaussian
Symposium, Munich, 1993, where Theorem 1 was reported by the author. Also, I
wish to thank referees and Communicating Editor for valuable comments.

## References

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  * [2] R.S. Ismagilov, Diameters of sets in normed spaces and the approximation of functions by trigonometric polynomials. Russian Math. Surveys 29 (1974) 169.
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  * [7] A. Kushpel, Estimates of the Bernstein widths and their analogs, Ukrainian Math. J. 45(1) (1993) 59-65.
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# Dynamical Properties and Effects of Quantum Diffraction on The Propagation
of E-A-Solitary Waves in Three-Component Fermi Plasma

Tamal Ghosh<EMAIL_ADDRESS>Department of Physics, Bidhannagar Govt.
College, Kolkata, West Bengal 700064, India  Suman Pramanick
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kharagpur, Kharagpur, West Bengal 721302, India  Soumya Sarkar
<EMAIL_ADDRESS>Department of Physics, National Institute of
Technology Karnataka, Karnataka 575025, India  Ankita Dey
<EMAIL_ADDRESS>Department of Physics, Lady Brabourne College,
Kolkata, West Bengal 700017, India  Swarniv Chandra  Govt. General Degree
College at Kushmandi, Dakshin Dinajpur, 733121, India

###### Abstract

Electron Acoustic Solitary structures in Fermi Plasma with two temperature
electrons have various applications in space and laboratory-made plasma.
Formulation of an adequate theory is important to understand various physical
systems with various physical parameters. The motion of two temperature
electrons in a quantum Fermi plasma system highly affects the solitary profile
of the system. We study the quantum Fermi plasma system with two temperature
electrons where the streaming velocities of the two-electron population are
opposite. We consider the quantum hydrodynamic model (QHD) and derive a linear
dispersion relation for the system. For the non-linear study of the system, we
use the standard perturbation technique to derive KdV-B equation and show the
evolution of a solitary profile with different plasma parameters. We analyze
the stable Rouge wave structure using NLSE and show simulation results. We
study the dynamical properties and phase plot for a two-stream quantum Fermi
plasma system with two temperature electrons.

KdV-B, Rouge Wave

††preprint: APS/123-QED

##  I Introduction

Plasma systems containing two distinct groups of electrons shows Electron
acoustic waves (EAWs). The distinction between two electron groups comes from
their energy. Two types of electrons are 1) Hot electrons and 2) Cold
electrons. The frequency of electron acoustic mode is higher than ion acoustic
frequencies of plasmas. Hot electrons can freely move with less viscous drag
and supply restoring force, whereas cold electron feels viscous drag and
produce inertia to the system. The thermal speed of hot electron is very large
in comparison to cold electron. The phase speed of electron acoustic wave is
smaller than the thermal speed of hot electrons but much larger than thermal
speed of cold electrons. In this system ions may be considered as uniform
neutralizing background. EAWs with two groups of electrons plays an important
role in space plasma (Ang and Zhang (2007)Ang. L.K. (2007); Barnes et al.
(2003)Barnes  _et al._ (2003); Feldman et al. (1975)Feldman  _et al._ (1975),
(1983a)Feldman  _et al._ (1983) as well as laboratory made plasmas (Ditmire et
al. (1998)Ditmire  _et al._ (1998); Sheridan et al. (1991)Gudmundsson  _et
al._ (2002); Armstrong et al. (1979)Armstrong R.J. (1979); Kadomtsev and
Pogutse (1971)Kadomtsev and Pogutse (1971); Henry and Trguier (1972)Henry and
Trguier (1972); Defler and Simonen (1969)Saberian and Esfandyari-Kalejahi
(2014)). The source of broadband electrostatic noises can be addressed with
EAWs and it has been used to explain those. The wave emission in different
regions of earth’s atmosphere can be explained with EAWs. For these studies
EAWs has become one of the important research areas in plasma physics. In
recent years there is a boost in the study of the nonlinear evolution of EAWs
(Bains et al. (2011)Bains  _et al._ (2011); Soultana and Kourakis
(2011)Sultana and Kourakis (2011); Kourakis and Shukla (2004)Kourakis and
Shukla (2004); Singh and Lakhina (2001)Singh and Lakhina (2001)). Various
space-craft missions, e,g, the FAST at the auroral region (Ergun et al.
(1998aErgun  _et al._ (1998a), 1998bErgun  _et al._ (1998b)); Delory et al.
(1998)Delory  _et al._ (1998); Pottelette et al. (1999)Pottelette  _et al._
(1999)) and the POLAR and GEOTAIL missions in the magnetosphere (Matsumoto et
al. (1994)Matsumoto  _et al._ (1994); Franz et al. (1998)Franz  _et al._
(1998); Cattell et al. (2003)Cattell  _et al._ (2003)) explained by EAW
related structures. Most of these application sites need theory and better
understanding of non-relativistic classical plasmas. However, there are
numbers of works on the theory of nonlinear propagation of electrostatic modes
in quantum plasmas with consideration of quantum hydrodynamic model of plasma
(Shukla and Eliasson (2006)Shukla and Eliasson (2006); Sahu and Roychoudhury
(2006)Sahu and Roychoudhury (2006); Ali and Shukla (2006)Shukla and Ali
(2005); Manfredi (2005)Manfredi (2005); Haas et al. (2003)Haas  _et al._
(2003); Gardner and Ringhofer (1996)Mamun and Shukla (2010)). The non-linear
wave structure of cold and hot electrons are investigated Chandra  _et al._
(2013)-Chandra and Ghosh (2012)

In this paper we study linear and nonlinear properties of EAWs in Fermi plasma
with two temperature electrons. We first assume basic hydrodynamic equations
for the system then we normalize them using suitable scaling for the system.
With the normalized equations on hand we linearize them to get linear
dispersion relation and linear dispersion characteristics are investigated.
The Korteweg-de Vries Burgers (KdV-B) equation is derived using standard
perturbation technique. We investigate dependence of soliton properties on
different plasma parameters. We study Rouge wave formation for the system. We
investigate the dynamical nature of the system. Then we conclude the paper
with some remarks and future work plans.

##  II Basic Formulation

The plasma system that we considered is unmagnetized consisting of two groups
of electrons. Two groups are 1) Hot electrons and 2) Cold electrons. The
thermal energy of hot electron is higher than cold electrons so the mobility
of hot electron is large compared to that of cold electrons. So, in the
momentum equation for hot electron we consider inertia term as zero, whereas
in the momentum equation of cold electrons we consider a viscous term as its
mobility is less and more responsive to the viscous forces.

$\frac{\partial n_{h}}{\partial
t}+\frac{\partial\left(n_{h}u_{h}\right)}{\partial x}=0$ (1) $\frac{\partial
n_{c}}{\partial t}+\frac{\partial\left(n_{c}u_{c}\right)}{\partial x}=0$ (2)
$0=\frac{e}{m_{e}}\frac{\partial\phi}{\partial
x}-\frac{1}{m_{e}n_{h}}\frac{\partial P_{h}}{\partial
x}+\frac{\hbar^{2}}{2m_{e}\gamma^{2}_{h}}\frac{\partial}{\partial
x}\left[\frac{1}{\sqrt{n_{h}}}\frac{\partial^{2}\sqrt{n_{h}}}{\partial
x^{2}}\right]$ (3)

This is momentum equation for hot electrons. The inertia term (left hand side
of the equation) is zero, as hot electrons are very mobile so its inertia can
be assumed to be zero. Here,
$\gamma_{h}=\left(1-\frac{u_{h}^{2}}{c^{2}}\right)^{-\frac{1}{2}}$

$\left(\frac{\partial}{\partial t}+u_{c}\frac{\partial}{\partial
x}\right)(u_{c}\gamma_{c})=\frac{e}{m_{e}}\frac{\partial\phi}{\partial x}\\\
-\frac{1}{m_{e}n_{c}}\frac{\partial P_{c}}{\partial
x}+\frac{\hbar^{2}}{2m_{e}^{2}\gamma_{c}^{2}}\frac{\partial}{\partial
x}\left[\frac{1}{\sqrt{n_{c}}}\frac{\partial^{2}\sqrt{n_{c}}}{\partial
x^{2}}\right]\\\ +\eta_{c}\frac{\delta^{2}u_{c}}{\delta x^{2}}$ (4)

This is momentum equation for cold electrons. The inertia term is non zero as
the mobility of cold electrons is very less. For our two-electron plasma
system cold electron produces the restoring force for electron-acoustic
oscillations. Cold electrons also got a viscous term. Here,
$\gamma_{c}=\left(1-\frac{u_{c}^{2}}{c^{2}}\right)^{-\frac{1}{2}}$

$\frac{\partial^{2}\phi}{\partial x^{2}}=4\pi
e\left(n_{c}+n_{h}-Z_{i}n_{i}\right)$ (5)

The subscript $i$ in the Poisson’s equation is for representing ions. The
above-mentioned equations are the governing equations for the plasma system,
where $n_{h}$ and $n_{c}$ are the density of hot and cold electrons
respectively. $\phi$ is the electrostatic potential. $P_{h}$ is the pressure
law for hot electrons. $\hbar$ is the plank constant divided by $2\pi$.
$m_{e}$ is the mass of electron and $e$ is the charge of electron and
$\eta_{c}$ is the viscous constant for cold electrons.

We considered Fermi pressure as the main pressure component for electrons.
Fermi pressure is

$P_{j}=\frac{m_{j}V_{Fj}^{2}}{3n_{j0}^{2}}n_{j}^{3}$ (6)

where subscript $j=h$ is for hot electrons and $j=c$ for cold electrons.
$n_{h0}$ is the initial hot electron density. The parameters need to be
normalized to get a good control of the equations and normalized equations are
easy to work with. Normalization means we need to define some suitable scaling
associated with our problem. With these scaling constants we can make our
parameters dimensionless.

For our problem, normalization has been done in following ways $x\rightarrow
x\omega_{j}/V_{Fj},t\rightarrow t\omega_{j},\phi\rightarrow
e\phi/2k_{B}T_{Fj},n_{h}\rightarrow n_{h}/n_{h0},n_{c}\rightarrow
n_{c}/n_{c0},,u_{h}\rightarrow u_{h}/V_{Fh},u_{c}\rightarrow u_{c}/V_{Fc}$
where $\omega_{j}=\sqrt{4\pi en_{c0}e^{2}/m_{e}}$,
$V_{Fh}=\sqrt{\frac{2K_{B}T_{Fh}}{m_{e}}}$ and
$H=\hbar\omega_{j}/2k_{B}T_{Fj}$ Changing the variables and parameters
accordingly we have Normalized Governing Equations.

$\frac{\partial n_{h}}{\partial
t}+\frac{\partial\left(n_{h}u_{h}\right)}{\partial x}=0$ (7) $\frac{\partial
n_{c}}{\partial t}+\frac{\partial\left(n_{c}u_{c}\right)}{\partial x}=0$ (8)
$0=\frac{\partial\phi}{\partial x}-n_{h}\frac{\partial n_{h}}{\partial
t}+\frac{H^{2}}{2\gamma_{h}^{2}}\frac{\partial}{\partial
x}\left[\frac{1}{\sqrt{n_{h}}}\frac{\partial^{2}\sqrt{n_{h}}}{\partial
x^{2}}\right]$ (9)

$\displaystyle\left(\frac{\partial}{\partial t}+u_{c}\frac{\partial}{\partial
x}\right)(u_{c}\gamma_{c})$ $\displaystyle=\frac{\partial\phi}{\partial x}$
(10) $\displaystyle+\frac{H^{2}}{2\gamma_{c}^{2}}\frac{\partial}{\partial
x}\left[\frac{1}{\sqrt{n_{c}}}\frac{\partial^{2}\sqrt{n_{c}}}{\partial
x^{2}}\right]$ $\displaystyle+\eta_{c}\frac{\delta^{2}u_{c}}{\delta x^{2}}$

$\frac{\partial^{2}\phi}{\partial
x^{2}}=n_{c}+\frac{n_{h}}{\delta}-\frac{\delta_{i}}{\delta}n_{i}$ (11)

where $\delta=\frac{n_{c0}}{n_{h0}}$, $\delta_{1}=Z_{i}\frac{n_{i0}}{n_{h0}}$.

##  III Linear Dispersion Relation

Figure 1: Dispersion relation plot for different $u_{0}$ keeping $\delta=0.3$
and $H=1$ Figure 2: Dispersion relation plot for different $\delta$ keeping
$u_{0}=0.25$ and $H=1$ Figure 3: Dispersion relation plot for different $H$
keeping $\delta=0.3$ and $u_{0}=0.25$

The system parameters can be expanded into following perturbation expansion
for $n_{h}$, $n_{c}$, $u_{h}$, $u_{c}$, and $\phi$

$\displaystyle\left(\begin{array}[]{l}n_{j}\\\ u_{j}\\\
\phi\end{array}\right)=\left(\begin{array}[]{c}1\\\ \pm u_{0j}\\\
\phi_{0}\end{array}\right)$
$\displaystyle+\epsilon^{1}\left(\begin{array}[]{l}n_{j}^{(1)}\\\
u_{j}^{(1)}\\\
\phi^{(1)}\end{array}\right)+\epsilon^{2}\left(\begin{array}[]{l}n_{j}^{(2)}\\\
u_{j}^{(2)}\\\ \phi^{(2)}\end{array}\right)$ (24)
$\displaystyle+\epsilon^{3}\left(\begin{array}[]{l}n_{j}^{(3)}\\\
u_{j}^{(3)}\\\ \phi^{(3)}\end{array}\right)+\cdots$ (28)

Here we assume existence of streaming velocities for hot and cold electrons
(+ve for hot electron and -ve for cold electrons) and equilibrium field in
constant field $\phi_{0}$. Substituting these expansions in the governing
relations and then taking only linear terms (linearizing) with the assumption
that all field variable varies periodically as $e^{i(kx-\omega t)}$, we have
following complex dispersion relation

$-k^{2}=\frac{1}{\left[\frac{H^{2}k^{2}}{4}\gamma_{c}^{2}-\left(\frac{\omega-
u_{0}k}{k}\right)^{2}\right]+i\eta_{c}\left(\omega-u_{0}k\right)}\\\
+\frac{\frac{1}{\delta}}{1+\frac{H^{2}k^{2}}{4}\gamma_{c}^{2}}$ (29)

Here we assume $u_{0h}=u_{0c}=u_{0}.$ Now we have to keep in mind that $k$
itself is a complex number, $k=$ $k_{1}+ik_{2}$. Putting this into the
expression of Complex dispersion relation we have two equations, one for real
part of the equation and other for imaginary part of the equation. The real
dispersion equation is:

$-1=\frac{\left(\frac{1}{\delta}+k_{1}^{2}+\frac{H^{2}k_{1}^{4}}{4}\gamma_{c}^{2}\right)\cdot\left[\frac{H^{2}k_{1}^{4}}{4}\gamma_{c}^{2}-\omega^{2}-u_{0}^{2}k_{1}^{2}+2\omega
u_{0}k_{1}\right]}{\left[k_{1}^{2}+\frac{H^{2}k_{1}^{2}}{4}\gamma_{c}^{2}\right]}$
(30)

With $k_{2}=0,$ if we plot $\omega$ vs $k_{1}$ then the plot will give us
dispersion plot. $H,u_{0},\delta$ are the parameters of the equation.

Fig. 1 shows dispersion plot for different values of equilibrium streaming
velocity $u_{0}.$ Plot shows with increasing $u_{0}$ the slope of $\omega$ vs
$k$ plot increases, i.e. an increase in EAW velocity.

Fig. 2. shows dispersion plot for different values of $\delta.$ Increase in
$\delta$ shows decrease in slope (for small $\omega$ this behavior is clear).
For $\delta=1$ i.e. the density of hot and cold electrons are same, we are
getting a linear behaviour in small $k$ regime.

Figure 4: 2D Solitary profile for different $H$ keeping $\delta$ and $u_{0}$
constant Figure 5: 3D Solitary profile for different $H$ keeping $\delta$ and
$u_{0}$ constant Figure 6: 2D Solitary profile for different $u_{0}$ keeping
$\delta$ and $H$ constant Figure 7: 3D Solitary profile for different $u_{0}$
keeping $\delta$ and $H$ constant Figure 8: 2D Solitary profile for different
$\delta$ keeping $H$ and $u_{0}$ constant Figure 9: 3D Solitary profile for
different $\delta$ keeping $H$ and $u_{0}$ constant

Increasing delta means more initial cold electrons compare to hot electrons.
So, according to our plots a greater number of initial cold electrons implies
a decrease in $k$ vs $\omega$ slop, that is increase in EAW velocity. Velocity
of sound waves, rather acoustic waves increases with increasing the rigidity
of the medium. Now a greater number of cold electrons means more viscous drag
and eventually more rigid medium, which can be the cause for increase in EAW
velocity.

Fig. 3. shows $\omega$ vs $k$ plot for different $H$ values. We change $H$
from 1 to 3 keeping $u_{0},\delta$ and constant. As $H$ increases the slope of
$\omega$ vs $k$ plot increases, that is also visible from the dispersion
relation. However, $H$ depends on plasma frequency $\omega_{c}$ linearly. So,
higher $H$ means higher plasma frequency. This leads to a conclusion that for
high plasma frequency the EAW has less wavenumber with same frequency.

##  IV Korteweg-de Vries Burgers Equation for Nonlinear Study

We used standard reductive perturbation technique with usual stretching of
space and time variables. We used two perturbation expansions to see the
behavior of the solution of KdV-B equation. Using the perturbation expansion
eq. (24) and, stretching of variables

$\xi=\epsilon^{\frac{1}{2}}\left(x-v_{0}t\right);\quad\eta=\eta_{0}\epsilon^{\frac{1}{2}};\quad\text{and}\quad\tau=\epsilon^{\frac{3}{2}}t$
(31)

Putting them into governing equations and taking the lower order terms with
power of epsilon we have the following KdV-Burger equation

$\frac{\partial\phi}{\partial\tau}+A\phi\frac{\partial\phi}{\partial\xi}+B\frac{\partial^{3}\phi}{\partial\xi^{3}}-C\frac{\partial^{2}\phi}{\partial\xi^{2}}=0$
(32)

The solution to (32) is

$\phi=\frac{12D}{N}\operatorname{sech}^{2}\xi-\frac{36R}{15N}\tanh\xi$ (33)

with,

$A=\frac{2DS_{1}R_{1}+Fn_{0c}R_{1}^{2}-D\left(M-u_{0c}\right)\frac{Q_{1}^{2}}{n_{0h}\delta}}{DS_{1}+n_{0c}R_{1}\left(1+\frac{3u_{0c}^{2}}{2c^{2}}\right)}$
(34)
$B=\frac{\left(D\left(M-u_{0c}\right)\left(\frac{T_{h}Q_{1}}{n_{0}h^{\delta}}\right)-T_{c}S_{1}n_{0c}-D\left(M-u_{0c}\right)\right)}{DS_{1}+n_{0c}R_{1}\left(1+\frac{3u_{0c}^{2}}{2c^{2}}\right)}$
(35) $C=\eta_{c}R_{1}$ (36)

With,

$D=M+\frac{3Mu_{0c}^{2}}{2c^{2}}-\frac{3u_{0c}^{3}}{2c^{2}}-u_{0c}$ (37)
$F=1+\frac{9u_{0c}^{2}}{2c^{2}}-\frac{3Mu_{0c}}{c^{2}}$ (38)
$T_{c}=\frac{H^{2}}{4n_{oc}^{2}}\left(1-\frac{u_{0c}^{2}}{c^{2}}\right)$ (39)
$T_{h}=\frac{H^{2}}{4n_{0h}^{2}}\left(1-\frac{u_{0h}^{2}}{c^{2}}\right)$ (40)
$S_{1}=\frac{n_{0c}}{\left.\left(M-u_{0c}\right)\left(u_{0c}+\frac{3u_{0c}^{3}}{2c^{2}}\right)-M\left(1+\frac{3u_{0c}^{2}}{2c^{2}}\right)\right]}$
(41)
$R_{1}=\frac{1}{\left(u_{0c}+\frac{3u_{0}^{3}c}{2c^{2}}\right)-M\left(1+\frac{3u_{0}^{2}c}{2c^{2}}\right)}$
(42) $Q_{1}=\frac{1}{n_{0k}}$ (43)

$\text{ For, }n_{0j}=1;u_{0j}=0;j=hc$

$S_{1}=-\frac{1}{M^{2}};R_{1}=-\frac{1}{M};Q_{1}=1;D=M;F=1;T=\frac{H^{2}}{4}$

The the value of

$A=\left(\frac{M^{3}}{2\delta}-\frac{3}{2M}\right)$ (44)
$B=\left(\frac{H^{2}}{8M}+\frac{M^{3}}{3}-\frac{M^{3}H^{2}}{4\delta}\right)$
(45) $C=-\frac{\eta_{0}}{M}$ (46)

FIG.4 and FIG.5 shows an increase in the peak of the solitary profile with
increasing $H$ For this plot $u_{0}$=0.5 and $\delta$=0.45. FIG.7 and FIG.8
show decrease in the peak of the solitary profile with increasing $\delta$.
For this plot $u_{0}$=0.5 and $H$=2

##  V Rogue wave

KdV-B equation gives the solitary wave structures which are created due to the
balance between the dispersive force and non-linear force in plasma. In case
the balance breaks the KdV-B starts to modify itself to a sudden wave having
amplitude almost twice or greater than the amplitude of normal waves at that
time. Like the sudden high amplitude, the suddenness of its timing makes it
more dangerous. The study of the Rogue wave is important as the interference
of the waves from different directions and the non-linear property make it
unpredictable. We study the stability of Rogue wave and its evolution from our
KdV-B equation

$F=\epsilon^{2}F_{0}+\sum_{s=1}^{\infty}\epsilon_{s}\left(F_{s}e^{is\psi}+F_{s}^{*}e^{-is\psi}\right)$
(47)

Where, $F$ is the field variable, is the smallness parameter, and $\psi$ is
the phase factor. Now, using this equation we can expand the field variable
i.e. the potential in KdV-B equation

$\phi=\epsilon^{2}\phi_{0}+\epsilon\phi_{1}e^{i\psi}+\epsilon\phi_{1}^{*}e^{-i\psi}+\epsilon^{2}\phi_{2}e^{2i\psi}+\epsilon^{2}\phi_{2}^{*}e^{-2i\psi}+\cdots$
(48)

The first harmonics ($\phi_{1}$) and second harmonics ($\phi_{2}$) can be
further expanded respectively

Figure 10: Rogue wave structure

$\phi_{1}=\phi_{1}^{(1)}+\epsilon\phi_{1}^{(2)}+\epsilon^{2}\phi_{1}^{(3)}+\ldots$
(49)
$\phi_{2}=\phi_{2}^{(1)}+\epsilon\phi_{2}^{(2)}+\epsilon^{2}\phi_{2}^{(3)}+\ldots$
(50)

$F_{0}$ and $F_{s}$ are assumed to vary very slowly with space and time. Now,
we use the change in variables
$\frac{\partial}{\partial\tau}=-is\omega-\epsilon
c\frac{\partial}{\partial\rho}+\epsilon^{2}\frac{\partial}{\partial\theta}\text{
and }\frac{\partial}{\partial\xi}=isk+\epsilon\frac{\partial}{\partial\rho}$
and put it in the Eq. (32) and equate the coefficients of the first order of
$\epsilon$, we get

$\omega=-Bk^{3}$ (51) $\frac{d\omega}{dk}=-3Bk^{2}$ (52)

Equating the coefficients of $e^{2i\psi}$ with $\epsilon^{2}$

$\varphi_{2}^{(1)}=\frac{A}{6Bk^{2}}\varphi_{1}^{(1)^{2}}$ (53)

Equating the terms independent of $\psi$ with $\epsilon^{3}$

$\varphi_{0}^{(1)}=\frac{A}{C}\varphi_{1}^{(1)}\varphi_{1}^{(1)^{*}}$ (54)

By equating the other higher terms we get

$\frac{\partial\varphi_{1}^{(1)}}{\partial\tau}+3ikB\frac{\partial^{2}\varphi_{1}^{(1)}}{\partial\rho^{2}}=-iAk\left(\varphi_{0}^{(1)}\varphi_{1}^{(1)}+\varphi_{2}^{(1)}\varphi_{1}^{(1)^{*}}\right)$
(55)
$i\frac{\partial\varphi_{1}^{(1)}}{\partial\tau}-3kB\frac{\partial^{2}\varphi_{1}^{(1)}}{\partial\rho^{2}}+\frac{A^{2}}{6Bk}\left(\varphi_{1}^{(1)^{2}}\varphi_{1}^{(1)^{*}}\right)=0$
(56)

Let’s assume $P=ic-3Bk$ and
$Q=A^{2}k\left[\frac{1}{6Bk^{2}+4ikc}-\frac{1}{3Bk^{2}}\right].$ The term
$\mathrm{PQ}$ is very important in defining the existence of a rogue wave. In
the region $PQ<0$ the rogue wave is stable and for $PQ>0$ the rogue wave
doesn’t exist. The solution of Eq. (56) we get as

$\phi(\rho,\theta)=\sqrt{\frac{2P}{Q}}\left[\frac{4(1+4iPQ)}{1+16P^{2}Q^{2}+4\rho^{2}}-1\right]\exp(2iP\theta)$
(57)

Here we plot the potential amplitude with the changed variable $\rho$ for
different values of $\theta$. And we take almost 500 values of $\theta$ and as
one can see that for only one value the potential suddenly gets abnormally
high with respect to the other. All of the solutions we get from the same
NLSE, where only one wave behaves dangerously. It perfectly depicts the
feature of a Rogue Wave, the amplitude is way higher than that of solitary
profiles in Fig. (4) to Fig. (9).

Here we take the viscosity coefficient $\eta_{c}=5$ and observed that the
amplitude of the rogue wave in the plot is lower than that of the rogue wave
with no viscous term.

##  VI Simulation of KdV-B and Rouge wave evolution

The static plots of two simulations of the solitary profiles and rogue wave
profile of the relativistic plasma are shown below.

Figure 11: Simulation result of evolution of solitary profile Figure 12:
Simulation result of evolution of Rogue wave profiles

A movie of KdV-B evolution can be found in this link:
https://doi.org/10.5281/zenodo.4366748 and a movie of Rouge wave evolution can
be found in this link: https://doi.org/10.5281/zenodo.4384499 .

In both FIG.11 and FIG.12, using the solutions of KdV-B and rouge wave, we
simulate the KDV solitary wave and rogue wave respectively. Here, $‘x’$
represents the space coordinate and $‘t’$ represents the time coordinate, and
$‘u’$ is the amplitude. In both cases, we see the changes in the waves in the
interval between $t=0s$ and $t=1s$. As you can see from the simulations that
rogue waves have much higher amplitudes than those of normal solitary waves
and also less frequent than normal waves.

##  VII Dynamical Study

To get the phase diagram of EAWs, we study a travelling wave solution followed
by the KdV-B equation. To do that we use the transformation $\eta=$
$\xi-M\tau$ with the proper boundary conditions at $\eta\rightarrow$
$\pm\infty,\psi,\delta\psi/\delta\eta$ and
$\left(\delta^{\wedge}2\psi\right)/\left(\delta\eta^{\wedge}2\right)\rightarrow
0$ where $M$ is the velocity of the wave. Applying a transformation like
Abdikian  _et al._ (2020) we can write the Non-linear Schrodinger equation as

Figure 13: $\psi$ vs $z$ plot (Phase plot) for dynamical behaviour.

$\frac{d^{2}\psi}{d\eta^{2}}=\left(\beta^{2}-\frac{1}{M_{1}l^{2}}\beta
V\right)\psi-\frac{M_{2}}{M_{1}l^{2}}$ (58)

Eq. (41) can be written in the following dynamical form

$\frac{d\psi}{d\eta}=z$ (59) $\frac{dz}{d\eta}=M_{1}\psi-M_{2}\psi^{3}$ (60)

Figure 14: $\psi$ vs $\eta$ plot for dynamical behaviour.

The phase plot gives the dynamical nature of the system (Fig.13) The
progressive nature of EAW can be seen from Fig.14. Plot shows a stable
solution. Different coloured plots in phase diagram correspond to different
energy solutions. Green one is the highest energy and blue one is the lowest.

##  VIII summary

Both the linear and nonlinear properties of electron-acoustic wave (EAW) have
been investigated in three component Fermi plasma consisting of two of Fermi
pressure for hot electrons. The stable models for EAWs are shown. The
dependence of electron-acoustic wave velocity on different plasma parameters
is shown and explained. The slope of the dispersion relation plots depends on
the velocity of EAW inversely. The dependence of this EAW velocity and EAW
frequency on different plasma parameters has been shown. Electron-acoustic
wave velocity increases with increasing $H$, $u_{0}$ and $\delta$ and
decreases with increasing viscous coefficient $\eta_{c}$ The formation of
rouge wave in the system has been studied. The evolution of solitary profile
and rouge wave has been studied. Dynamical properties of the system has also
been studied and found a stable dynamical structure.

###### Acknowledgements.

SP would like to acknowledge the support of institute fellowship by IIT
Kharagpur and the scholarship provided by DST-INSPIRE. SC would like to thank
the Institute of Natural Sciences and Applied Technology, the Physics
departments of Jadavpur University and Government General Degree College at
Kushmandi for providing facilities to carry this work.

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# Properties of Primitive Galaxies

Sara R. Heap1 I. Hubeny2 J.-C. Bouret3 T. Lanz4
J. Brinchmann5 1Emerita scientist, NASA Goddard Space Flight Center,
Greenbelt, MD,
2Steward Observatory, University of Arizona, Tucson, AZ,
3Aix-Marseille Univ. CNRS, CNES, LAM, Marseille, France,
4Observatoire de Côte d’Azur, Nice, France,
5University of Porto, Porto, Portugal

(2022)

###### Abstract

We report on a study of 9 nearby star-forming, very low-metallicity galaxies
observed by Hubble’s COS far-UV spectrograph that can serve as templates of
high-z galaxies to be observed by JWST. We find that the nebular spectra of
these primitive galaxies show evidence of irradiation by X-ray emitters.
Following Thuan et al. (2004), we identify the sources of X-ray emission as
massive X-ray binaries containing a massive accreting stellar black hole. We
further find that the lower the metallicity, the higher the probability of
strong X-irradiation. Following Heger et al. (2003), we suggest that these
accreting black holes are produced by direct collapse of stars having initial
masses greater than $\sim 50\,M_{\odot}$. Our models of young star clusters
with an embedded stellar black hole produce effects on the surrounding gaseous
medium that are consistent with the observed spectra. We conclude that
primitive galaxies are qualitatively different from more metal-rich galaxies
in showing evidence of hard radiation that can best be explained by the
presence of one or more embedded stellar black holes.

††volume: 361††journal: Massive Stars Near & Far

##  1 Introduction

We present a study of far-UV spectra of nearby primitive galaxies that could
give insights into processes operating in the high-z galaxies thought to be
responsible for the re-ionization of the universe. By “primitive galaxies,” we
mean galaxies having a low stellar mass, $\log(M_{\ast}/M_{\odot})\leq 8$ and
low gas metallicity, $\log O/H+12\leq 8$ and an even lower iron abundance,
whether they are local or at high redshift.

Our main goal here is to gain insight into the role of metallicity in
influencing the physical properties, evolutionary paths, and destinies of
massive stars and through them, with the help of models, an understanding of
low-metallicity galaxies both near and far. One approach is to examine the
high-velocity winds of hot, massive stars. Another approach is to test the
importance of X-ray radiation in very low metallicity galaxies, which was
first pointed out by Thuan et al. (2004) who found that the three most metal-
deficient galaxies are ultra-luminous X-ray emitters (ULX). A ULX is defined
by having an X-ray luminosity, $L_{x}>1\times 10^{39}$ erg s-1. They
identified these X-ray sources as massive X-ray binaries (MXRB’s), each
containing a massive, accreting stellar black hole. The optical spectral
region also shows evidence of hard radiation. Shirazi & Brinchmann (2012)
found that spectra of primitive galaxies observed by the SDSS show nebular He
II $\lambda 4686$ in emission, the result of ionization to He III and
subsequent recombination. Recently, spectroscopic surveys by Izotov et al.
(2021) found six compact star-forming galaxies with 12+log O/H=7.46-7.88 all
showing evidence of hard radiation in the form of high-ionization emission
lines such as [Ne V].

##  2 Observations of primitive galaxies

The far-UV spectral region is the most informative region for studying stellar
winds of primitive stars as it has resonance lines of abundant elements in
several ionization stages. There is empirical evidence from far-UV spectra of
hot, massive stars in the galaxy and Magellanic Clouds that the strength of
the stellar wind increases with increasing metallicity as $Z^{0.7}$ (Vink et
al. 2001), but this relation has not been tested below $\log Z\sim 8.0$. We
have made predictions of the far-UV spectra of such primitive stars based on
the grid of non-LTE photospheric models by Lanz & Hubeny (2003, 2007) and
unified spectra (photosphere + wind) calculated by Bouret using the CMFGEN
code (Hillier & Miller, 1998). Figure 1 shows predicted spectra of massive
stars demonstrating that even at $\log Z/Z_{\odot}=-2$, the N V doublet and C
IV doublet are still detectable stellar-wind features. When compared to the
spectrum of the most metal-deficient galaxy in our sample, I Zw 18-NW, there
is qualitative agreement between theory and observation, given that we are
comparing the spectra of a galaxy to that of a single star.

Figure 1: CMFGEN models of the far-UV spectra of main-sequence O-type stars.
The stellar parameters are given above each model. They are: effective
temperature, log g, log Z, terminal velocity of the wind in km/s, log mass-
loss rate in solar masses per year. In a single star, the N V doublet and C IV
doublet cannot both be strong, whereas in a star cluster or galaxy, both
features can be strong. As He II emission is formed in the stellar wind, He II
$\lambda$1640 is a broad feature, easily distinguished from nebular emission
lines which are narrow.

Far-UV spectra are similarly informative about the circumstellar,
interstellar, and circumgalactic medium of primitive galaxies. Figures 2-3
show the far-UV spectra of 9 galaxies having nebular oxygen abundances, log
O/H + 12 = 7.0 - 8.3. Spectral regions encompassing the C IV $\lambda\lambda
1548,1550$ doublet, He II $\lambda 1640$, and O III] $\lambda\lambda
1661,1666$ are highlighted in yellow. The spectra are arranged in order of
increasing oxygen abundance. In Figure 2, covering log Z = 7.0-7.8, not only
nebular He II $\lambda 1640$ (IP=54 eV) is in emission but also nebular C IV
doublet (IP=64 eV) in emission superimposed on the stellar spectrum, in which
C IV doublet has a P Cygni profile characteristic of a radiatively driven,
high-velocity stellar wind. In Figure 3, covering log Z=7.8-8.2, nebular He II
$\lambda 1640$ emission weakens with increasing metallicity to the point that
it is undetectable at log Z=8.2. At the same time, broad stellar emission,
characteristic of Wolf-Rayet stars, becomes strong in two of the galaxies.

Figure 2: Hubble/COS far-UV spectra of 9 low-Z galaxies galaxies. Spectral
regions encompassing the C IV $\lambda\lambda 1548,1550$ doublet, He II
$\lambda 1640$, and O III] $\lambda\lambda 1661,1666$ are highlighted. The
galaxy ID is given on the Y-axis of the plot, and its redshift, at the bottom
right of the plot. The nebular abundance of oxygen, log O/H+12, is listed at
the bottom left, denoted as log Z. The spectra are arranged in order of
increasing metal abundance. The spectra of I Zw 18-NWwere obtained directly in
programs ID=11523, 12028 (PI:Green), and most of the others in program
ID=14120 (PI:Brinchmann). The two exceptions are SBS 0335-053 and SBS
1415+437, which were obtained from the Hubble archive (ID 15193, PI: Aloisi).

Figure 3: Far-UV spectra of galaxies spanning log Z=7.83-8.23.

##  3 Discussion and Conclusions

How are we to understand this metallicity sequence of spectra? To find out, we
used CLOUDSPEC (Hubeny et al. 2000), a combination of CLOUDY (Ferland et al.
2017) and SYNSPEC (Hubeny & Lanz, 1995) to model the far-UV spectra of
galaxies for 3 scenarios. In each scenario, there is a 5 Myr-old primitive
stellar population (called “SPop” in Fig. 4) like I Zw 18-NW. In the first
scenario (red), there is a dormant black-hole companion or no black hole at
all. In the second (green), there is is a black-hole disk radiating X-rays at
a rate of $10^{39}$ erg/s, and in the third (blue), a X-ray source ten times
more luminous. Figure 4 shows the results. Nebular emission by He II $\lambda
1640$ is not present when there is no black hole, it is detectable in the case
of $L_{x}=10^{39}$ erg/s, and it quite strong in the $L_{x}=10^{40}$ erg/s
case. The C IV doublet is seen in absorption when there is no black hole, the
absorption lines are partially filled in for the case of $L_{x}=10^{39}$
erg/s, and strongly in emission at $L_{x}=10^{40}$ erg/s. Although the
statistics are limited, we conclude that the presence of C IV emission is
associated with galaxies whose metallicity,$\ logZ\leq 7.83$, while nebular He
II emission is still present in galaxies until $\log Z>8.0$.

Figure 4: Sample CLOUDSPEC model (Hubeny et al. 2000) of a primitive galaxy
like I Zw 18-NWshowing the effects of an embedded stellar black hole. Inputs
to CLOUDSPEC include: full SED of a stellar cluster at age=5 Myr, log Z=7.2;
BH parameters: $L_{x}$ as shown, temperature of accretion disk, $T=10^{6}$ K;
nebular parameters: election density, $N_{e}=10\,\rm{cm}^{-3}$, H I column
density =$2\times 10^{21}$ cm-2, covering fraction=0.4, no dust.

Why should the presence of a black hole be correlated with metallicity? One
answer comes from a paper Heger et al. (2003), titled: “How Massive Single
Stars End Their Life”. It suggests that stars having a final mass greater than
about 40 $M_{\odot}$ collapse directly into a black hole with little to no
mass lost in an explosion. Since very low metallicity stars lose very little
mass during their lifetime because of such weak stellar winds, their final
mass is 80% of their initial mass (e.g. Groh et al. 2019), so stars formed
with an initial mass of 50 $M_{\odot}$ or more will eventually collapse to a
massive stellar black hole.

In this interpretation, we are viewing the former secondary orbiting a black
hole, which formed from the collapsed primary, and the former secondary is now
feeding the black hole via a black hole disk. The black hole disk is hot,
particularly at its inner edge where it is 1-2 million degrees, so its
emission is centered on the X-ray spectral region. The reason why the disk is
hot is because a stellar-mass black hole, as opposed to a supermassive black
hole, forms a very deep potential well, so that accreted material is heated to
such high temperatures.

There are several COS far-UV spectra of primitive galaxies from the CLASSY
program (Berg et al. 2022) in the Hubble archives that could be included. And
combination with optical spectra from the SDSS and other ground-based
telescopes would greatly benefit the statistics on primitive galaxies,
especially if could account for primitive galaxies not showing evidence of
hard radiation as well as ones that do. There are also theoretical issues to
be cleared up, mainly the evolution of massive stars in binary systems, since
most hot, massive stars are in binary systems (Sana et al., 2012). Other
factors such as the rotation and the C/O ratio should be explored.

We conclude that massive, extremely low metallicity galaxies are extensions
from those of low metallicity, e.g. spectral lines are generally weaker, and
stellar winds are weaker. However, primitive galaxies are qualitatively
different from more metal-rich galaxies in showing evidence of hard radiation
that can best be explained by the presence of one or more stellar black holes.

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