Requirements

Install alpine-2.1.3 or newer (see these notes if you are unsure how to do it).

Decide which serial port to use and plug you modem there.
In below examples replace ttyS0 with the ttyS? that applies for your needs.
(/dev/ttyS0 is same as COM1, ttyS1=COM2, ttyS2=COM3, ttyS3=COM4, ...)

Install packages

Install required packages

apk add ppp

Load modules

Load some needed modules and make sure they get automatically loaded at next reboot

modprobe ppp_generic echo "ppp_generic" >> /etc/modules-load.d/ppp.conf

Configfiles

/etc/ppp/peers/serialmodem

(The filename 'serialmodem' can be changed to whatever is appropriate for you, but you will need to remember it when running pon/poff command)

debug
/dev/ttyS0
115200
modem
crtscts
asyncmap 0
defaultroute
lock
noauth
user '{login_id}'
connect '/usr/sbin/chat -v -f /etc/ppp/chat-serialmodem'

/etc/ppp/chat-serialmodem

(The filename 'chat-serialmodem' can be changed to whatever is appropriate for you, but you will need modify above configfile to reflect your decition)

ABORT 'BUSY'
ABORT 'ERROR'
ABORT 'NO ANSWER'
ABORT 'NO CARRIER'
ABORT 'NO DIALTONE'
ABORT 'Invalid Login'
ABORT 'Login incorrect'
REPORT 'CONNECT'
TIMEOUT '60'
'' 'ATZ'
OK 'ATDT{phonenumber}'
CONNECT '\d\c'
Username: '{login_id}'
Password: '{your_password}'

/etc/ppp/pap-secrets

When you look at the logs and see pppd report something like this:

daemon.debug pppd[5665]: rcvd [LCP ConfReq id=0xf6 <asyncmap 0xa0000> <auth pap> <magic 0xa239b2b1> <pcomp> <accomp>]

(Note the "<auth pap>" section)
Then you might need to use pap-secrets file (or chap-secrets depending on what pppd reports in the logs).
You file might in this case look something like this

# client     server  secret            IP addresses
{login_id}   *       {your_password}   *

If you are using 'pap-secrets' (or 'chat-secrets') you should most likely comment out 'Username:' and 'Password:' lines in your '/etc/ppp/chat-serialmodem' config.

Note for above example configs

Note: Replace above words {login_id}, {your_password} and {phonenumber} with what you received from your ISP. The characters { and } should also be removed.

Note: You might need to replace the 'Username:' and 'Password:' parts to reflect the words your ISP is using to ask you to enter your credentials

Tip: You might need to replace "CONNECT '\d\c'" with "CONNECT 'CLIENT'" in your chat-serialmodem config

References:

Start/Stop

Start connection

pon serialmodem

Stop connection

poff serialmodem

If something goes wrong...

Check if process is running

pidof pppd

Logfile might give you a clue on what went wrong

egrep "pppd|chat" /var/log/messages

Check nic information

ifconfig ppp0

pppd has a statusinformation function that could come in handy

pppstats