Step 1:
Asikni is Chenab River in the northwestern Himalayas.
Step 2:
Narmada lies in central India, while Krishna and Tungabhadra flow further south.
Step 3:
Periyar (Purna) is located in Kerala and is the southernmost among them.
Step 4:
Thus, from North to South:
\[
\boxed{B \rightarrow C \rightarrow D \rightarrow A \rightarrow E}
\]
Hence, the correct option is:
\[
\boxed{\text{(1)}}
\]