Update the search component to send event for updating the query.

lib/decentralised_book_index_web/live/author_live/index.ex

@@ -186,4 +186,13 @@ defmodule DecentralisedBookIndexWeb.AuthorLive.Index do
 
     {:noreply, push_patch(socket, to: ~p"/authors?#{params}")}
   end
+
+  @impl true
+  def handle_event("search", %{"query" => query}, socket) do
+    params =
+      socket.assigns.params
+      |> Map.put("query", query)
+
+    {:noreply, push_patch(socket, to: ~p"/authors?#{params}")}
+  end
 end

lib/decentralised_book_index_web/live/book_live/index.ex

@@ -250,4 +250,14 @@ defmodule DecentralisedBookIndexWeb.BookLive.Index do
         )
     end
   end
+
+  @impl true
+  def handle_event("search", %{"query" => query}, socket) do
+    params =
+      socket.assigns.params
+      |> Map.put("query", query)
+
+    {:noreply, push_patch(socket, to: ~p"/books?#{params}")}
+  end
+
 end