Police arrested Paul Alfred Atkinson, 45, on a weapons charge Wednesday after he previously reported his girlfriend Leona Dawn Perryman, of Dorset, England, missing, then disappeared himself.
Atkinson told police Perryman had previously left for days during their trip without telling him where she was going, Santa Monica police Sgt. Jay Trisler said.
When Atkinson last saw her, she said she was going to an Internet cafe or the beach, Trisler said.
Police were optimistic they would find her.
"Our belief is that she's still out there. We are asking for public assistance and hope she will find out and contact her parents or us," Trisler said.
Perryman was last seen shortly before midnight on Dec. 18. She arrived in the U.S. on Oct. 13 with Atkinson.
Atkinson was arrested at a restaurant in Santa Monica after being found with a concealed knife, Trisler said. He remained in custody on the weapons charge, with bail set at $20,000. He is scheduled to appear in court Monday.
The couple, visiting on a 90-day visa, has round-trip plane tickets and are scheduled to return to England on Jan. 7.
Atkinson was the first person to report Perryman missing.