In a surprise visit to Kherson, President Zelensky vowed Monday to sweep Russian soldiers from the rest of occupied Ukraine. “You see our strong army,” he told reporters gathered in Freedom Square in the center of the newly liberated provincial capital. “We are step-by-step coming through our country, through the temporarily occupied territories.” 

Ukrainian morale is high and cross-river artillery duels have already started. But a cross-river amphibious assault promises to be costly. In 1943, 1.5 million Soviet soldiers were killed or wounded crossing the Dnipro River to recapture western Ukraine from Nazi rule. Last week, Russian soldiers fled east across this river, blowing up bridges behind them. Now Ukrainian soldiers face the challenge of crossing the Dnipro, a barrier often called Ukraine’s Mississippi.

As winter descends on southern Ukraine, though, both sides are increasingly exposed. Tree leaves, the age-old cover for soldiers, have fallen. Modern drone-mounted thermal imaging equipment now discloses the locations of power generators and soldiers in trenches. Bracing for a cross-river assault, Russia’s army is showing reporters its frontline trenches protected by concrete obstacles called “dragons’ teeth.” 

This may all be a matter of public relations. The river front line is a long one: 150 miles from the Dnipro delta to Enerhodar, home to the Russia-controlled Zaporizhzhia nuclear power plant.  Historically, the track record for such fortifications is spotty. In the 1930s, the French spent a decade and $9 billion in today’s dollars to build the 280-mile Maginot Line. When Germany attacked France in June 1940, they simply drove their tanks around the line.

After de-mining Kherson  and consolidating their forces, Ukrainian military leaders are expected to start probing Russian defenses, looking for weak spots. Soldiers can cross on motorboats or pontoon bridges. The challenge, though, is the same one that Soviet military planners faced nearly 80 years ago: getting heavy equipment — tanks, armored personnel, and their munitions — across the river.

To date, Ukraine’s battlefield success is largely due to two factors. One is the Ukrainians’ iron will to recover their country. The second is military aid from America and other members of the North Atlantic Treaty. Without the aid, Ukrainians probably would have been forced to resort to waging the kind of deadly partisan warfare they fought in western Ukraine between 1945 and 1954.

Biden administration officials hint that winter is a good time to talk. Both Washington and Kyiv agree on the goal of expelling Russia from Ukrainian territories taken after President Putin ordered the invasion in February. But after returning to the pre-war lines, the two governments diverge. President Zelensky vows to go further, and expel Russia from the two areas it occupied in 2014 — Crimea and Donetsk.       

Washington does not recognize these occupations, but prefers a negotiated settlement. Ukraine could strengthen its hand by retaking the Kherson region, key to controlling land routes to Crimea. Loss of the eastern half of the region would deprive Mr. Putin of a cherished goal — a 250-mile, east-west land bridge connecting Russia with the Crimean peninsula. Loss of Kherson would sever a canal bringing water to the semi-arid peninsula.

Cut off from water and highway access and within firing range of Ukraine’s highly accurate, American-supplied artillery, Crimeans might start thinking about a diplomatic solution — such as, say, an autonomous, free-port status under nominal Ukrainian sovereignty. Crimea may be dear to Ukraine, but the fact is that many of its 2 million residents do not want to be liberated by Ukraine. In  2014, 97 percent of voters in a sham referendum opted to join Russia.

Before that vote, polls showed that about two-thirds of Crimeans wanted to join Russia. Since the vote, the population has been brainwashed with Russian propaganda. Crimean Tatars, the historic ethnic group on the peninsula, are opposed to Russian rule. Yet they account for only 10 percent of the population. Meantime, Mr. Putin, during eight years of occupation, turned Crimea into an armed camp.

The main point of entry from the mainland is an easily defendable isthmus only five miles wide. In a stronger argument against a military attack, the Russian people love Crimea as much as Ukrainians do. So it would be shortsighted to suggest that the war is over. This spring, a Ukrainian attack on Crimea could wake the Russian bear from hibernation, and once again the Mississippi of Ukraine could run red with blood.

So the best solution could prove to be to roll up to the gate of the bear den, and then apply maximum pressure for a non-military outcome. In what could be a first step, the CIA director, William Burns, met on Monday in Turkey with his Russian counterpart, Sergei Naryshkin, the foreign intelligence chief. It was the first high-level, face-to-face meeting between Russian and American officials since Mr. Putin attacked Ukraine nine months ago.