The Benjamin-Ono equation describes internal waves in deep stratified fluids. Its N-soliton solutions are explicit: N Lorentzian peaks that interact elastically, emerging from collisions with only phase shifts. As N grows, the solutions become more complex but remain exact.
The authors construct infinite-order multisoliton solutions — solutions with countably many soliton peaks, arranged so the total energy is finite. The construction passes from the finite N-soliton formulas to N = infinity through a careful limiting procedure that controls the convergence of the infinite product representations.
The infinite soliton is not a mathematical curiosity. It appears naturally in the soliton resolution conjecture: the claim that generic solutions to integrable dispersive equations decompose asymptotically into a sum of solitons plus radiation. For the Benjamin-Ono equation, the soliton resolution must accommodate the possibility that the soliton content is infinite — that the solution decomposes into infinitely many solitons as time goes to infinity.
The construction verifies this: the infinite-order multisoliton solutions are legitimate solutions to the Benjamin-Ono equation, and they represent the soliton content of initial data whose spectral decomposition produces infinitely many discrete eigenvalues. Each eigenvalue generates one soliton, and the solution is the nonlinear superposition of all of them.
Finite energy, infinite solitons. The solution is a countable superposition that converges because each successive soliton is smaller, arranged so the total energy sums. The integrable structure guarantees exact superposition — no interaction energy, no binding.