ON NON-UNIQUENESS OF SOLITARY WAVES ON TWO-DIMENSIONAL ROTATIONAL FLOW

The paper proves non-uniqueness near the first bifurcation point on the solitary-wave branch by an abstract bifurcation analysis with the physical branch parameter t, covering both possibilities: (a) a local turning point of R(t) or (b) a monotone R(t) with secondary bifurcating curves. This is encapsulated in Theorems 1.9 and 1.10 and the surrounding discussion, which explicitly note that secondary bifurcation is guaranteed when the point is not a turning point and also describe the turning-point alternative . By contrast, the candidate solution asserts unconditionally that a fold in R occurs and that R has a strict local extremum at the degeneracy, giving exactly two solutions on the branch for each nearby R. That claim depends on (i) misidentifying the eigenvalue crossing zero as the smallest/principal one and (ii) incorrectly invoking positivity of the kernel eigenfunction at the fold. In the paper’s notation, the lowest eigenvalue µ0(t) is negative and simple, while the first degeneracy involves µ1(t) crossing 0 (the second eigenvalue on (−∞,ν0)), hence its eigenfunction need not be positive (Proposition 1.6 and the setup around (1.24)–(1.25)) . The candidate’s derivation of ∂RG≠0 from positivity therefore fails. The paper deliberately avoids that assumption and proves non-uniqueness in either case (turning or secondary branches), aligning with the abstract framework and compactness properties developed in Sections 2–4 and formalized in Theorems 1.9–1.10 . Hence the paper’s argument is correct for its claims; the model’s fold claim and positivity-based step are not justified.