Stability and Interaction of Compactons in the Sublinear KdV Equation

The paper studies the sublinear defocusing KdV equation u_t − α|u|^{α−1}u_x + u_{xxx} = 0 for α ∈ (0,1), proves the compacton profile U(x) = ±a1 sin^{2/(1−α)}(x) on [0,π] with c1 = 4/(1−α)^2 and scaling a(c), λ(c) (equations (5),(8),(9)), and shows U ∈ C^3 only when α>1/3 (equation (10)) . It then defines the linearized operator L1 = −∂_x^2 − 4/(1−α)^2 + 2α(1+α)/((1−α)^2 sin^2 x), proves essential self-adjointness and discreteness for α>1/5, and computes the simple spectrum µ1<µ2=0<µ3<⋯ with U′ as the zero eigenfunction (equations (16)–(21) and (19)) . Finally, it establishes the VK slope dP/dc<0 via scaling and concludes energetic stability as a non-degenerate minimizer of E at fixed P in the symmetric, compact, same-support class (equation (24) and the ensuing discussion) . The candidate solution reproduces these points: the compacton ansatz, C^3 threshold α>1/3, the exact operator L1 with the 1/sin^2 potential and parameter A=αp(p−1), the location of the simple eigenvalues (µ1<0, µ2=0), and the VK slope from the c-scaling; it also restricts stability to symmetric, compact perturbations supported on the same interval, precisely as in the paper. Differences are minor and mostly stylistic: the candidate uses a Hardy-inequality/closed-form approach to selfadjointness instead of Weyl/Frobenius, and there is a small sign slip in the differentiated travelling-wave ODE (the correct sign comes from differentiating (5) and matches the PDE (4)). Overall, both arguments align on substance and conclusions.