Periodic Waves in the Fractional Modified Korteweg–de Vries Equation

Periodic waves in the modified Korteweg–de Vries (mKdV) equation are revisited setting of fractional Laplacian. Two families solutions local case given by sign-definite dnoidal and sign-indefinite cnoidal solutions. Both can be characterized general as global minimizers quadratic part energy functional subject to fixed \(L^4\) norm: (sign-indefinite) obtained subspace even (odd) functions. Morse index is computed for both spectral stability criterion derived. We show numerically that family has a generic fold bifurcation Laplacian lower regularity symmetry-breaking cases.