Rogue periodic waves of the modified KdV equation

Abstract

Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg-de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine's breather) of the nonlinear Schrödinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.

Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

“…We use a general method of nonlinearization of Lax equations (originally proposed in [13] and developed for the NLS equation in [37,38]) and apply it to the case of two eigenvalues compared to the case of one eigenvalue considered in [15,17]. Although the exact solutions for rogue waves in the focusing NLS equation are more complicated than the corresponding solutions for the modified Korteweg-de Vries equation (which have been analyzed in [14,16]), efficient computational methods are developed to visualize the Lax spectrum associated with the double-periodic solutions, the admissible eigenvalues, and rogue waves on the double-periodic background. We also study magnification of such rogue waves that depends on parameters of the double-periodic background.…”

Darboux transformation and soliton solutions of the semi-discrete massive Thirring model

“…It is however difficult to compute relations between the new and old potentials from these four equations. Therefore, we will obtain the relations between (R n , Q n ) and (R [1] n , Q [1] n ) by using dressing methods from Appendix A in [5]. Expanding Eq.…”

“…In order to determine a n (t), b n (t), c n (t), and d n (t), we use the symmetry properties of the Lax pair (5). This allows us to find simultaneously both the coefficients of T (λ) and the transformations between the potentials U, Q, R and U [1] , Q [1] , R [1] .…”

“…Because the passage from the discrete system (1) to the continuum limit (3) involves the change of the coordinates U (x, t) = u(x, t − x) and Q(x, t) = v(x, t − x), the initial-value problem for the semi-discrete MTM system (1) does not represent the initial-value problem for the continuous MTM system (3) in time variable t. In addition, numerical explorations of the semi-discrete system (1) are challenging because the last two constraints in the system (1) may lead to appearance of bounded but non-decaying sequences {R n } n∈Z and {Q n } n∈Z in response to the bounded and decaying sequence {U n } n∈Z . On the other hand, since the semi-discrete MTM system (1) has the Lax pair of linear equations (5), it is integrable by the inverse scattering transform method which implies existence of infinitely many conserved quantities, exact solutions, transformations between different solutions, and reductions to other integrable equations [10]. These properties of integrable systems were not explored for the semi-discrete MTM system (1) in the previous work [11].…”

“…Theorem 1. Let Φ n (λ 1 ) = (f n , g n ) T be a nonzero solution of the Lax pair (5) with λ = λ 1 and (U n , R n , Q n ) be a solution of the semi-discrete MTM system (1). Another solution of the semi-discrete MTM system (1) is given by…”