Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

Abstract

In this paper, we utilize generalized Darboux transformation to study higher-order rogue wave solutions of the three-wave resonant interaction equation, which describes the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media. A general Nthorder rogue wave solution with two characteristic velocities structural parameters and 3N independent parameters under a determined plane-wave background and a specific parameter condition is derived. As an application, we show that four fundamental rogue waves with fundamental, two kinds of line and quadrilateral patterns, or six fundamental rogue waves with fundamental, triangular, two kinds of quadrilateral and circular patterns can emerge in the secondorder rogue waves. Moreover, several important wave characteristics including the maximum values, the corresponding coordinate positions of the humps, and the stability problem for some special higher-order rogue wave solutions such as the fundamental and quadrilateral cases are discussed.

Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

“…Another popular method to derive exact solutions to soliton equation theoretically is the Darboux transformation [53][54][55][56][57][58][59][60], but we have demonstrated that the bilinear method is a feasible scheme too. Examples of these solutions include solitons, breathers, rogue waves, and many other types of rational solutions.…”

Bright and dark soliton solutions to the partial reverse space–time nonlocal Mel’nikov equation

“…Besides, another popular method to derive a variety of solutions to soliton equation theoretically is the Darboux transformation [52,53,54,55,56,57,58,59], but these obtained solutions have demonstrated that the bilinear method is a feasible scheme in computing different types solutions. Examples of these solutions include solitons, breathers, rogue waves, and many other types of rational solutions.…”

A Neumann System of the Third Order Differential Operator Associated with the Boussinesq Equation

“…A Neumann system of the Boussinesq equation associated with the third order differential operator is obtained in this paper, which is the extension of the famous KdV Neumann system associated with the second order differential operator. There are many methods to deal with the integrability and involutivity [12] [13] [14]. A generating function method starting from the Lax-Moser matrix [15]- [20] is used to give an effective way to prove the involutivity of integrals.…”