Elliptic waves in two-component long-wave–short-wave resonance interaction system in one and two dimensions

• Jacobi elliptic solutions for (1+11+1)D and (2+12+1)D LSRI system in terms of Lamé polynomials order 1 and 2.
• Three types of Jacobi elliptic solutions: similar, mixed, superposed solutions.
• Identified special in-phase and out-of-phase periodic wave train solutions.
• Demonstrated the appearance of various types of solitons in the hyperbolic limit.
• Identified special solitary waves: bi-soliton and anti-dark soliton wave at the order 2.

We consider (2+1)(2+1)- and (1+1)(1+1)-dimensional long-wave–short-wave resonance interaction systems. We construct an extensive set of exact periodic solutions of these systems in terms of Lamé polynomials of order one and two. The periodic solutions are classified into three categories as similar, mixed, superposed elliptic solutions. We also discuss the hyperbolic solutions as limiting cases.