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Orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equation
Xiaoxiao Zheng,Jie Xin,Xiaoming Peng
Keywords:generalized Long-Short wave equation; periodic traveling waves; orbital stability
Abstract:
      This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equation iεt+ εxx = nε + α|ε|2ε, nt = (|ε|2)x, x ∈ R. Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Long-Short wave equation. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lamé equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period L. In the sense of limit, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equation. In particular, as α = 0, we can also obtain the orbital stability results of periodic wave solution and solitary wave solution for the long-short wave resonance equations. The result in the present paper improve and extends the previous stability results of long-shore wave equations and its extension equations.