Orbital stability of solitary waves for Kundu equation
The Kundu equation \[ i u_t + u_{xx} + c_1 |u|^2 u + c_2 |u|^4 u + c_3 i (|u|^2 u)_x + c_4 i u (|u|^2)_x = 0 \] includes several nonlinear evolution equations integrable by inverse scattering such the derivative nonlinear Schrödinger equation, the Chen--Lee--Lin equation, and the Gerdjikov--Ivanov equation. Solitary waves in all these equations can be obtained in the explicit form by integrating a system of ordinary differential equations. The authors study orbital stability of solitary waves in the Kundu equation. Since the Kundu equation can not be formulated in a standard symplectic Hamiltonian form, it is difficult to apply the Grillakis--Shatah--Strauss theory on orbital stability of solitary waves in Hamiltonian systems. The authors overcome this difficulty with the spectral stability analysis and the construction of three invariants of motion. As a result, they derive sufficient conditions for orbital stability of solitary waves, which generalize the previously found conditions of \textit{B. Guo} and \textit{Y. Wu} [J. Differ. Equations 123, No. 1, 35--55 (1995; Zbl 0844.35116)] and \textit{M. Colin} and \textit{M. Ohta} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 5, 753--764 (2006; Zbl 1104.35050)].
derivative nonlinear Schrödinger equation
scientific article; zbMATH DE number 5593950
scientific article