Instability of solitary waves for nonlinear Schr\"odinger equations of derivative type
speaker: Masahito Ohta (Tokyo University of Science, Japan)abstract: We consider a nonlinear Schr¥"odinger equation of derivative type: ¥i¥partial_tu+¥partial_x^2u+i where ¥(b¥ge 0¥). It has a two parameter family of solitary wave solutions ¥u_{¥omega}(t,x)=e^{i¥omega_0 t}¥phi_{¥omega}(x-¥omega_1 t),¥ where ¥(¥omega=(¥omega0,¥omega1) ¥in ¥Omega:=¥{(¥omega0,¥omega1)¥in R2: ¥omega12<4 ¥omega0¥}¥), ¥¥phi_{¥omega}(x)=¥tilde{¥phi}_{¥omega}(x) ¥exp ¥left( i¥frac{¥omega_1}{2} x-¥frac{i}{4} ¥int_{-¥infty}^{x} ¥¥tilde{¥phi}_{¥omega}(x)=¥left¥{¥frac{4¥omega_0-¥omega_1^2}{-¥frac{¥omega_1}{2} +¥sqrt{¥omega_0+¥frac{4}{3}b(4¥omega_0-¥omega_1^2)} ¥, ¥cosh (¥sqrt{4¥omega_0-¥omega_1^2}¥,x)}¥right¥}^{1/2}.¥ The orbital stability of solitary waves ¥(u{¥omega}(t)¥) has been studied by Guo and Wu (1995) and Colin and Ohta (2006) for the case ¥(b=0¥). In this talk, we consider the case ¥(b>0¥), and prove the orbital instability of ¥(u{¥omega}(t)¥) for some ¥(¥omega¥).
timetable:
Wed 28 May, 11:15 - 12:00, Aula Dini
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