A Riemann-Hilbert approach for a new type coupled nonlinear Schrodinger equations

作者：Guo, Boling; Liu, Nan^*; Wang, Yufeng

来源：Journal of Mathematical Analysis and Applications, 2018, 459(1): 145-158.

DOI：10.1016/j.jmaa.2017.10.064

• Coupled nonlinear Schrodinger equations
• Riemann-Hilbert problem
• Inverse scattering transform
• Solitons

We develop a Riemann-Hilbert approach to the Cauchy problem on the line for a new type of coupled nonlinear SchrOdinger (CNLS) equations
iq(1,t) + q(1,xx) + 2(vertical bar q(1)vertical bar(2) - 2 vertical bar q(2)vertical bar(2))q(1) - 2q(2)(2)q(1)* = 0
iq(2,t) + q(2,xx) + 2(2 vertical bar q(1)vertical bar(2) - vertical bar q(2)vertical bar(2))q(2) + 2q(1)(2)q(2)* = 0
This approach allows us to give a representation of the solution to the Cauchy problem of the CNLS equations in terms of the solution of a 4 x 4 Riemann-Hilbert problem formulated in the complex k-plane. Due to the energy conservation law of above system is integral(+infinity)(-infinity) (vertical bar q(1)vertical bar(2) - vertical bar q(2)vertical bar(2))dx, it is difficult to obtain a solution for this system by using the energy estimate method of PDE's. Therefore, this approach efficiently provides a new way in studying the nonlinear problems that PDE's theory can't solve. Furthermore, this representation is then used for retrieving the soliton solutions.

• 出版日期2018-3-1
• 单位中国工程物理研究院; 北京应用物理与计算数学研究所; 中央民族大学