散焦型非线性薛定谔方程的Whitham调制理论及其间断初值问题解的分类和演化  

Whitham modulation theory of defocusing nonlinear Schrödinger equation and the classification and evolutions of solutions with initial discontinuity

作　　者：公睿智 王灯山 Gong Rui-Zhi;Wang Deng-Shan(Laboratory of Mathematics and Complex Systems,Ministry of Education,School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China)

机构地区：[1]北京师范大学数学科学学院,数学与复杂系统教育部重点实验室,北京100875

出　　处：《物理学报》2023年第10期148-166,共19页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11971067)资助的课题。

摘　　要：Whitham调制理论自1965年被首次提出后,由于其在研究色散流体动力学和处理间断初值问题上的独特优势得到了人们的广泛关注.本文发展了散焦型非线性薛定谔方程的Whitham调制理论,研究它的间断初值问题解的分类和演化,并利用直接数值模拟验证结果的正确性.具体地,推导出稀疏波和色散冲击波解及其相应的Whitham方程,详细讨论了每种分类中黎曼不变量和色散流体的密度分布.最后,分析了色散流体的活塞问题,发现了新奇的波状涌潮结构.Since the Whitham modulation theory was first proposed in 1965,it has been widely concerned because of its superiority in studying dispersive fluid dynamics and dealing with discontinuous initial value problems.In this paper,the Whitham modulation theory of the defocusing nonlinear Schrödinger equation is developed,and the classification and evolution of the solutions of discontinuous initial value problem are studied.Moreover,the dispersive shock wave region,the rarefaction wave region,the unmodulated wave region and the plateau region are distinguished.Particularly,the correctness of the results is verified by direct numerical simulation.Specifically,the solutions of 0-phase and 1-phase and their corresponding Whitham equations are derived by the finite gap integration method.Also the Whitham equation of genus N corresponding to the N-phase periodic wave solution is derived.The basic structures of rarefaction wave and dispersive shock wave are given,in which the boundaries of the regions are calculated in detail.The Riemann invariants and density distributions of dispersive fluids in each case are discussed.When the initial value is fixed as a special one,the vacuum point is considered and analyzed in detail.In addition,the oscillating front and the soliton front in the dispersive shock wave are considered.In fact,the Whitham modulation theory has many wonderful applications in real physics and engineering.The dam problem is investigated as a special Riemann problem,the piston problem of dispersive fluid is analyzed,and the novel undular bores are found.

关 键 词：Whitham调制理论 LAX对 黎曼问题 稀疏波 色散冲击波 

分 类 号：O175.29[理学—数学]