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On the exact solution of the geometric optics approximation of the defocusing nonlinear Schrödinger equation

✍ Scribed by Otis C. Wright; M. Gregory Forest; K.T.-R. McLaughlin

Book ID
104337474
Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
63 KB
Volume
257
Category
Article
ISSN
0375-9601

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✦ Synopsis

The implicit solution of the geometric optics equations i.e. the modulation equations arising from the WKB Ansatz of Ž . the defocusing nonlinear Schrodinger NLS equation is known to be expressible in terms of the classical hodograph ẗransform.

In this note, the solution procedure for the 2 = 2 system of quasilinear modulation equations is implemented, analogous to the implicit solution of the inviscid Burgers' equation, for smooth monotone initial data consistent with the modulation Ansatz. The implicit system is solved exactly using a classical method of Riemann. The relevant Riemann-Green functions can be found explicitly, hence allowing the exact location and time of shock formation to be calculated. The entire evolution of the exact solution can be observed through the shock formation.

📜 SIMILAR VOLUMES

Exponentially small asymptotics of solut
Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation
✍ A.H. Vartanian 📂 Article 📅 2003 🏛 Elsevier Science 🌐 English ⚖ 492 KB

The matrix Riemann-Hilbert factorization approach is used to derive the leading-order, exponentially small asymptotics as t --\* +oo such that x/t ~ O(1) of solutions to the Cauchy problem for the defocusing nonlinear Schr6dinger equation, iOtu + 02xu -2([ul 2 -1)u = 0, with finite density initial d