Relativistic self-focusing and channel formation in laser-plasma interactions

Ritchie, Burke

doi:10.1103/physreve.50.r687

Cited by 188 publications

(76 citation statements)

References 15 publications

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“…This implies (47). Hence (48) follows from (32). To prove (49), observing (34) and applying the Cauchy Schwarz inequality, it remains to give a uniform L 2 -bound for {uk G(u, Du, D 2 u).…”

Section: Solution Of the Linearized Equationmentioning

confidence: 93%

“…The case g(s)= (1+s) 1Â2 in (1) models the self-channeling of a high-power ultra short laser in matter (cf. [4], [5], [6], [7], [8], [32]). Equations of the form (1) appear in plasma physics and fluid mechanics [22], [24], [30], in the theory of Heisenberg ferromagnet and magnons [3], [18], [31], [34], in dissipative quantum mechanics [12] and in condensed matter theory [23].…”

Section: Introductionmentioning

confidence: 98%

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On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

Poppenberg¹

2001

Journal of Differential Equations

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A class of quasilinear Schro dinger equations is studied which contain strongly singular nonlinearities including derivatives of second order. Such equations have been derived as models of several physical phenomena. The results of this paper apply to the superfluid film equation in fluid mechanics. The local well posedness for smooth solutions of the Cauchy problem is proved in arbitrary space dimension without any smallness assumption on the initial value. This improves results in the literature which cover the one-dimensional case only. Almost global well posedness results are obtained as well. The proof combines Nash Moser techniques on implicit function theorems in Fre chet spaces with hyperbolic semigroup theory including evolution systems using the intersection of all Sobolev spaces as basic function space. The nondissipativity is overcome by a transformation procedure yielding the evolution operator and a priori estimates of higher order Sobolev norms for the linearized problem. Here a certain trace has to vanish in view of apparent integrability conditions. The transformations and the regularity theory are substantially different from the easier one-dimensional case. In particular, continuity estimates are proved for the operator norms of the linear evolution operator in negative Sobolev spaces. Academic Press

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“…This implies (47). Hence (48) follows from (32). To prove (49), observing (34) and applying the Cauchy Schwarz inequality, it remains to give a uniform L 2 -bound for {uk G(u, Du, D 2 u).…”

Section: Solution Of the Linearized Equationmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 98%

On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

Poppenberg¹

2001

Journal of Differential Equations

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“…Quasilinear equations of form (1) appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of h, the superfluid film equation in plasma physics by Kurihara in [13] (cf. [14]) for h(s) = s. In the case h(s) = (1 + s) 1/2 , (1) models the self-channeling of a high-power ultra short laser in matter; see [4], [6], [8], [23] and the references in [5]. Equation (1) also appears in plasma physics and fluid mechanics [13], [14], [17], [19], [21], in the theory of Heisenberg ferromagnets and magnons [2], [12], [15], [22], [25], in dissipative quantum mechanics [10] and in condensed matter theory [18].…”

Section: Introductionmentioning

confidence: 99%

Soliton solutions for quasilinear Schrödinger equations, I

Liu¹,

Wang²

2002

Proc. Amer. Math. Soc.

357

135

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“…Such equations have been derived as models for several phenomena in plasma physics and fluid mechanics [15,18,24] and in the theory of Heisenberg ferromagnet [1,11,25,30]. The case g s = s models the time evolution of the wave function in superfluid films [12,13], and the case g s = 1 + s 1/2 describes the self-channeling of a high-power ultra short laser in matter [2,3,4,5,6,27].…”

Section: Introductionmentioning

confidence: 99%

An Inverse Function Theorem in Sobolev Spaces and Applications to Quasi-Linear Schrödinger Equations

Poppenberg

2001

Journal of Mathematical Analysis and Applications

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A Nash-Moser type inverse function theorem in Banach spaces with loss of derivatives is proved, and applications are given to singular quasi-linear Schrödinger equations like the superfluid film equation in plasma physics. Based on an implicit function theorem in Sobolev spaces, a linearization method is introduced for the local and global well-posedness of the Cauchy problem for nonlinear evolution equations. The technique compensates for a loss of derivatives in the linearized problem. This is illustrated by an application to strongly singular quasi-linear Schrödinger equations where the nonlinearities include derivatives of second order. The local well-posedness of the Cauchy problem is proved in Sobolev spaces for arbitrary space dimension without assuming smallness assumptions on the initial value. Here the linearized problem is solved using hyperbolic semigroup theory, including evolution systems.

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Relativistic self-focusing and channel formation in laser-plasma interactions

Cited by 188 publications

References 15 publications

On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

On the Local Well Posedness of Quasilinear Schrödinger Equations in Arbitrary Space Dimension

Soliton solutions for quasilinear Schrödinger equations, I

An Inverse Function Theorem in Sobolev Spaces and Applications to Quasi-Linear Schrödinger Equations

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