Nonlinear electromagnetic-spin waves

Soliton solutions for quasilinear Schrödinger equations, I

“…( (2) in [20]. We emphasize here that for α > 1 2 , 2α2 * > 2 * and the condition 4α ≤ p + 1 < 2α2 * can be satisfied for all dimensions N .…”

“…Solutions of (2) will be shown to exist as minimizers of the above minimization problems which are called ground state solutions of (2). Under (V1) or (V2) we have that the embedding from X into L p+1 (R N ) is compact (e.g., [1], [11], [24], [26]).…”

“…Remark 1.4. It would be interesting to know whether solutions exist for all λ > 0 in (2). We shall discuss this in a forthcoming paper.…”

“…[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]. In the mathematical literature very few results are known about equations of the form (1).…”

Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

“…In the case ( ) = (1 + ) 1/2 , problem (2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity, and this leads to interesting new nonlinear wave equations; see [3][4][5][6]. For more physical motivations and more references dealing with applications, we can refer to [7][8][9][10][11][12][13][14] and references therein.…”