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The aim of this paper was to propose a systematic study of a ( 1 + 1 ) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis and here present exact solutions of the equation.
The aim of this paper was to propose a systematic study of a ( 1 + 1 ) -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a Lie symmetry analysis and here present exact solutions of the equation.
Let L = − △ + V L=-\bigtriangleup +V be the Schrödinger operator on R n {{\mathbb{R}}}^{n} , where V ≠ 0 V\ne 0 is a non-negative function satisfying the reverse Hölder class R H q 1 R{H}_{{q}_{1}} for some q 1 > n ⁄ 2 {q}_{1}\gt n/2 . △ \bigtriangleup is the Laplacian on R n {{\mathbb{R}}}^{n} . Assume that b b is a member of the Campanato space Λ ν θ ( ρ ) {\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with L L is ℐ β L {{\mathcal{ {\mathcal I} }}}_{\beta }^{L} . We study the boundedness of the commutators [ b , ℐ β L ] \left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b ∈ Λ ν θ ( ρ ) b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces M p → , φ α , V {M}_{\overrightarrow{p},\varphi }^{\alpha ,V} , vanishing generalized mixed Morrey spaces V M p → , φ α , V V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V} , and L M p → , φ α , V , { x 0 } L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [ b , ℐ β L ] \left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when b b belongs to Λ ν θ ( ρ ) {\Lambda }_{\nu }^{\theta }\left(\rho ) with θ > 0 \theta \gt 0 , 0 < ν < 1 0\lt \nu \lt 1 , and ( φ 1 , φ 2 ) \left({\varphi }_{1},{\varphi }_{2}) satisfying certain requirements are bounded from L M p → , φ 1 α , V , { x 0 } L{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V,\left\{{x}_{0}\right\}} to L M q → , φ 2 α , V , { x 0 } L{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V,\left\{{x}_{0}\right\}} ; from M p → , φ 1 α , V {M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to M q → , φ 2 α , V {M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V} , and from
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