scholarly journals Ill-posedness for a system of quadratic nonlinear Schrödinger equations in two dimensions

2016 ◽  
Vol 271 (1) ◽  
pp. 136-163
Author(s):  
Tsukasa Iwabuchi ◽  
Takayoshi Ogawa ◽  
Kota Uriya
Keyword(s):  
Two Dimensions ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

scholarly journals Infinitely many positive solutions of nonlinear Schrödinger equations

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Positive Solutions ◽  
Radial Symmetry ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u \in H^1({\mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_\infty |$$ | a ( x ) - a ∞ | is uniformly small in $${\mathbb {R}}^N$$ R N , etc. ....

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

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