Solitary Waves For Some Nonlinear Schrödinger Systems

In this paper we study the existence of radially symmetric positive solutions in Hrad 1 (RN) × Hrad 1 (RN) of the elliptic system:- Δ u + u - (α u2 + β v2) u = 0,- Δ v + ω2 v - (β u2 + γ v2) v = 0,N = 1, 2, 3, where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form (φ{symbol}, 0) and (0, ψ) where φ{symbol} and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters α, β, γ, ω has been studied recently by several authors [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys. 255 (2005) 629-653; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys., Erratum, in press; L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, preprint; B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in RN, preprint; J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Physica D 108 (1997) 92-112]. For N = 2, 3, perhaps the most general existence result has been proved in [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458] under conditions which are equivalent to ours. Motivated by some numerical computations, we return to this problem and, using our approach, we give a more detailed description of the regions of parameters for which existence can be proved. In particular, based also on numerical evidence, we show that the shape of the region of the parameters for which existence of solution can be proved, changes drastically when we pass from dimensions N = 1, 2 to dimension N = 3. Our approach differs from the ones used before. It relies heavily on the spectral theory for linear elliptic operators. Furthermore, we also consider the case N = 1 which has to be treated more extensively due to some lack of compactness for even functions. This case has not been treated before. © 2007 Elsevier Masson SAS. All rights reserved.251149161Ambrosetti, A., Colorado, E., Bound and ground states of coupled nonlinear Schrödinger equations (2006) C. R. Acad. Sci. Paris, Ser. I, 342, pp. 453-458Bonorino, L., Brietzke, E., Lukaszczyk, J.P., Taschetto, C., Properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation (2005) J. Differential Equations, 214, pp. 156-175Lin, T.C., Wei, J., Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3 (2005) Comm. Math. Phys., 255, pp. 629-653Reed, M., Simon, B., (1972) Methods of Modern Mathematical Physics, IV, Analysis of Operators, , Academic Press, New YorkReed, M., Simon, B., (1972) Methods of Modern Mathematical Physics, II, Fourier Analysis, , Academic Press, New YorkYang, J., Classification of the solitary waves in coupled nonlinear Schrödinger equations (1997) Physica D, 108, pp. 92-11

Elliptic systems with nonlinearities of arbitrary growth

In this paper we study the existence of nontrivial solutions for a system of coupled semilinear Poisson equations. The growth of one nonlinearity is polynomial, while no growth restriction is imposed on the second nonliearity. The existence of a strong solution is proved

Non-variational elliptic systems in dimension two : a priori bounds and existence of positive solutions

We establish a priori bounds for positive solutions of semilinear elliptic systems in two dimenions. We obtain results concerning such bounds when the nonlinearities depend exponentially on the variables. Based on these bounds, existence of positive solutions is proved

Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio

Fully dynamic recognition of proper circular-arc graphs

We present a fully dynamic algorithm for the recognition of proper circular-arc (PCA) graphs. The allowed operations on the graph involve the insertion and removal of vertices (together with its incident edges) or edges. Edge operations cost O(log n) time, where n is the number of vertices of the graph, while vertex operations cost O(log n + d) time, where d is the degree of the modified vertex. We also show incremental and decremental algorithms that work in O(1) time per inserted or removed edge. As part of our algorithm, fully dynamic connectivity and co-connectivity algorithms that work in O(log n) time per operation are obtained. Also, an O(\Delta) time algorithm for determining if a PCA representation corresponds to a co-bipartite graph is provided, where \Delta\ is the maximum among the degrees of the vertices. When the graph is co-bipartite, a co-bipartition of each of its co-components is obtained within the same amount of time.Comment: 60 pages, 15 figure

Fungal Planet description sheets: 1042–1111

Novel species of fungi described in this study include those from various countries as follows: Antarctica, Cladosporium arenosum from marine sediment sand. Argentina, Kosmimatamyces alatophylus (incl. Kosmimatamyces gen. nov.) from soil. Australia, Aspergillus banksianus, Aspergillus kumbius, Aspergillus luteorubrus, Aspergillus malvicolor and Aspergillus nanangensis from soil, Erysiphe medicaginis from leaves of Medicago polymorpha, Hymenotorrendiella communis on leaf litter of Eucalyptus bicostata, Lactifluus albopicri and Lactifluus austropiperatus on soil, Macalpinomyces collinsiae on Eriachne benthamii, Marasmius vagus on soil, Microdochium dawsoniorum from leaves of Sporobolus natalensis, Neopestalotiopsis nebuloides from leaves of Sporobolus elongatus, Pestalotiopsis etonensis from leaves of Sporobolus jacquemontii, Phytophthora personensis from soil associated with dying Grevillea mccutcheonii. Brazil, Aspergillus oxumiae from soil, Calvatia baixaverdensis on soil, Geastrum calycicoriaceum on leaf litter, Greeneria kielmeyerae on leaf spots of Kielmeyera coriacea. Chile, Phytophthora aysenensis on collar rot and stem of Aristotelia chilensis. Croatia, Mollisia gibbospora on fallen branch of Fagus sylvatica. Czech Republic, Neosetophoma hnaniceana from Buxus sempervirens. Ecuador, Exophiala frigidotolerans from soil. Estonia, Elaphomyces bucholtzii in soil. France, Venturia paralias from leaves of Euphorbia paralias. India, Cortinarius balteatoindicus and Cortinarius ulkhagarhiensis on leaf litter. Indonesia, Hymenotorrendiella indonesiana on Eucalyptus urophylla leaf litter. Italy, Penicillium taurinense from indoor chestnut mill. Malaysia, Hemileucoglossum kelabitense on soil, Satchmopsis pini on dead needles of Pinus tecunumanii. Poland, Lecanicillium praecognitum on insects' frass. Portugal, Neodevriesia aestuarina from saline water. Republic of Korea, Gongronella namwonensis from freshwater. Russia, Candida pellucida from Exomias pellucidus, Heterocephalacria septentrionalis as endophyte from Cladonia rangiferina, Vishniacozyma phoenicis from dates fruit, Volvariella paludosa from swamp. Slovenia, Mallocybe crassivelata on soil. South Africa, Beltraniella podocarpi, Hamatocanthoscypha podocarpi, Coleophoma podocarpi and Nothoseiridium podocarpi (incl. Nothoseiridium gen. nov.)from leaves of Podocarpus latifolius, Gyrothrix encephalarti from leaves of Encephalartos sp., Paraphyton cutaneum from skin of human patient, Phacidiella alsophilae from leaves of Alsophila capensis, and Satchmopsis metrosideri on leaf litter of Metrosideros excelsa. Spain, Cladophialophora cabanerensis from soil, Cortinarius paezii on soil, Cylindrium magnoliae from leaves of Magnolia grandiflora, Trichophoma cylindrospora (incl. Trichophoma gen. nov.) from plant debris, Tuber alcaracense in calcareus soil, Tuber buendiae in calcareus soil. Thailand, Annulohypoxylon spougei on corticated wood, Poaceascoma filiforme from leaves of unknown Poaceae. UK, Dendrostoma luteum on branch lesions of Castanea sativa, Ypsilina buttingtonensis from heartwood of Quercus sp. Ukraine, Myrmecridium phragmiticola from leaves of Phragmites australis. USA, Absidia pararepens from air, Juncomyces californiensis (incl. Juncomyces gen. nov.) from leaves of Juncus effusus, Montagnula cylindrospora from a human skin sample, Muriphila oklahomaensis (incl. Muriphila gen. nov.)on outside wall of alcohol distillery, Neofabraea eucalyptorum from leaves of Eucalyptus macrandra, Diabolocovidia claustri (incl. Diabolocovidia gen. nov.)from leaves of Serenoa repens, Paecilomyces penicilliformis from air, Pseudopezicula betulae from leaves of leaf spots of Populus tremuloides. Vietnam, Diaporthe durionigena on branches of Durio zibethinus and Roridomyces pseudoirritans on rotten wood. Morphological and culture characteristics are supported by DNA barcodes

Fungal Planet description sheets: 1042–1111

Novel species of fungi described in this study include those from various countries as follows: Antarctica, Cladosporium arenosum from marine sediment sand. Argentina, Kosmimatamyces alatophylus (incl. Kosmimatamyces gen. nov.) from soil. Australia, Aspergillus banksianus, Aspergillus kumbius, Aspergillus luteorubrus, Aspergillus malvicolor and Aspergillus nanangensis from soil, Erysiphe medicaginis from leaves of Medicago polymorpha, Hymenotorrendiella communis on leaf litter of Eucalyptus bicostata, Lactifluus albopicri and Lactifluus austropiperatus on soil, Macalpinomyces collinsiae on Eriachne benthamii, Marasmius vagus on soil, Microdochium dawsoniorum from leaves of Sporobolus natalensis, Neopestalotiopsis nebuloides from leaves of Sporobolus elongatus, Pestalotiopsis etonensis from leaves of Sporobolus jacquemontii, Phytophthora personensis from soil associated with dying Grevillea mccutcheonii. Brazil, Aspergillus oxumiae from soil, Calvatia baixaverdensis on soil, Geastrum calycicoriaceum on leaf litter, Greeneria kielmeyerae on leaf spots of Kielmeyera coriacea. Chile, Phytophthora aysenensis on collar rot and stem of Aristotelia chilensis. Croatia, Mollisia gibbospora on fallen branch of Fagus sylvatica. Czech Republic, Neosetophoma hnaniceana from Buxus sempervirens. Ecuador, Exophiala frigidotolerans from soil. Estonia, Elaphomyces bucholtzii in soil. France, Venturia paralias from leaves of Euphorbia paralias. India, Cortinarius balteatoindicus and Cortinarius ulkhagarhiensis on leaf litter. Indonesia, Hymenotorrendiella indonesiana on Eucalyptus urophylla leaf litter. Italy, Penicillium taurinense from indoor chestnut mill. Malaysia, Hemileucoglossum kelabitense on soil, Satchmopsis pini on dead needles of Pinus tecunumanii. Poland, Lecanicillium praecognitum on insects' frass. Portugal, Neodevriesia aestuarina from saline water. Republic of Korea, Gongronella namwonensis from freshwater. Russia, Candida pellucida from Exomias pellucidus, Heterocephalacria septentrionalis as endophyte from Cladonia rangiferina, Vishniacozyma phoenicis from dates fruit, Volvariella paludosa from swamp. Slovenia, Mallocybe crassivelata on soil. South Africa, Beltraniella podocarpi, Hamatocanthoscypha podocarpi, Coleophoma podocarpi and Nothoseiridium podocarpi (incl. Nothoseiridium gen. nov.)from leaves of Podocarpus latifolius, Gyrothrix encephalarti from leaves of Encephalartos sp., Paraphyton cutaneum from skin of human patient, Phacidiella alsophilae from leaves of Alsophila capensis, and Satchmopsis metrosideri on leaf litter of Metrosideros excelsa. Spain, Cladophialophora cabanerensis from soil, Cortinarius paezii on soil, Cylindrium magnoliae from leaves of Magnolia grandiflora, Trichophoma cylindrospora (incl. Trichophoma gen. nov.) from plant debris, Tuber alcaracense in calcareus soil, Tuber buendiae in calcareus soil. Thailand, Annulohypoxylon spougei on corticated wood, Poaceascoma filiforme from leaves of unknown Poaceae. UK, Dendrostoma luteum on branch lesions of Castanea sativa, Ypsilina buttingtonensis from heartwood of Quercus sp. Ukraine, Myrmecridium phragmiticola from leaves of Phragmites australis. USA, Absidia pararepens from air, Juncomyces californiensis (incl. Juncomyces gen. nov.) from leaves of Juncus effusus, Montagnula cylindrospora from a human skin sample, Muriphila oklahomaensis (incl. Muriphila gen. nov.)on outside wall of alcohol distillery, Neofabraea eucalyptorum from leaves of Eucalyptus macrandra, Diabolocovidia claustri (incl. Diabolocovidia gen. nov.)from leaves of Serenoa repens, Paecilomyces penicilliformis from air, Pseudopezicula betulae from leaves of leaf spots of Populus tremuloides. Vietnam, Diaporthe durionigena on branches of Durio zibethinus and Roridomyces pseudoirritans on rotten wood. Morphological and culture characteristics are supported by DNA barcodes