The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

Zhijie Chen; Ting Jung Kuo; Chang Shou Lin

doi:10.1016/j.matpur.2019.05.004

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

Zhijie Chen, Ting Jung Kuo^*, Chang Shou Lin

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y^″(z)=[∑k=03n_k(n_k+1)℘(z+ [Formula presented] |τ)+B]y(z),n_k∈Z_≥0 from the monodromy aspect. We prove the existence of a pre-modular form Z_r,s ⁿ(τ) of weight [Formula presented] ∑n_k(n_k+1) such that the monodromy data (r,s) is characterized by Z_r,s ⁿ(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n₁=n₂=n₃=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+e^u=16πδ₀andΔu+e^u=8π∑k=13δ_{[Formula presented]} on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

Original language	English
Pages (from-to)	251-272
Number of pages	22
Journal	Journal des Mathematiques Pures et Appliquees
Volume	132
DOIs	https://doi.org/10.1016/j.matpur.2019.05.004
Publication status	Published - 2019 Dec

Keywords

Generalized Lamé equation
Mean field equation
Pre-modular form

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1016/j.matpur.2019.05.004

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title = "The geometry of generalized Lam{\'e} equation, II: Existence of pre-modular forms and application",

abstract = "In this paper, the second in a series, we continue to study the generalized Lam{\'e} equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lam{\'e} case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.",

keywords = "Generalized Lam{\'e} equation, Mean field equation, Pre-modular form",

author = "Zhijie Chen and Kuo, {Ting Jung} and Lin, {Chang Shou}",

note = "Publisher Copyright: {\textcopyright} 2019 Elsevier Masson SAS",

year = "2019",

month = dec,

doi = "10.1016/j.matpur.2019.05.004",

language = "English",

volume = "132",

pages = "251--272",

journal = "Journal des Mathematiques Pures et Appliquees",

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TY - JOUR

T1 - The geometry of generalized Lamé equation, II

T2 - Existence of pre-modular forms and application

AU - Chen, Zhijie

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

PY - 2019/12

Y1 - 2019/12

N2 - In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

AB - In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

KW - Generalized Lamé equation

KW - Mean field equation

KW - Pre-modular form

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The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

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Keywords

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