We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schr"odinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.
{"title":"Long-time Asymptotics for the Integrable Discrete Nonlinear Schrödinger Equation: the Focusing Case","authors":"H. Yamane","doi":"10.1619/FESI.62.227","DOIUrl":"https://doi.org/10.1619/FESI.62.227","url":null,"abstract":"We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schr\"odinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2015-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.62.227","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67431745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.
{"title":"Confluence of Singularities in Hypergeometric Systems","authors":"Martin Klimeš","doi":"10.1619/fesi.63.153","DOIUrl":"https://doi.org/10.1619/fesi.63.153","url":null,"abstract":"A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2015-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67432136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X))partial derivative(x) + (z + b(X))partial derivative(y) + c(X)partial derivative(z), where X = (x, y, z) is an element of C-3. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.
研究了原点为幂零线性部分的全纯奇异向量场的约简问题，P = (y + a(X))偏导数(X) + (z + b(X))偏导数(y) + c(X)偏导数(z)，其中X = (X, y, z)是c -3中的一个元素。通过引入a . Shirai在上一篇论文[M-S]中给出的拟估值的概念，我们刻画了一类可以通过坐标的形式变换简化成更简单的类，这类类在定理a和定理B中允许有各种变化。
{"title":"Reduced Forms of a Singular Vector Field in C 3 with Nilpotent Linear Part","authors":"M. Miyake","doi":"10.1619/FESI.58.253","DOIUrl":"https://doi.org/10.1619/FESI.58.253","url":null,"abstract":"We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X))partial derivative(x) + (z + b(X))partial derivative(y) + c(X)partial derivative(z), where X = (x, y, z) is an element of C-3. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2015-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78059217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: rho u(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to(2))(gamma)Delta u + u(t) = 0, rho v(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to 2)(gamma)Delta v + v(t) = 0, with homogeneous Dirichlet boundary condition and rho > 0 and gamma > 0. When either the coefficient rho or the initial data are appropriately small, we prove the global existence theorem by using several identities and the energy decay. Moreover, under the same assumption for rho and the initial data, we derive the decay estimates of the solutions and their second order derivatives.
{"title":"Global Existence and Decay Properties of Solutions for Coupled Degenerate Dissipative Hyperbolic Systems of Kirchhoff Type","authors":"K. Ono","doi":"10.1619/FESI.57.319","DOIUrl":"https://doi.org/10.1619/FESI.57.319","url":null,"abstract":"Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: rho u(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to(2))(gamma)Delta u + u(t) = 0, rho v(tt) - (parallel to del u(t)parallel to(2) + parallel to del v(t)parallel to 2)(gamma)Delta v + v(t) = 0, with homogeneous Dirichlet boundary condition and rho > 0 and gamma > 0. When either the coefficient rho or the initial data are appropriately small, we prove the global existence theorem by using several identities and the energy decay. Moreover, under the same assumption for rho and the initial data, we derive the decay estimates of the solutions and their second order derivatives.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91064088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we construct solutions e it D j of the Schro ¨ dinger equation on R N which have nontrivial asymptotic properties simultaneously on di¤erent time and space scales. More precisely, given m A ð 0 ; N Þ and b b 1 = 2 we consider the set o bm ; r ð j Þ of limit points in L r ð R N Þ as t ! y of t m = 2 ½ e it D j (cid:1)ð(cid:2) t b Þ . We show in particular that, given 0 < n < N and an arbitrary countable set S H ð n ; N Þ , there exists an initial value f such that o bm ; r ð f Þ ¼ L r ð R N Þ for all m A ð 0 ; N Þ and b b 1 = 2 such that m = 2 b A S , and all su‰ciently large r . We also establish a result of a similar nature for a nonlinear Schro¨dinger equation.
． 在这篇文章,我们构造it解决方案e D j Schro之¨丁格equation on R N哪有nontrivial asymptotic财产simultaneously on在¤erent时间和空间scales。更多precisely,赐予m Að0;NÞb和b型1 = 2我们认为《套o bm;ðr jÞ限额之分在美国洛杉矶rðr NÞt !y t m = 2½e它的D j (cid: 1)ð(cidÞb: 2) t。我们在社会这一点,给节目0 < n < n和an arbitrary countable套S Hðn;NÞ,有exists价值f如此那名字的首字母o的bm;ðr fÞ¼L r为所有m Aððr NÞ0;NÞ和b b 1 = 2这样那m = 2 b A S,和全苏‰ciently大r。我们还建立非线性a a类似的论点自然为Schro¨丁格equation。
{"title":"Multiscale Asymptotic Behavior of the Schrödinger Equation","authors":"D. Fang, Jian Xie, T. Cazenave","doi":"10.1619/FESI.54.69","DOIUrl":"https://doi.org/10.1619/FESI.54.69","url":null,"abstract":". In this paper, we construct solutions e it D j of the Schro ¨ dinger equation on R N which have nontrivial asymptotic properties simultaneously on di¤erent time and space scales. More precisely, given m A ð 0 ; N Þ and b b 1 = 2 we consider the set o bm ; r ð j Þ of limit points in L r ð R N Þ as t ! y of t m = 2 ½ e it D j (cid:1)ð(cid:2) t b Þ . We show in particular that, given 0 < n < N and an arbitrary countable set S H ð n ; N Þ , there exists an initial value f such that o bm ; r ð f Þ ¼ L r ð R N Þ for all m A ð 0 ; N Þ and b b 1 = 2 such that m = 2 b A S , and all su‰ciently large r . We also establish a result of a similar nature for a nonlinear Schro¨dinger equation.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79645782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.
{"title":"Resurgent Analysis of the Witten Laplacian in One Dimension","authors":"A. Getmanenko","doi":"10.1619/FESI.54.383","DOIUrl":"https://doi.org/10.1619/FESI.54.383","url":null,"abstract":"The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85370719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on Hs(T), s > s0 where s0 = –11/6 + $sqrt{61}$/6 ≈ –0.5316 < –1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the Duhamel formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szego equation under randomization of initial data.
研究了一类具有随机初始数据的色散偏微分方程的柯西问题。特别地，我们通过在Duhamel公式的第二次迭代上表现出随机化下的非线性平滑，对初始数据在支持s > 50 (s = -11/6 + $sqrt{61}$/6≈-0.5316 < -1/2)的平均零高斯测度的情况下，几乎肯定地构造了平均零周期KdV的局部时解。我们还证明了初始数据随机化时无色散三次Szego方程不存在非线性平滑。
{"title":"Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation","authors":"Tadahiro Oh","doi":"10.1619/FESI.54.335","DOIUrl":"https://doi.org/10.1619/FESI.54.335","url":null,"abstract":"We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on Hs(T), s > s0 where s0 = –11/6 + $sqrt{61}$/6 ≈ –0.5316 < –1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the Duhamel formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szego equation under randomization of initial data.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2010-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1619/FESI.54.335","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72430340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.
{"title":"Parametric Transformations between the Heun and Gauss Hypergeometric Functions","authors":"R. Vidunas, G. Filipuk","doi":"10.1619/fesi.56.271","DOIUrl":"https://doi.org/10.1619/fesi.56.271","url":null,"abstract":"The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2009-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88154331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give some functional equations with a parameter c for Lauricella’s hypergeometric functions; they can be regarded as multivariable versions of the Gauss quadratic transformation formula for the hypergeometric function. These functional equations for c = 1 are utilized for the study of arithmetic-geometric means of several terms.
{"title":"Some transformation formulas for Lauricella's hypergeometric functions FD","authors":"Keiji Matsumoto, Katsuyoshi Ohara","doi":"10.1619/FESI.52.203","DOIUrl":"https://doi.org/10.1619/FESI.52.203","url":null,"abstract":"In this paper, we give some functional equations with a parameter c for Lauricella’s hypergeometric functions; they can be regarded as multivariable versions of the Gauss quadratic transformation formula for the hypergeometric function. These functional equations for c = 1 are utilized for the study of arithmetic-geometric means of several terms.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2009-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80117977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a three-component system of nonlinear Schrodinger equations related to the Raman amplification in a plasma. In dimension N ≤ 3, we study the orbital instability of standing wave solution of the form (0,0,e2iωtψ), where ψ is a ground state of scalar nonlinear Schrodinger equation. Using time derivative instead of space derivatives to estimate nonlinear terms, we improve an instability result in our previous paper [4], and also give a simpler proof.
{"title":"Instability of Standing Waves for a System of Nonlinear Schrodinger Equations with Three-Wave Interaction","authors":"M. Colin, T. Colin, Masahito Ohta","doi":"10.1619/FESI.52.371","DOIUrl":"https://doi.org/10.1619/FESI.52.371","url":null,"abstract":"We consider a three-component system of nonlinear Schrodinger equations related to the Raman amplification in a plasma. In dimension N ≤ 3, we study the orbital instability of standing wave solution of the form (0,0,e2iωtψ), where ψ is a ground state of scalar nonlinear Schrodinger equation. Using time derivative instead of space derivatives to estimate nonlinear terms, we improve an instability result in our previous paper [4], and also give a simpler proof.","PeriodicalId":55134,"journal":{"name":"Funkcialaj Ekvacioj-Serio Internacia","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73194430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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