February  2012, 32(2): 353-379. doi: 10.3934/dcds.2012.32.353

Lie's reduction method and differential Galois theory in the complex analytic context

1. 

Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia

2. 

Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain

Received  December 2010 Revised  April 2011 Published  September 2011

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.
Citation: David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353
References:
[1]

C. Athorne and T. Hartl, Solvable structures and hidden symmetries,, J. Phys. A, 27 (1995), 3463.   Google Scholar

[2]

C. Athorne, Symmetries of linear ordinary differential equations,, J. Phys. A, 30 (1997), 4639.  doi: 10.1088/0305-4470/30/13/015.  Google Scholar

[3]

C. Athorne, On the Lie symmetry algebra of general ordinary differential equation,, J. Phys. A, 31 (1998), 6605.  doi: 10.1088/0305-4470/31/31/008.  Google Scholar

[4]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58,, Contemp. Math., 509 (2010).   Google Scholar

[5]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem,, J. Lie Theory, 20 (2010), 483.   Google Scholar

[6]

R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'', Lectures at the R.G.I. in Park City, (1991).   Google Scholar

[7]

A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'', Lecture Notes in Mathematics, 1226 (1986).   Google Scholar

[8]

J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'', Napoli Series on Physics and Astrophysics, (2000).   Google Scholar

[9]

J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237.   Google Scholar

[10]

J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle,, Acta Appl. Math., 66 (2001), 67.  doi: 10.1023/A:1010743114995.  Google Scholar

[11]

J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008).   Google Scholar

[12]

J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory,, Acta Appl. Math., 70 (2002), 43.  doi: 10.1023/A:1013913930134.  Google Scholar

[13]

G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières,, (French) [Singular foliations of codimension one, 56 (2006), 735.   Google Scholar

[14]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation,, Algebraic, analytic and geometric aspects of complex differential equations and their deformations, (2007), 15.   Google Scholar

[15]

G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité,, Comment. Math. Helv., 83 (2008), 471.  doi: 10.4171/CMH/133.  Google Scholar

[16]

G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'', (French) [Lessons on the general theory of surfaces. I, (1914).   Google Scholar

[17]

W. Fulton and J. Harris, "Representation Theory. A First Course,'', Graduate Texts in Mathematics, 129 (1991).   Google Scholar

[18]

A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales,, Compt. Rend. Acad. Sci. Paris, (1893), 964.   Google Scholar

[19]

M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action,, J. Math. Phys., 40 (1999), 3104.  doi: 10.1063/1.532749.  Google Scholar

[20]

J. E. Humphreys, "Linear Algebraic Groups,'', Graduate Texts in Mathematics, (1975).   Google Scholar

[21]

I. Kaplansky, "An Introduction to Differential Algebra,'', Actualités Sci. Ind., (1251).   Google Scholar

[22]

E. R. Kolchin, Galois theory of differential fields,, Amer. J. Math., 75 (1953), 753.  doi: 10.2307/2372550.  Google Scholar

[23]

E. R. Kolchin, "Differential Algebra and Algebraic Groups,'', Pure and Applied Mathematics, 54 (1973).   Google Scholar

[24]

A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures,, J. Dynam. Control Systems, 1 (1995), 91.  doi: 10.1007/BF02254657.  Google Scholar

[25]

S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten,, Math. Ann. Bd., 25 (1885), 71.  doi: 10.1007/BF01446421.  Google Scholar

[26]

S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'', Lepziger Berichte, (1893).   Google Scholar

[27]

S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'', (German) Bearbeitet und herausgegeben von Georg Scheffers, (1893).   Google Scholar

[28]

S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales,, Compt. Rend. Acad. Sci. Paris, (1893), 1233.   Google Scholar

[29]

S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'', Reprinted by Chelsea books, (1967).   Google Scholar

[30]

J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités,, J. Math. Pures Appl., 4 (1839), 423.   Google Scholar

[31]

B. Malgrange, Le groupoïde de Galois d'un feuilletage,, (French) [The Galois groupoid of a foliation], 38 (2001), 465.   Google Scholar

[32]

B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219.  doi: 10.1142/S0252959902000213.  Google Scholar

[33]

B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'', (French) [Lie Pseudogroups and Differential Galois Theory], (2010).   Google Scholar

[34]

P. Malliavin, "Géométrie Différentielle Intrinsèque,'', (French) [Intrinsic Differential Geometry], (1972).   Google Scholar

[35]

M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases,, Phys. Rev. A (3), 52 (1995), 936.  doi: 10.1103/PhysRevA.52.936.  Google Scholar

[36]

J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[37]

J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.   Google Scholar

[38]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'', Progress in Mathematics, 179 (1999).   Google Scholar

[39]

J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845.   Google Scholar

[40]

K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants,, Nagoya Math. J., 113 (1989), 173.   Google Scholar

[41]

K. Nishioka, General solutions depending algebraically on arbitrary constants,, Nagoya Math. J., 113 (1989), 1.   Google Scholar

[42]

K. Nishioka, Lie extensions,, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82.  doi: 10.3792/pjaa.73.82.  Google Scholar

[43]

K. Nomizu, "Lie Groups and Differential Geometry,'', The Mathematical Society of Japan, (1956).   Google Scholar

[44]

J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation,, (French) [Differential Galois theory and resummation], (1990), 117.   Google Scholar

[45]

M. Rosenlicht, A remark on quotient spaces,, An. Acad. Brasil. Ci., 35 (1963), 487.   Google Scholar

[46]

C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory],, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16 (2001).   Google Scholar

[47]

J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.   Google Scholar

[48]

J.-P. Serre, Espaces fibrés algebriques (French),, Séminaire Claude Chevalley, 3 (1958), 1.   Google Scholar

[49]

Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'', Translated from the Japanese by the author, 82 (1990).   Google Scholar

[50]

S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles,, J. Math. Phys., 25 (1984), 3155.  doi: 10.1063/1.526085.  Google Scholar

[51]

M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory,, IEEE Trans. Automat. Control, 30 (1985), 266.  doi: 10.1109/TAC.1985.1103934.  Google Scholar

[52]

H. Umemura, On the irreducibility of the first differential equation of Painlevé,, in, (1988), 771.   Google Scholar

[53]

H. Umemura, Galois theory of algebraic and differential equations,, Nagoya Math. J., 144 (1996), 1.   Google Scholar

[54]

H. Umemura, Differential Galois theory of infinite dimension,, Nagoya Math. J., 144 (1996), 59.   Google Scholar

[55]

H. Umemura, Sur l'équivalence des théories de Galois différentielles générales,, (French) [On the equivalence of general differential Galois theories], 346 (2008), 1155.   Google Scholar

[56]

M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328 (2003).   Google Scholar

[57]

E. Vessiot, Sur l'intégration des equations différentielles linéaires, (French), 9 (1892), 197.   Google Scholar

[58]

E. Vessiot, Sur une classe d'équations différentielles,, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53.   Google Scholar

[59]

E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires,, Compt. Rend. Acad. Sci. Paris, (1893), 1112.   Google Scholar

[60]

E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales,, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894).   Google Scholar

[61]

E. Vessiot, Sur la théorie de Galois et ses diverses généralisations, (French), 21 (1904), 9.   Google Scholar

[62]

E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif, (French), 57 (1940), 1.   Google Scholar

show all references

References:
[1]

C. Athorne and T. Hartl, Solvable structures and hidden symmetries,, J. Phys. A, 27 (1995), 3463.   Google Scholar

[2]

C. Athorne, Symmetries of linear ordinary differential equations,, J. Phys. A, 30 (1997), 4639.  doi: 10.1088/0305-4470/30/13/015.  Google Scholar

[3]

C. Athorne, On the Lie symmetry algebra of general ordinary differential equation,, J. Phys. A, 31 (1998), 6605.  doi: 10.1088/0305-4470/31/31/008.  Google Scholar

[4]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Differential Galois theory of algebraic Lie-Vessiot systems, in "Differential Algebra, Complex Analysis and Orthogonal Polynomials,'' 1-58,, Contemp. Math., 509 (2010).   Google Scholar

[5]

D. Blázquez-Sanz and J. J. Morales-Ruiz, Local and global aspects of Lie superposition theorem,, J. Lie Theory, 20 (2010), 483.   Google Scholar

[6]

R. L. Bryant, "An introduction to Lie Groups and Symplectic Geometry,'', Lectures at the R.G.I. in Park City, (1991).   Google Scholar

[7]

A. Buium, "Differential Function Fields and Moduli of Algebraic Varieties,'', Lecture Notes in Mathematics, 1226 (1986).   Google Scholar

[8]

J. F. Cariñena, J. Grabowski and G. Marmo, "Lie-Scheffers Systems: A Geometric Approach,'', Napoli Series on Physics and Astrophysics, (2000).   Google Scholar

[9]

J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237.   Google Scholar

[10]

J. F. Cariñena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle,, Acta Appl. Math., 66 (2001), 67.  doi: 10.1023/A:1010743114995.  Google Scholar

[11]

J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008).   Google Scholar

[12]

J. F. Cariñena and A. Ramos, A new geometric approach to Lie systems and physical applications. Symmetry and perturbation theory,, Acta Appl. Math., 70 (2002), 43.  doi: 10.1023/A:1013913930134.  Google Scholar

[13]

G. Casale, Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières,, (French) [Singular foliations of codimension one, 56 (2006), 735.   Google Scholar

[14]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation,, Algebraic, analytic and geometric aspects of complex differential equations and their deformations, (2007), 15.   Google Scholar

[15]

G. Casale, Le groupoïde de Galois de $P_1$ et son irréductibilité,, Comment. Math. Helv., 83 (2008), 471.  doi: 10.4171/CMH/133.  Google Scholar

[16]

G. Darboux, "Leçons sur la théorie générale des surfaces. I, II,'', (French) [Lessons on the general theory of surfaces. I, (1914).   Google Scholar

[17]

W. Fulton and J. Harris, "Representation Theory. A First Course,'', Graduate Texts in Mathematics, 129 (1991).   Google Scholar

[18]

A. Guldberg, Sur les équations différentialles ordinaries qui possèdent un système fondamental d'intègrales,, Compt. Rend. Acad. Sci. Paris, (1893), 964.   Google Scholar

[19]

M. Havlíček, S. Pošta and P. Winternitz, Nonlinear superposition formulas based on imprimitive group action,, J. Math. Phys., 40 (1999), 3104.  doi: 10.1063/1.532749.  Google Scholar

[20]

J. E. Humphreys, "Linear Algebraic Groups,'', Graduate Texts in Mathematics, (1975).   Google Scholar

[21]

I. Kaplansky, "An Introduction to Differential Algebra,'', Actualités Sci. Ind., (1251).   Google Scholar

[22]

E. R. Kolchin, Galois theory of differential fields,, Amer. J. Math., 75 (1953), 753.  doi: 10.2307/2372550.  Google Scholar

[23]

E. R. Kolchin, "Differential Algebra and Algebraic Groups,'', Pure and Applied Mathematics, 54 (1973).   Google Scholar

[24]

A. G. Khovanskiĭ, Topological obstructions to the representability of functions by quadratures,, J. Dynam. Control Systems, 1 (1995), 91.  doi: 10.1007/BF02254657.  Google Scholar

[25]

S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuierliche endliche Gruppe gestatten,, Math. Ann. Bd., 25 (1885), 71.  doi: 10.1007/BF01446421.  Google Scholar

[26]

S. Lie, "Über Differentialgleichungen die Fundamentalintegrale besitzen,'', Lepziger Berichte, (1893).   Google Scholar

[27]

S. Lie, "Vorlesungen über continuierliche Gruppen mit Geometrischen und anderen Anwendungen,'', (German) Bearbeitet und herausgegeben von Georg Scheffers, (1893).   Google Scholar

[28]

S. Lie, Sur les équations différentielles ordinaries, qui possèddent des systemes fondamentaux d'integrales,, Compt. Rend. Acad. Sci. Paris, (1893), 1233.   Google Scholar

[29]

S. Lie and G. Scheffers, "Vorlesungen über Differentialgleichungen mit bekannten Infinitesimalen Transformationen,'', Reprinted by Chelsea books, (1967).   Google Scholar

[30]

J. Liouville, Mémoire sur l'integration de une classe de équations différentielles du second ordre en quantités finies explicités,, J. Math. Pures Appl., 4 (1839), 423.   Google Scholar

[31]

B. Malgrange, Le groupoïde de Galois d'un feuilletage,, (French) [The Galois groupoid of a foliation], 38 (2001), 465.   Google Scholar

[32]

B. Malgrange, On nonlinear differential Galois theory,, Chinese Ann. Math. Ser. B, 23 (2002), 219.  doi: 10.1142/S0252959902000213.  Google Scholar

[33]

B. Malgrange, "Pseudogroupes de Lie et Théorie de Galois Différentielle,'', (French) [Lie Pseudogroups and Differential Galois Theory], (2010).   Google Scholar

[34]

P. Malliavin, "Géométrie Différentielle Intrinsèque,'', (French) [Intrinsic Differential Geometry], (1972).   Google Scholar

[35]

M. Maamache, Ermakov systems, exact solution, and geometrical angles and phases,, Phys. Rev. A (3), 52 (1995), 936.  doi: 10.1103/PhysRevA.52.936.  Google Scholar

[36]

J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, II,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[37]

J. J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential,, Methods Appl. Anal., 8 (2001), 113.   Google Scholar

[38]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,'', Progress in Mathematics, 179 (1999).   Google Scholar

[39]

J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations,, Ann. Sci. cole Norm. Sup. (4), 40 (2007), 845.   Google Scholar

[40]

K. Nishioka, Differential algebraic function fields depending rationally on arbitrary constants,, Nagoya Math. J., 113 (1989), 173.   Google Scholar

[41]

K. Nishioka, General solutions depending algebraically on arbitrary constants,, Nagoya Math. J., 113 (1989), 1.   Google Scholar

[42]

K. Nishioka, Lie extensions,, Proc. Japan Acad. Ser. A Math. Sci., 73 (1997), 82.  doi: 10.3792/pjaa.73.82.  Google Scholar

[43]

K. Nomizu, "Lie Groups and Differential Geometry,'', The Mathematical Society of Japan, (1956).   Google Scholar

[44]

J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation,, (French) [Differential Galois theory and resummation], (1990), 117.   Google Scholar

[45]

M. Rosenlicht, A remark on quotient spaces,, An. Acad. Brasil. Ci., 35 (1963), 487.   Google Scholar

[46]

C. Sancho de Salas, "Grupos Algebraicos y Teoría de Invariantes,'' (Spanish) [Algebraic Groups and Invariant Theory],, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 16 (2001).   Google Scholar

[47]

J.-P. Serre, Géométrie algébrique et géométrie analytique, (French), 6 (): 1955.   Google Scholar

[48]

J.-P. Serre, Espaces fibrés algebriques (French),, Séminaire Claude Chevalley, 3 (1958), 1.   Google Scholar

[49]

Y. Sibuya, "Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation,'', Translated from the Japanese by the author, 82 (1990).   Google Scholar

[50]

S. Shnider and P. Winternitz, Classification of systems of nonlinear ordinary differential equations with superposition principles,, J. Math. Phys., 25 (1984), 3155.  doi: 10.1063/1.526085.  Google Scholar

[51]

M. Sorine and P. Winternitz, Superposition laws for solutions of differential matrix Riccati equations arising in control theory,, IEEE Trans. Automat. Control, 30 (1985), 266.  doi: 10.1109/TAC.1985.1103934.  Google Scholar

[52]

H. Umemura, On the irreducibility of the first differential equation of Painlevé,, in, (1988), 771.   Google Scholar

[53]

H. Umemura, Galois theory of algebraic and differential equations,, Nagoya Math. J., 144 (1996), 1.   Google Scholar

[54]

H. Umemura, Differential Galois theory of infinite dimension,, Nagoya Math. J., 144 (1996), 59.   Google Scholar

[55]

H. Umemura, Sur l'équivalence des théories de Galois différentielles générales,, (French) [On the equivalence of general differential Galois theories], 346 (2008), 1155.   Google Scholar

[56]

M. van der Put and M. F. Singer, "Galois Theory of Linear Differential Equations,'', Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328 (2003).   Google Scholar

[57]

E. Vessiot, Sur l'intégration des equations différentielles linéaires, (French), 9 (1892), 197.   Google Scholar

[58]

E. Vessiot, Sur une classe d'équations différentielles,, Ann. Sci. École Norm. Sup. (3), 10 (1893), 53.   Google Scholar

[59]

E. Vessiot, Sur une classe systèmes d'équations différentielles ordinaires,, Compt. Rend. Acad. Sci. Paris, (1893), 1112.   Google Scholar

[60]

E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales,, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894).   Google Scholar

[61]

E. Vessiot, Sur la théorie de Galois et ses diverses généralisations, (French), 21 (1904), 9.   Google Scholar

[62]

E. Vessiot, Sur la réductibilité des systèmes automorphes dont le groupe d'automorphie est un groupe continu fini simplement transitif, (French), 57 (1940), 1.   Google Scholar

[1]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280

[2]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[3]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[4]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[5]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[6]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[7]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[8]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[9]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[10]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[11]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[12]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[13]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[14]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[15]

Aisling McGlinchey, Oliver Mason. Observations on the bias of nonnegative mechanisms for differential privacy. Foundations of Data Science, 2020, 2 (4) : 429-442. doi: 10.3934/fods.2020020

[16]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[17]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[18]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[19]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[20]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]