American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 923-935. doi: 10.3934/proc.2015.0923

A functional-analytic technique for the study of analytic solutions of PDEs

Eugenia N. Petropoulou 1, and  Panayiotis D. Siafarikas 2,

1. 

Department of Civil Engineering, University of Patras, 26500 Patras, Greece
2. 

Department of Mathematics, University of Patras, 26500 Patras, Greece

Received  August 2014 Revised  December 2014 Published  November 2015

A functional-analytic method is used to study the existence and the uniqueness of bounded, analytic and entire complex solutions of partial differential equations. As a benchmark problem, this method is applied to the nonlinear Benjamin--Bona--Mahony equation and the associated to this, linear equation. The predicted solutions are in power series form and two concrete examples are given for specific initial conditions.
Keywords: Benjamin--Bona--Mahony equation., series solutions, Bounded solutions, entire solutions, analytic solutions.
Mathematics Subject Classification: Primary: 35A01, 35A02; Secondary: 35B08, 35C10, 35G05, 35G20, 35Q5.
Citation: Eugenia N. Petropoulou, Panayiotis D. Siafarikas. A functional-analytic technique for the study of analytic solutions of PDEs. Conference Publications, 2015, 2015 (special) : 923-935. doi: 10.3934/proc.2015.0923
References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47.   Google Scholar

[2]

G. Caciotta and F. Nicoló, Local and global analytic solutions for a class of characteristic problems of the Einstein vacuum equations in the "double null foliation gauge",, Ann. Henri Poincare, 13 (2012), 1167.   Google Scholar

[3]

G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and singularity formation for the Peakon $b$-family equations,, Acta Appl. Math., 122 (2012), 419.   Google Scholar

[4]

C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, in Global Analysis (Proc. Sympos. Pure Math., (1970), 61.   Google Scholar

[5]

L. C. Evans, Partial differential equations,, $2^{nd}$ edition, (2010).   Google Scholar

[6]

N. Hayashi and K. Kato, Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearity,, Comm. Partial Differential Equations, 22 (1997), 773.   Google Scholar

[7]

A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV,, J. Differential Equations, 253 (2012), 3101.   Google Scholar

[8]

E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space,, J. Mathematical Phys., 12 (1971), 1961.   Google Scholar

[9]

E. K. Ifantis, Analytic solutions for nonlinear differential equations,, J. Math. Anal. Appl., 124 (1987), 339.   Google Scholar

[10]

E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation,, J. Math. Anal. Appl., 124 (1987), 381.   Google Scholar

[11]

J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type,, Math. Balkanica, 2 (1972), 76.   Google Scholar

[12]

T. Kusano and S. Oharu, Bounded entire solutions of second order semilinear elliptic equations with application to a parabolic initial value problem,, Indiana Univ. Math. J., 34 (1985), 85.   Google Scholar

[13]

E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations,, Dynam. Systems Appl. 13 (2004), 13 (2004), 283.   Google Scholar

[14]

E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations,, Commun. Pure Appl. Anal. 8 (2009), 8 (2009), 1053.   Google Scholar

[15]

E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs,, J. Math. Anal. Appl. 331 (2007), 331 (2007), 279.   Google Scholar

[16]

E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs II,, Numer. Funct. Anal. Optim. 30 (2009), 30 (2009), 613.   Google Scholar

[17]

E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane,, Numer. Funct. Anal. Optim. 34 (2013), 34 (2013), 770.   Google Scholar

[18]

I. G. Petrovsky, Lecture on partial differential equations., Translated from the Russian by A. Shenitzer, (1957).   Google Scholar

[19]

A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing,, Phys. Rev. A 45 (1992), 45 (1992), 1943.   Google Scholar

[20]

G. Zampieri, A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbbR^{2}$,, Rend. Sem. Mat. Univ. Padova 63 (1980), 63 (1980), 83.   Google Scholar

[21]

G. Zampieri, Analytic solutions of P.D.E.'s., Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), XLV (1999), 365.   Google Scholar

show all references

References:
[1]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47.   Google Scholar

[2]

G. Caciotta and F. Nicoló, Local and global analytic solutions for a class of characteristic problems of the Einstein vacuum equations in the "double null foliation gauge",, Ann. Henri Poincare, 13 (2012), 1167.   Google Scholar

[3]

G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and singularity formation for the Peakon $b$-family equations,, Acta Appl. Math., 122 (2012), 419.   Google Scholar

[4]

C. J. Earle and R. S. Hamilton, A fixed point theorem for holomorphic mappings, in Global Analysis (Proc. Sympos. Pure Math., (1970), 61.   Google Scholar

[5]

L. C. Evans, Partial differential equations,, $2^{nd}$ edition, (2010).   Google Scholar

[6]

N. Hayashi and K. Kato, Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearity,, Comm. Partial Differential Equations, 22 (1997), 773.   Google Scholar

[7]

A. A. Himonas and G. Petronilho, Analytic well-posedness of periodic gKdV,, J. Differential Equations, 253 (2012), 3101.   Google Scholar

[8]

E. K. Ifantis, Solution of the Schrödinger equation in the Hardy-Lebesgue space,, J. Mathematical Phys., 12 (1971), 1961.   Google Scholar

[9]

E. K. Ifantis, Analytic solutions for nonlinear differential equations,, J. Math. Anal. Appl., 124 (1987), 339.   Google Scholar

[10]

E. K. Ifantis, Global analytic solutions of the radial nonlinear wave equation,, J. Math. Anal. Appl., 124 (1987), 381.   Google Scholar

[11]

J. Kajiwara, Holomorphic solutions of a partial differential equation of mixed type,, Math. Balkanica, 2 (1972), 76.   Google Scholar

[12]

T. Kusano and S. Oharu, Bounded entire solutions of second order semilinear elliptic equations with application to a parabolic initial value problem,, Indiana Univ. Math. J., 34 (1985), 85.   Google Scholar

[13]

E. N. Petropoulou and P. D. Siafarikas, Analytic solutions of some non-linear ordinary differential equations,, Dynam. Systems Appl. 13 (2004), 13 (2004), 283.   Google Scholar

[14]

E. N. Petropoulou and P. D. Siafarikas, Polynomial solutions of linear partial differential equations,, Commun. Pure Appl. Anal. 8 (2009), 8 (2009), 1053.   Google Scholar

[15]

E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs,, J. Math. Anal. Appl. 331 (2007), 331 (2007), 279.   Google Scholar

[16]

E. N. Petropoulou, P. D. Siafarikas and E. E. Tzirtzilakis, A "discretization" technique for the solution of ODEs II,, Numer. Funct. Anal. Optim. 30 (2009), 30 (2009), 613.   Google Scholar

[17]

E. N. Petropoulou and E. E. Tzirtzilakis, On the logistic equation in the complex plane,, Numer. Funct. Anal. Optim. 34 (2013), 34 (2013), 770.   Google Scholar

[18]

I. G. Petrovsky, Lecture on partial differential equations., Translated from the Russian by A. Shenitzer, (1957).   Google Scholar

[19]

A. Vourdas, Analytic representations in the unit disc and applications to phase states and squeezing,, Phys. Rev. A 45 (1992), 45 (1992), 1943.   Google Scholar

[20]

G. Zampieri, A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $\mathbbR^{2}$,, Rend. Sem. Mat. Univ. Padova 63 (1980), 63 (1980), 83.   Google Scholar

[21]

G. Zampieri, Analytic solutions of P.D.E.'s., Ann. Univ. Ferrara-Sez. VII-Sc. Mat. XLV (1999), XLV (1999), 365.   Google Scholar

[1]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[2]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[3]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[4]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[5]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[6]

Fang-Di Dong, Wan-Tong Li, Shi-Liang Wu, Li Zhang. Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1031-1060. doi: 10.3934/dcdsb.2020152
[7]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[8]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[10]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[11]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[12]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[13]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495
[14]

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
[15]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[16]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[17]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[18]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[19]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345
[20]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

 Impact Factor: 

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences