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Amin Esfahani

Associate Professor of Pure Mathematics

Education

  • Ph.D. 2005-2008

    High Dimensional Nonlinear Dispersive Models

    Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ – Brasil

  • M.Sc. 2003

    The Green functions and (non)-existence of biharmonic problems at critical growth

    Sharif University of Technology, Tehran, Iran

  • B.Sc.2001

    Mathematics

    Damghan University, Damghan, Iran

Teaching

  • Calculus
  • PDE
  • Theory of differential equations
  • Differential geometry
  • Geometry of manifolds
  • Philosophy of Mathematics

Selected Publications

Esfahani, A. Existence of solutions of a class of nonlinear nonlocal problems involving the potential (2019) Chaos, Solitons and Fractals, 119, pp. 1-7.

DOI: 10.1016/j.chaos.2018.12.013

In this paper, we consider a class of nonlinear nonlocal problem involving an anisortropic operator and an external potential. We show the existence of positive and sign-changing solutions of this problem via the variational methods. © 2018 Elsevier Ltd

AUTHOR KEYWORDS: Positive and nodal solutions; Potential BO-ZK equation; Variational method
INDEX KEYWORDS: Existence of Solutions; External potential; Nodal solutions; Nonlocal problems; Potential BO-ZK equation; Sign changing solutions; Variational methods, Mathematical techniques
PUBLISHER: Elsevier Ltd

Wang, H., Esfahani, A. Well-posedness result for the Ostrovsky, Stepanyams and Tsimring equation at the critical regularity (2018) Nonlinear Analysis: Real World Applications, 44, pp. 347-364.

DOI: 10.1016/j.nonrwa.2018.05.011

Studied here is the Ostrovsky, Stepanyams and Tsimring (OST) equation ut+uxxx−εH(u+uxx)x+uux=0. It is showed that the associated initial value problem is locally analytically well-posed in the critical Sobolev spaces H−3∕2(R). This improves the previous results on this equation in the Sobolev spaces. © 2018 Elsevier Ltd

AUTHOR KEYWORDS: Besov–Bourgain space; Cauchy problem; Critical regularity; OST equation
INDEX KEYWORDS: Initial value problems, Cauchy problems; Critical regularity; OST equation; Wellposedness, Sobolev spaces
PUBLISHER: Elsevier Ltd

Esfahani, A., Levandosky, S. Solitary waves of a coupled KdV system with a weak rotation (2018) Journal of Differential Equations, 265 (10), pp. 4835-4872.

DOI: 10.1016/j.jde.2018.06.023

Studied here is a coupled KdV system with a weak rotation. The existence of ground state solutions is proved, and the continuous dependence on the parameter and asymptotic behavior of ground state solutions are established. The orbital stability of ground states is also investigated. © 2018 Elsevier Inc.

AUTHOR KEYWORDS: KdV system with rotation; Orbital stability; Solitary waves; Variational problems
PUBLISHER: Academic Press Inc.

Esfahani, A., Esfahani, S.E. Positive and nodal solutions of the generalized BO-ZK equation (2018) Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 112 (4), pp. 1381-1390.

DOI: 10.1007/s13398-017-0435-2

In this article, we establish the existence of a least-energy nodal solution and a positive ground state for nonlinear nonlocal problem involving the BO-ZK operator. By using constrained minimization on the Nehari-type manifolds, we prove the generalized BO-ZK equation has a least-energy nodal solution with its energy exceeding twice the least energy, and a positive ground state which both solutions are block-radial. © 2017, Springer-Verlag Italia S.r.l.

AUTHOR KEYWORDS: BO-ZK operator; Positive and nodal solutions; Variational method
PUBLISHER: Springer-Verlag Italia s.r.l.

Esfahani, A., Pastor, A. Two dimensional solitary waves in shear flows (2018) Calculus of Variations and Partial Differential Equations, 57 (4), art. no. 102, .

DOI: 10.1007/s00526-018-1383-1

In this paper we study existence and asymptotic behavior of solitary-wave solutions for the generalized Shrira equation, a two-dimensional model appearing in shear flows. The method used to show the existence of such special solutions is based on the mountain pass theorem. One of the main difficulties consists in showing the compact embedding of the energy space in the Lebesgue spaces; this is dealt with interpolation theory. Regularity and decay properties of the solitary waves are also established. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PUBLISHER: Springer New York LLC

Esfahani, A. Liouville-type theorem for a nonlocal operator on the half plane (2018) Monatshefte fur Mathematik, 186 (3), pp. 439-452.

DOI: 10.1007/s00605-018-1195-6

In this article we consider the following integral equation associated to the BO–ZK operator in the half plane. By combining the lifting regularity and the moving planes method for integral forms, we demonstrate that there is no positive solution for this integral equation. © 2018, Springer-Verlag GmbH Austria, part of Springer Nature.

AUTHOR KEYWORDS: BO–ZK operator; Half plane; Integral equation; Nonexistence
PUBLISHER: Springer-Verlag Wien

Carvajal, X., Esfahani, A., Panthee, M. Well-Posedness Results and Dissipative Limit of High Dimensional KdV-Type Equations (2017) Bulletin of the Brazilian Mathematical Society, 48 (4), pp. 505-550.

DOI: 10.1007/s00574-017-0034-z

Considered in this work is an n-dimensional dissipative version of the Korteweg–de Vries (KdV) equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study well-posedness behavior of this equation when the dissipative effects are reduced. © 2017, Sociedade Brasileira de Matemática.

AUTHOR KEYWORDS: Dispersive-dissipative models; Ill-posedness; Initial value problem; KdV equation; Well-posedness
PUBLISHER: Springer New York LLC

Esfahani, A., Mohammadi, H.B. Global Existence and Asymptotic Behavior of Solutions for the Cauchy Problem of a Dissipative Boussinesq-Type Equation (2017) Mathematical Modelling and Analysis, 22 (4), pp. 441-463. 

DOI: 10.3846/13926292.2017.1319882

We consider the Cauchy problem for a Boussinesq-type equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted spaces are established by the contraction mapping principle. © 2017, © Vilnius Gediminas Technical University, 2017.

AUTHOR KEYWORDS: asymptotic behavior; dissipative Boussinesq equation; Sobolev spaces
PUBLISHER: Taylor and Francis Ltd.

Esfahani, A. Qualitative properties of solutions of an integral equation associated with the Benjamin–Ono–Zakharov–Kuznetsov operator (2017) Indagationes Mathematicae, 28 (2), pp. 601-611.

DOI: 10.1016/j.indag.2017.01.008

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions. © 2017 Royal Dutch Mathematical Society (KWG)

AUTHOR KEYWORDS: BO–ZK operator; Integral equation; Regularity
PUBLISHER: Elsevier B.V.

Esfahani, A., Pastor, A. Sharp Constant of an Anisotropic Gagliardo–Nirenberg-Type Inequality and Applications (2017) Bulletin of the Brazilian Mathematical Society, 48 (1), pp. 171-185.

DOI: 10.1007/s00574-016-0017-5

In this paper we establish the best constant of an anisotropic Gagliardo–Nirenberg-type inequality related to the Benjamin–Ono–Zakharov–Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space. © 2016, Sociedade Brasileira de Matemática.

AUTHOR KEYWORDS: BO-ZK equation; Fractional Sobolev-Liouville inequality; Gagliardo–Nirenberg inequality
PUBLISHER: Springer New York LLC

Esfahani, A., Levandosky, S. Stability of solitary waves of the kadomtsev{petviashvili equation with a weak rotation (2017) SIAM Journal on Mathematical Analysis, 49 (6), pp. 5096-5133.

DOI: 10.1137/16M1103865

We consider the stability of solitary wave solutions of the rotation-generalized Kadomtsev{Petviashvili (RGKP) equation. Using an iteration method developed by Petviashvili, we numerically compute the solitary waves over a range of the parameters c and and use these to determine the concavity of the function d(c) that determines their stability. In spite of the absence of any scaling invariance property for the RGKP equation, we prove the strong instability of the solitary waves. © 2017 Society for Industrial and Applied Mathematics.

AUTHOR KEYWORDS: Rotating uid; Solitary waves; Stability; Variational method
INDEX KEYWORDS: Convergence of numerical methods; Iterative methods; Stability, Iteration method; Rotating uid; Scaling invariance; Soli-tary wave solutions; Variational methods; Weak rotation, Solitons
PUBLISHER: Society for Industrial and Applied Mathematics Publications

Esfahani, A. Anisotropic Gagliardo–Nirenberg inequality with fractional derivatives (2015) Zeitschrift fur Angewandte Mathematik und Physik, 66 (6), pp. 3345-3356.

DOI: 10.1007/s00033-015-0586-y

Studied here is anisotropic Gagliardo–Nirenberg inequality in n-dimensional case. A fractional version of this inequality will be proved. Its sharp constant is also elicited in terms of the ground states of the associated nonlocal equation. © 2015, Springer Basel.

AUTHOR KEYWORDS: Anisotropic Gagliardo-Nirenberg inequality; Sharp constant; Sobolev-Liouville spaces
INDEX KEYWORDS: Ground state, Fractional derivatives; Non-local equations, Anisotropy
PUBLISHER: Birkhauser Verlag AG

Esfahani, A., Pastor, A., Bona, J.L. Stability and decay properties of solitary-wave solutions to the generalized BO-ZK equation (2015) Advances in Differential Equations, 20 (9-10), pp. 801-834.

Studied here is the generalized Benjamin-Ono-Zakharov-Kuznetsov equation ut + upux + αH uxx + εuxyy = 0, (x, y) ∈ R2, t ∈ R+, in two space dimensions. Here, H is the Hilbert transform and sub-scripts denote partial differentiation. We classify when this equation possesses solitary-wave solutions in terms of the signs of the constants α and ε appearing in the dispersive terms and the strength of the non-linearity. Regularity and decay properties of these solitary wave are determined and their stability is studied.

PUBLISHER: Khayyam Publishing

Wang, H., Esfahani, A. Global rough solutions to the sixth-order Boussinesq equation (2014) Nonlinear Analysis, Theory, Methods and Applications, 102, pp. 97-104.

DOI: 10.1016/j.na.2014.02.006

We show that the Cauchy problem for the sixth-order Boussinesq equation with cubic nonlinearity is globally well-posed in Hs<(R) provided 3/2 < s < 2. © 2014 Published by Elsevier Ltd.


AUTHOR KEYWORDS: Almost conservation law; Boussinesq equation; Global rough solution; Modified energy functional
INDEX KEYWORDS: Boussinesq equations; Cauchy problems; Conservation law; Cubic nonlinearities; Energy functionals, Nonlinear analysis, Mathematical techniques

Pourgholi, R., Esfahani, A., Kumar, S. A numerical algorithm for solving an inverse semilinear wave problem (2014) International Journal of Computing Science and Mathematics, 5 (1), pp. 1-15.

DOI: 10.1504/IJCSM.2014.059378

The problem of identifying the solution (k(x, t),U(x, t)) in an inverse semilinear wave problem is considered. It is shown that under certain conditions of data φ, ψ, there exists a unique solution (k(x, t),U(x, t)) of this problem. Furthermore a numerical algorithm for solving the inverse semilinear wave problem is proposed. The approach for this inverse problem is given by using the semi-discretisation method. A polynomial function is proposed to approximate U(x, t) then the finite difference method is applied to approximate unknown k(x, t). Numerical results show efficiency of our method. Copyright © 2014 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Existence; Finite difference method; Inverse semilinear wave problem; Polynomial function; Stability; Uniqueness

Esfahani, A., Pourgholi, R. Well-posedness of the ADMB-KdV equation in Sobolev spaces of negative indices (2014) Acta Mathematica Vietnamica, 39 (2), pp. 237-251.

DOI: 10.1007/s40306-014-0050-7

In this paper, we study the ADMB-KdV equation. We show that the associated initial value problem is locally well posed in Sobolev spaces Hs (ℝ2) for s > -1/2. We also prove that our result is sharp in some sense. © 2013 Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer.

AUTHOR KEYWORDS: ADMB-KdV equation; Cauchy problem; Sobolev spaces

Esfahani, A., Pourgholi, R. Dynamics of Solitary Waves of the Rosenau-RLW Equation (2014) Differential Equations and Dynamical Systems, 22 (1), pp. 93-111.

DOI: 10.1007/s12591-013-0174-6

In this paper we study the solitary waves of the Rosenau-RLW equation. By using some trigonometric function methods, a family of stable solitary wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency and wave speed, for what should be an equation relevant to modeling in a number of fields. © 2013 Foundation for Scientific Research and Technological Innovation.

AUTHOR KEYWORDS: Rosenau-RLW equation; Solitary waves; Stability

Esfahani, A. Instability of the stationary solutions of generalized dissipative Boussinesq equation (2014) Applications of Mathematics, 59 (3), pp. 345-358.

DOI: 10.1007/s10492-014-0059-1

In this work we study the generalized Boussinesq equation with a dissipation term. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive sufficient conditions for the blow-up of the solution to the problem. Furthermore, the instability of the stationary solutions of this equation is established.

AUTHOR KEYWORDS: Damped Boussinesq equation; Instability; Stationary solution
INDEX KEYWORDS: Mathematical techniques; Plasma stability, Blow-up; Boussinesq equations; Global solutions; Stationary solutions; Suitable conditions, Initial value problems
PUBLISHER: Kluwer Academic Publishers

Esfahani, A., Wang, H. A bilinear estimate with application to the sixth-order Boussinesq equation (2014) Differential and Integral Equations, 27 (5-6), pp. 401-414.

In this work, we study the Cauchy problem of the sixth-order Boussinesq equation. We shall use the [k; Z]-multiplier norm method to get a bilinear estimate on nonlinear term, and the local well-posedness on H s(ℝ) is obtained for s > -3/4 .

PUBLISHER: Khayyam Publishing

Esfahani, A. Sharp well-posedness of the Ostrovsky, Stepanyams and Tsimring equation (2013) Mathematical Communications, 18 (2), pp. 323-335.

In this paper, we study the Ostrovsky, Stepanyams and Tsimring equation. We show that the associated initial value problem is locally well-posed in Sobolev spaces Hs (ℝ) for s > -3/2. We also prove that our result is sharp in the sense that the flow map of this equation fails to be C2 in Hs(ℝ) for s < -3/2. © 2013 Department of Mathematics, University of Osijek.


AUTHOR KEYWORDS: Local well-posedness; OST equation; Sobolev spaces

Esfahani, A., Pourgholi, R. The ADMB-KdV equation in a time-weighted space (2013) Annali dell'Universita di Ferrara, 59 (2), pp. 269-283.

DOI: 10.1007/s11565-013-0178-8

This paper studies the ADM-BKdV equation. The associated initial value problem will be proved to be locally well-posed in anisotropic Sobolev spaces Hs1,s2 (ℝ2) for s1>-3/2, s2>-1/2 and 2s1+4s2>-1. © 2013 Università degli Studi di Ferrara.


AUTHOR KEYWORDS: ADMB-KdV equation; Anisotropic Sobolev spaces; Cauchy problem

Esfahani, A. Scattering of solutions and stability of solitary waves for the generalized BBM-ZK equation (2013) Communications in Mathematical Sciences, 12 (2), pp. 293-315.

DOI: 10.4310/CMS.2014.v12.n2.a5

In this paper, a two-dimensional version of the BBM equation will be considered. The existence and scattering of global small amplitude solutions to this equation will be studied. The orbital stability of solitary wave solutions of this equation will be also investigated. © 2014 International Press.

AUTHOR KEYWORDS: BBM-ZK equation; Orbital stability; Scattering

Pourgholi, R., Esfahani, A., Foadian, S., Parehkar, S. Resolution of an inverse problem by haar basis and legendre wavelet methods (2013) International Journal of Wavelets, Multiresolution and Information Processing, 11 (5), art. no. 1350034, .

DOI: 10.1142/S0219691313500343

In this paper, two numerical methods are presented to solve an ill-posed inverse problem for Fisher's equation using noisy data. These two methods are the Haar basis and the Legendre wavelet methods combined with the Tikhonov regularization method. A sensor located at a point inside the body is used and u(x, t) at a point x = a, 0 < a < 1 is measured and these methods are applied to the inverse problem. We also show that an exponential rate of convergence of these methods. In fact, this work considers a comparative study between the Haar basis and the Legendre wavelet methods to solve some ill-posed inverse problems. Results show that an excellent estimation of the unknown function of the inverse problem which have been obtained within a couple of minutes CPU time at Pentium(R) Dual-Core CPU 2.20 GHz. © World Scientific Publishing Company.


AUTHOR KEYWORDS: error analysis; Haar basis method; Ill-posed inverse problems; Legendre wavelet method; noisy data
INDEX KEYWORDS: Comparative studies; Exponential rates; Fisher's equation; Haar basis method; ILL-posed inverse problem; Legendre wavelet methods; Noisy data; Tikhonov regularization method, Differential equations; Error analysis, Inverse problems

Wang, H., Esfahani, A. Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation (2013) Nonlinear Analysis, Theory, Methods and Applications, 89, pp. 267-275.

DOI: 10.1016/j.na.2013.04.011

We consider the initial value problem associated to the nonlinear periodic sixth-order Boussinesq equation. For given initial data in Sobolev spaces Hs(T), the local well-posedness of this equation is established for s>-1/2. © 2013 Elsevier Ltd. All rights reserved.


AUTHOR KEYWORDS: Boussinesq equation; Cauchy problem; Well-posedness
INDEX KEYWORDS: Boussinesq equations; Cauchy problems; Local well-posedness; Wellposedness, Initial value problems, Partial differential equations

Pourgholi, R., Esfahani, A. An efficient numerical method for solving an inverse wave problem (2013) International Journal of Computational Methods, 10 (3), art. no. 1350009, .

DOI: 10.1142/S0219876213500096

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. © 2013 World Scientific Publishing Company.


AUTHOR KEYWORDS: Existence and uniqueness; stability; SVD method; the Tikhonov regularization method
INDEX KEYWORDS: Efficient numerical method; Existence and uniqueness; Initial and boundary conditions; Inverse wave problems; Singular value decomposition method; SVD method; Tikhonov regularization; Tikhonov regularization method, Convergence of numerical methods; Numerical methods; Singular value decomposition, Inverse problems

Pourgholi, R., Esfahani, A., Rahimi, H., Tabasi, S.H. Solving an inverse initial-boundary-value problem using basis function method (2013) Computational and Applied Mathematics, 32 (1), pp. 27-40.

DOI: 10.1007/s40314-013-0005-y

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example. © 2013 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.


AUTHOR KEYWORDS: Existence; Inverse initial-boundary-value problem; The L-curve method; Tikhonov regularization method; Uniqueness

Pourgholi, R., Esfahani, A., Abtahi, M. A numerical solution of a two-dimensional IHCP (2013) Journal of Applied Mathematics and Computing, 41 (1-2), pp. 61-79.

DOI: 10.1007/s12190-012-0592-6

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e.; the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method. © 2012 Korean Society for Computational and Applied Mathematics.


AUTHOR KEYWORDS: Consistency; Convergence; Existence; Finite difference method; Inverse heat conduction problem; Least-square method; Stability; Tikhonov regularization method; Uniqueness
INDEX KEYWORDS: Consistency; Convergence; Existence; Inverse heat conduction problem; Least square methods; Tikhonov regularization method; Uniqueness, Convergence of numerical methods; Finite difference method; Linear systems; Singular value decomposition; Two dimensional, Least squares approximations

Pourgholi, R., Esfahani, A., Dana, H. Real valued genetic algorithm for solving an inverse hyperbolic problem: Multi-core parallelisation approach (2013) International Journal of Mathematical Modelling and Numerical Optimisation, 4 (4), pp. 410-424.

DOI: 10.1504/IJMMNO.2013.059206

In this paper, a numerical approach combining the use of the least squares method and the genetic algorithm (sequential and multi-core parallelisation approach) is proposed for the determination of temperature in an inverse hyperbolic heat conduction problem (IHHCP). Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that an excellent estimation can be obtained by implementation sequential genetic algorithm within a CPU with clock speed 2.4 GHz, and parallel genetic algorithm within a 16-core CPU with clock speed 2.4 GHz for each core. © 2013 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Genetic algorithm; IHHCP; Inverse hyperbolic heat; Inverse hyperbolic heat conduction problem; Multi-core parallelisation; The least squares method
PUBLISHER: Inderscience Enterprises Ltd.

Esfahani, A., Levandosky, S. Solitary waves of the rotation-generalized benjamin-ono equation (2013) Discrete and Continuous Dynamical Systems- Series A, 33 (2), pp. 663-700.

DOI: 10.3934/dcds.2013.33.663

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

AUTHOR KEYWORDS: RGBO equation; Solitary waves; Stabilit

Esfahani, A. Cauchy problem for a model of nonlinear waves generated in viscous films (2012) Mathematical Methods in the Applied Sciences, 35 (16), pp. 1931-1950.

DOI: 10.1002/mma.2556

This work studies a two-dimensional version of the Korteweg-de Vries equation, which is the so-called anisotropic dissipation-modified Boussinesq-Korteweg-de Vries equation. The local well-posedness for the Cauchy problem associated with this equation is proven when the initial value belongs to the Sobolev space H s(ℝ 2), for all s > - 1/6. A global existence result will be obtained under suitable conditions. Copyright © 2012 John Wiley & Sons, Ltd.

AUTHOR KEYWORDS: Boussinesq-KdV equation; Cauchy problem; dissipation
INDEX KEYWORDS: Boussinesq-KdV equation; Cauchy problems; Global existence; Initial values; Local well-posedness; Nonlinear waves; Viscous films; Work study, Energy dissipation; Korteweg-de Vries equation, Partial differential equations

Solaymani Fard, O., Esfahani, A., Vahidian Kamyad, A. On solution of a class of fuzzy BVPs (2012) Iranian Journal of Fuzzy Systems, 9 (1), pp. 49-60.

This paper investigates the existence and uniqueness of solutions to first-order nonlinear boundary value problems (BVPs) involving fuzzy differential equations and two-point boundary conditions. Some sufficient conditions are presented that guarantee the existence and uniqueness of solutions under the approach of Hukuhara differentiability.

AUTHOR KEYWORDS: Boundary value problems; Fuzzy differential equations; Fuzzy numbers

Esfahani, A., Farah, L.G., Wang, H. Global existence and blow-up for the generalized sixth-order Boussinesq equation (2012) Nonlinear Analysis, Theory, Methods and Applications, 75 (11), pp. 4325-4338.

DOI: 10.1016/j.na.2012.03.019

In this paper we prove local well-posedness in L2(R) and H1(R) for the generalized sixth-order Boussinesq equation utt= uxx+βuxxxx+uxxxxxx+(|u|αu)xx. Our proof relies in the oscillatory integrals estimates introduced by Kenig et al. (1991) [14]. We also show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem. © 2012 Elsevier Ltd. All rights reserved.

AUTHOR KEYWORDS: Boussinesq equation; Sobolev spaces; Well-posedness
INDEX KEYWORDS: Blow-up; Boussinesq equations; Global existence; Global solutions; Local well-posedness; Oscillatory integral; Sufficient conditions; Wellposedness, Initial value problems; Sobolev spaces, Partial differential equations

Esfahani, A. The Cauchy problem for the dissipation-modified Kadomtsev-Petviashvili equation (2012) Journal of Mathematical Analysis and Applications, 390 (2), pp. 439-455.

DOI: 10.1016/j.jmaa.2012.02.006

Considered herein is the dissipation-modified Kadomtsev-Petviashvili equation in two space-dimensional case. It is established that the Cauchy problem associated to this equation is locally well-posed in anisotropic Sobolev spaces. It is also shown in some sense that this result is sharp. In addition, the global well-posedness for this equation under suitable conditions is proved. © 2012 Elsevier Inc.

AUTHOR KEYWORDS: Anisotropic Sobolev spaces; DMKP equation; Well-posedness

Esfahani, A., Levandosky, S. Stability of Solitary Waves for the Generalized Higher-Order Boussinesq Equation (2012) Journal of Dynamics and Differential Equations, 24 (2), pp. 391-425.

DOI: 10.1007/s10884-012-9250-9

This work studies the stability of solitary waves of a class of sixth-order Boussinesq equations. © 2012 Springer Science+Business Media, LLC.

AUTHOR KEYWORDS: Boussinesq equation; Solitary waves; Stability

Esfahani, A. Remarks on a two dimensional bbm type equation (2012) Communications on Pure and Applied Analysis, 11 (3), pp. 1111-1127.

DOI: 10.3934/cpaa.2012.11.1111

In this work, we study a two-dimensional version of the BBM equation. We prove that the Cauchy problem for this equation is globally well-posed in a natural space. We also show that the orbital stability of the solitary waves of the equation. Furthermore, we establish that if the solution of the Cauchy problem has a compact support for all times, then this solution vanishes identically.

AUTHOR KEYWORDS: BBM equation; Cauchy problem; Stability; Unique continuation

Esfahani, A., Farah, L.G. Local well-posedness for the sixth-order Boussinesq equation (2012) Journal of Mathematical Analysis and Applications, 385 (1), pp. 230-242.

DOI: 10.1016/j.jmaa.2011.06.038

In this work we study the local well-posedness of the initial value problem for the nonlinear sixth-order Boussinesq equation utt=uxx+βuxxxx+uxxxxxx+(u2)xx, where β=±1. We prove the local well-posedness with initial data in non-homogeneous Sobolev spaces Hs(R{double-struck}) for negative indices of s∈R{double-struck}. © 2011 Elsevier Inc.

AUTHOR KEYWORDS: Boussinesq equation; Local well-posedness; Sobolev spaces

Esfahani, A., Pastor, A. On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations (2011) Bulletin of the London Mathematical Society, 43 (6), pp. 1130-1140.

DOI: 10.1112/blms/bdr048

We prove unique continuation results for the Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations. Our method of proof goes back to the complex variable techniques introduced by Bourgain. The approach is quite similar to that used by Panthee for the Kadomtsev-Petviashvili-II and Zakharov-Kuznetsov equations. © 2011 London Mathematical Society.

Esfahani, A. Remarks on solitary waves of the generalized two dimensional Benjamin-Ono equation (2011) Applied Mathematics and Computation, 218 (2), pp. 308-323.

DOI: 10.1016/j.amc.2011.05.065

In this paper we study the generalized 2D-BO equation in two dimensions:(ut+βHuxx+upux) x+uyy=0,(x,y)∈R2,t≥0.We classify the existence and non-existence of solitary waves depending on the sign of , β and on the nonlinearity. We also prove nonlinear stability and some regularity properties of such waves. © 2011 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Nonlinear PDE; Regularity; Solitary wave solution
INDEX KEYWORDS: Non linear PDE; Non-existence; Non-linear stabilities; Non-Linearity; Regularity; Regularity properties; Solitary wave solution; Two-dimension, Control nonlinearities; Solitons, Nonlinear equations

Esfahani, A. Solitary wave solutions for generalized Rosenau-KdV equation (2011) Communications in Theoretical Physics, 55 (3), pp. 396-398.

DOI: 10.1088/0253-6102/55/3/04

In this work, we study the generalized Rosenau-KdV equation. We shall use the sech-ansätze method to derive the solitary wave solutions of this equation. © 2011 Chinese Physical Society and IOP Publishing Ltd.

AUTHOR KEYWORDS: Ansätze method; KdV equation; Rosenau equation; Solitons

Esfahani, A. Instability of solitary waves of the generalized higher-order KP equation (2011) Nonlinearity, 24 (3), pp. 833-846.

DOI: 10.1088/0951-7715/24/3/006

This paper deals with the generalized higher-order Kadomtsev-Petviashvili (KP) equation. The strong instability of solitary wave solutions of this equation will be proved. © 2011 IOP Publishing Ltd & London Mathematical Society.

Esfahani, A. Traveling wave solutions for generalized bretherton equation (2011) Communications in Theoretical Physics, 55 (3), pp. 381-386.

DOI: 10.1088/0253-6102/55/3/01

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He's semi-inverse method (He's variational method). Various traveling wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency, and wave speed. © 2011 Chinese Physical Society and IOP Publishing Ltd.

AUTHOR KEYWORDS: Bretherton equation; traveling wave; trigonometric function method; variational method

Esfahani, A. Solitary waves for the perturbed nonlinear KleinGordon equation (2011) Applied Mathematics Letters, 24 (2), pp. 204-209.

DOI: 10.1016/j.aml.2010.09.004

In this work, we study the perturbed nonlinear KleinGordon equation. We shall use the sech-anstze method to derive the solitary wave solutions of this equation. © 2010 Elsevier Ltd. All rights reserved.

AUTHOR KEYWORDS: Anstze method; KleinGordon equation; Solitons
INDEX KEYWORDS: Anstze method; Klein-Gordon equation; Nonlinear klein-Gordon equation; Solitary wave; Solitary wave solution, Solitons, Nonlinear equations

Esfahani, A. On the ZK equation with a directional dissipation (2011) Applied Mathematics and Computation, 217 (10), pp. 4911-4927.

DOI: 10.1016/j.amc.2010.11.039

This paper studies the generalized Zakharov-Kuznetsov-Burgers equation. The initial value problem associated to this equation will be investigated in the nonhomogeneous Sobolev spaces and some suitable weighted spaces, under appropriate conditions. Moreover, an ill-posedness result (in some sense) will be proved in the anisotropic Sobolev spaces. Furthermore some exact traveling wave solutions of this equation will be obtained. © 2010 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Initial value problem; Traveling wave solutions; ZKB equation
INDEX KEYWORDS: Burgers equations; Exact traveling waves; Ill-posedness; Non-homogeneous; Traveling wave solution; Weighted space; ZKB equation, Differential equations; Initial value problems, Sobolev spaces

Esfahani, A., Pastor, A. Ill-posedness results for the (generalized) benjamin-ono-zakharov-kuznetsov equation (2011) Proceedings of the American Mathematical Society, 139 (3), pp. 943-956.

DOI: 10.1090/S0002-9939-2010-10532-4

Here we consider results concerning ill-posedness for the Cauchy problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation, namely, (IVP) { ut - Huxx + uxyy + ukux = 0, (x, y) ε ℝ2, t ε ℝ+, u(x, y, 0) = Φ(x, y). For k = 1, (IVP) is shown to be ill-posed in the class of anisotropic Sobolev spaces Hs1,s2 (ℝ2), s1, s 2 ε ℝ, while for k ≥ 2 ill-posedness is shown to hold in Hs1,s2 (ℝ2), 2s1 + s2 < 3/2 - 2/k. Furthermore, for k = 2, 3, and some particular values of s1, s2, a stronger result is also established. © 2010 American Mathematical Society.


AUTHOR KEYWORDS: Cauchy problem; Ill-posedness; Nonlinear PDE
PUBLISHER: American Mathematical Society

Esfahani, A. Decay properties of the traveling waves of the rotation-generalized Kadomtsev-Petviashvili equation (2010) Journal of Physics A: Mathematical and Theoretical, 43 (39), art. no. 395201, .

DOI: 10.1088/1751-8113/43/39/395201

In this paper, we study the decay properties of the traveling-wave solutions of the rotation-generalized Kadomtsev-Petviashvili equation. © 2010 IOP Publishing Ltd.

Esfahani, A. On the generalized Kadomtsev-Petviashvili equation with generalized evolution and variable coefficients (2010) Physics Letters, Section A: General, Atomic and Solid State Physics, 374 (35), pp. 3635-3645.

DOI: 10.1016/j.physleta.2010.07.015

In this Letter, the existence of the solitary wave solution of the Kadomtsev-Petviashvili equation with generalized evolution and time-dependent coefficients will be studied. We use the solitary wave ansätze-method to derive these solutions. A couple of conserved quantities are also computed. Moreover, some figures are plotted to see the effects of the coefficient functions on the propagation and asymptotic characteristics of the solitary waves. © 2010 Elsevier B.V. All rights reserved.


AUTHOR KEYWORDS: Ansätze method; Conservation laws; Kadomtsev-Petviashvili equation; Traveling wave

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