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204
Analytical and numerical aspects of certain nonlinear evolution equations
 Journal of Computational Physics
, 1984
"... Various numerical methods are used in order to approximate the Kortewegde Vries equation, namely: (i) ZabuskyKruskal scheme, (ii) hopscotch method, (iii) a scheme due to Goda, (iv) a proposed local scheme, (v) a proposed global scheme, (vi) a scheme suggested by Kruskal, (vii) split step Fourier m ..."
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Cited by 73 (3 self)
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Various numerical methods are used in order to approximate the Kortewegde Vries equation, namely: (i) ZabuskyKruskal scheme, (ii) hopscotch method, (iii) a scheme due to Goda, (iv) a proposed local scheme, (v) a proposed global scheme, (vi) a scheme suggested by Kruskal, (vii) split step Fourier method by Tappert, (viii) an improved split step Fourier method, and (ix) pseudospectral method by Fornberg and Whitham. Comparisons between our proposed scheme, which is developed using notions of the inverse scattering transform, and the other utilized schemes are obtained. 1.
Multidimensional quadrilateral lattices are integrable, Phys. Lett. A 233
, 1997
"... The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in part ..."
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Cited by 47 (15 self)
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The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial–boundary data which define the lattice uniquely.
A refined global wellposedness result for Schrödinger equations with derivative
 SIAM J. Math. Anal
, 2002
"... Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense that unif ..."
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Cited by 39 (16 self)
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Abstract. In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally wellposed in H s, for s> 1 2 for data small in L2. To understand the strength of this result one should recall that for s < 1 the Cauchy problem is illposed, in the 2 sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the “Imethod ” used by the same authors. The same argument can be used to prove that any to obtain global wellposedness for s> 2 3 quintic nonlinear defocusing Schrödinger equation on the line is globally wellposed for large data in H s, for s> 1 2. 1.
Existence and nonexistence of solitary wave solutions to highorder model evolution equations
 SIAM J. Math. Anal
"... Abstract. The problem of existence of solitary wave solutions to some higherorder model evolution equations arising from water wave theory is discussed. A simple direct method for finding monotone solitary wave solutions is introduced, and by exhibiting explicit necessary and sufficient conditions, ..."
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Cited by 31 (3 self)
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Abstract. The problem of existence of solitary wave solutions to some higherorder model evolution equations arising from water wave theory is discussed. A simple direct method for finding monotone solitary wave solutions is introduced, and by exhibiting explicit necessary and sufficient conditions, it is illustrated that a model admit exact sech solitary wave solutions. Moreover, it is proven that the only fifthorder perturbations of the KortewegdeVries equation that admit solitary wave solutions reducing to the usual onesoliton solutions in the limit are those admitting families of explicit sech solutions. Key words, solitary wave, nonlinear evolution equation, water waves, singular perturbation AMS(MOS) subject classifications. 76B25, 35Q51, 35Q53, 35B25, 76B15
Homoclinic Orbits in Reversible Systems and Their Applications in Mechanics, Fluids and Optics
, 1997
"... This survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuoustime dynamical systems, where the homoclinic orbit and the equilibrium are both reversible. The focus is on evenorder reversible systems in four or more dimensions. Local theory, gener ..."
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Cited by 27 (3 self)
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This survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuoustime dynamical systems, where the homoclinic orbit and the equilibrium are both reversible. The focus is on evenorder reversible systems in four or more dimensions. Local theory, generic argument, and global existence theories are examined for each qualitatively distinct linearisation. Several recent results, such as coalescence caused by nontransversality and the reversible orbitflip bifurcation are covered. A number of open problems are highlighted. Applications are reviewed to systems arising from a variety of disciplines. With the aid of numerical methods, three examples are presented in detail, one of which is infinite dimensional. 1 Introduction Classical Hamiltonian dynamical systems with quadratic kinetic energy are reversible in the sense that they are invariant under a reversal of time and all momentum variables. The concept of reversible systems in their ow...
An rMatrix Approach to Nonstandard Classes of Integrable Equations
 Publ. RIMS, Kyoto Univ
, 1993
"... Three different decompositions of the algebra of pseudodifferential operators and the corresponding rmatrices are considered. Three associated classes of nonlinear integrable equations in 1+1 and 2+1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. Th ..."
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Cited by 24 (4 self)
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Three different decompositions of the algebra of pseudodifferential operators and the corresponding rmatrices are considered. Three associated classes of nonlinear integrable equations in 1+1 and 2+1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2+1dimensional hierarchies are associated with the KadomtsevPetviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2+1dimensional equations as well as to a variety of integrable equations in 1+1 dimensions. It is shown, how the multiHamiltonian structure of the 1+1dimensional equations can be obtained from the underlying rmatrices. Further, intimate relations between the equations associated with the three different rmatrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connect...
Perturbative Symmetry Approach
 Journal of Physics A, Vol
, 2002
"... In the Symmetry Approach the existence of infinite hierarchies of higher symmetries and/or local conservation laws is taken as a definition of integrability. The main aims of the approach are to obtain easily verifiable necessary conditions of integrability, to identify integrable cases and even ..."
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Cited by 22 (6 self)
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In the Symmetry Approach the existence of infinite hierarchies of higher symmetries and/or local conservation laws is taken as a definition of integrability. The main aims of the approach are to obtain easily verifiable necessary conditions of integrability, to identify integrable cases and even
Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKdV equation
, 1995
"... I. Tori of revolution and the mKDV equation ..."
NonAbelian Integrable Systems of the Derivative Nonlinear Schrödinger Type
 Inverse Problems
, 1998
"... this paper we classify integrable nonabelian generalizations of the integrable systems (3). Namely, we consider the systems of the form ..."
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Cited by 16 (3 self)
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this paper we classify integrable nonabelian generalizations of the integrable systems (3). Namely, we consider the systems of the form