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409
Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation
 J. Comput. Phys
, 1984
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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 tha ..."
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Cited by 58 (18 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.
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 58 (22 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.
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 53 (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 52 (8 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...
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 39 (9 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
Internal Solitary Waves
, 2000
"... The basic theory of internal solitary waves is developed, with the main emphasis on environmental situations, such as the many occurrences of suchwaves in shallow coastal seas and in the atmospheric boundary layer. Commencing with the equations of motion for an inviscid, incompressible densitystrat ..."
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Cited by 30 (19 self)
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The basic theory of internal solitary waves is developed, with the main emphasis on environmental situations, such as the many occurrences of suchwaves in shallow coastal seas and in the atmospheric boundary layer. Commencing with the equations of motion for an inviscid, incompressible densitystratified fluid, we describe asymptotic reductions to model longwave equations, such as the wellknown Kortewegde Vries equation. We then describe various solitary wave solutions, and propose a variablecoefficient modified Kortewegde Vries equations as an appropriate evolution equation to describe internal solitary waves in environmental situations, when the effects of a variable background and dissipation need to be taken into account. 1. INTRODUCTION Solitary waves are finiteamplitude waves of permanent form which owe their existence to a balance between nonlinear wavesteepening processes and linear wave dispersion. Typically, they consist of a single isolated wave of elevation, or depr...
Quasilinear stokes phenomenon for the Painlevé first equation
 J. Phys. A: Math. Gen
"... Abstract. Using the RiemannHilbert approach, the Ψfunction corresponding to the solution of the first Painlevé equation yxx = 6y 2 + x with the asymptotic behavior y ∼ ± √ −x/6 as x  → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in ..."
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Cited by 27 (1 self)
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Abstract. Using the RiemannHilbert approach, the Ψfunction corresponding to the solution of the first Painlevé equation yxx = 6y 2 + x with the asymptotic behavior y ∼ ± √ −x/6 as x  → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the powerlike expansion to the latter are found. The Painlevé first equation [1] 1.
Bethe ansatz and inverse scattering transform in a periodic boxball system
 Nucl. Phys. B [PM
"... Abstract. We formulate the inverse scattering method for a periodic boxball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansätze at q = 1 and q = 0, which provides the ultradiscrete analogue of quasiperiodic solutions in soliton equations, e.g., a ..."
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Cited by 26 (10 self)
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Abstract. We formulate the inverse scattering method for a periodic boxball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansätze at q = 1 and q = 0, which provides the ultradiscrete analogue of quasiperiodic solutions in soliton equations, e.g., actionangle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions. 1.