Solitons in Mathematics and Physics

Newell, Alan C.

doi:10.1137/1.9781611970227

Cited by 1,044 publications

(921 citation statements)

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“…(II.15) corresponds to the first harmonic, and all other harmonics are small with respect to the parameter λ. This is the condition under which the nonlinear Schrödinger equation is derived (see, for example, [15,24,25]). In this case, at leading order in λ, we obtain the stationary NLSE (compare with [15,10,26])…”

Section: Basic Equationsmentioning

confidence: 99%

Deep-water internal solitary waves near critical density ratio

Агафонцев

Dias

Кузнецов

2007

Physica D: Nonlinear Phenomena

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Bifurcations of solitary waves propagating along the interface between two ideal fluids are considered. The study is based on a Hamiltonian approach. It concentrates on values of the density ratio close to a critical one, where the supercritical bifurcation changes to the subcritical one. As the solitary wave velocity approaches the minimum phase velocity of linear interfacial waves (the bifurcation point), the solitary wave solutions transform into envelope solitons. In order to describe their behavior and bifurcations, a generalized nonlinear Schrödinger equation describing the behavior of solitons and their bifurcations is derived. In comparison with the classical NLS equation this equation takes into account three additional nonlinear terms: the so-called Lifshitz term responsible for pulse steepening, a nonlocal term analogous to that first found by Dysthe for gravity waves and the six-wave interaction term. We study both analytically and numerically two solitary wave families of this equation for values of the density ratio ρ that are both above and below the critical density ratio ρcr. For ρ > ρcr, the soliton solution can be found explicitly at the bifurcation point. The maximum amplitude of such a soliton is proportional to √ ρ − ρcr, and at large distances the soliton amplitude decays algebraically. A stability analysis shows that solitons below the critical ratio are stable in the Lyapunov sense in the wide range of soliton parameters. Above the critical density ratio solitons are shown to be unstable with respect to finite perturbations.

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Section: Basic Equationsmentioning

confidence: 99%

Deep-water internal solitary waves near critical density ratio

Агафонцев

Dias

Кузнецов

2007

Physica D: Nonlinear Phenomena

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“…they express the nonlinear transform between the potential Φ and the physical fields v and b; they are reminiscent of the logarithmic Hopf-Cole transform 19) which maps the Burgers equation into a linear problem, and of similar transforms appearing in the calculation of multisoliton solutions to various integrable equations; in the latter context, the nonlinear potential Φ is called a τ function (Newell 1985). The difference is that the residues are only partially constrained by the SR. Other differences will appear next.…”

Section: The Complete Truncation: a Poor τ Function?mentioning

confidence: 99%

Meromorphy and topology of localized solutions in the Thomas–MHD model

Fournier

Galtier

2001

J. Plasma Phys.

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Abstract. The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245(1968] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas.

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“…Isospectral solutions and solitons. We briefly reinterpret the calculation [12] of the KdV-hierarchy in order to point out some general aspects. The original problem reads: let v xx + (λ + q(x, t))v = 0 (λ ∈ R, v = v(x, t)) (6.4) be the eigenvalue problem depending on a parameter t and our task is to determine such evolution equations v t = P (P = A(λ; ··, q r , ··)v + B(λ; ··, q r , ··)v x ) (q r = ∂ r q ∂x r ) (6.5) that the compatibility conditions of the system (6.4, 6.5) are of the special kind q t = Q(··, q r , ··).…”

Section: Further Perspectivesmentioning

confidence: 99%

On the internal approach to differential equations 3. Infinitesimal symmetries

Chrastinová

Tryhuk

2016

Mathematica Slovaca

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Abstract. The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher-order symmetries. The classical Lie's approach is appropriately adapted. PrefaceIf the invertible higher-order transformations of differential equations are accepted as a reasonable subject, the common Lie-Cartan's methods are insufficient for complete solution of the symmetry problem. We recall that even the structure of all higher-order symmetries of the trivial (empty) systems of differential equations (that is, of the infinite-order jet spaces without any differential constraits) is unknown [1,2,3]. The same can be said for the "linearized theory" of the higher-order infinitesimal transformations treated in this article.Let us outline the core of the subject. We start with surfaceslying in the space R m+n with coordinates x 1 , . . . , x n , w 1 , . . . , w m . The higherorder transformations are defined by formulaē, ··) (i, i ′ = 1, . . . , n; j, j ′ = 1, . . . , m) (1.1)2010 Mathematics Subject Classification. 54A17, 58J99, 35A30.

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Solitons in Mathematics and Physics

Cited by 1,044 publications

References 0 publications

Deep-water internal solitary waves near critical density ratio

Deep-water internal solitary waves near critical density ratio

Meromorphy and topology of localized solutions in the Thomas–MHD model

On the internal approach to differential equations 3. Infinitesimal symmetries

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