On asymptotic behavior of solutions to Korteweg–de Vries type equations related to vortex filament with axial flow

Segata, Jun Ichi

doi:10.1016/j.jde.2008.03.031

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…Furthermore, our lines of approach are justified by the important outcomes of Kato's theory [40][41][42][43], that the blow-up occurs in any H k ,  k 2-norm, if it occurs at all; see also [14]. The global well-posedness of the Cauchy problem for ENLS type equations in  H k ( ),  k 2 has been addressed in [50], with most recent results those of [39,60] for the global well posedness for   k 1 2. We remark that the importance of the rigorous justification of conservation laws and energy equations has been also underlined in [5][6][7], and in [56] which considers the conservation laws for the Cauchy problem of the higher dimensional NLS equation.…”

Section: Structure Of the Presentation And Main Findingsmentioning

confidence: 99%

Conservation laws, exact traveling waves and modulation instability for an extended nonlinear Schrödinger equation

Achilleos¹,

Diamantidis²,

Frantzeskakis³

et al. 2015

J. Phys. A: Math. Theor.

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We study various properties of solutions of an extended nonlinear Schrödinger (ENLS) equation, which arises in the context of geometric evolution problems -including vortex filament dynamicsand governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak H 2 (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis. 1991 Mathematics Subject Classification. 35Q53, 35Q55, 35B45, 35B65, 37K40. 1 Sec. 3]. Briefly speaking, this derivation begins from a natural generalization of the localized induction equation (LIE) which governs the velocity of the vortex filament. The generalization takes account of the axial-flow effect up to the second-order (see [27, Eqs. (3.1)-(3.2)]). Then, the Hirota equation is derived by repeating the original Hasimoto's procedure [33], which proved the equivalence between LIE and the integrable NLS equation iφ t + φ xx + 1 2 |φ| 2 φ = 0 (see also [47,48]). This equivalence implies that LIE is completely integrable. Thus, since the generalized LIE of [27] preserves integrability, the equivalent evolution equation which is in this case ENLS equation (1.1), should be also integrable. This is the reason why condition (1.2) is essential for the association of Eq. (1.1) with the vortex filament dynamics.On the other hand, non-integrable versions of (1.1), i.e., when the balance condition (1.2) is violated, can be derived in the context of geometric evolution equations. In particular, it was shown in [53] that certain differential equations of one-dimensional dispersive flows into compact Riemann surfaces, may be reduced by a definition of a generalized Hasimoto transform, to ENLS type equations. See also [54,55], and references therein.Another important physical context, where certain versions of the ENLS equation have important applications, is that of nonlinear optics: such models have been used to describe femtosecond pulse pr...

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Section: Structure Of the Presentation And Main Findingsmentioning

confidence: 99%

Conservation laws, exact traveling waves and modulation instability for an extended nonlinear Schrödinger equation

Achilleos¹,

Diamantidis²,

Frantzeskakis³

et al. 2015

J. Phys. A: Math. Theor.

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“…Onodera [5,6] proved the unique solvability of the Cauchy problem for a geometrically generalized equation. Segata [8] proved the unique solvability and showed the asymptotic behavior in time of the solution to the Hirota equation, given by iq t = q xx + 1 2 |q| 2 q + iα − q xxx + |q| 2 q x , which can be obtained by applying the Hasimoto transformation to the vortex filament equation. Although there are many literatures regarding Schrödinger type equations, for (1.4), the boundary condition does not transfer into a form that is manageable, so we decided to work with the vortex filament equation directly.…”

Section: Introductionmentioning

confidence: 98%

Solvability of an Initial-Boundary Value Problem for a Second Order Parabolic System with a Third Order Dispersion Term

Aiki¹,

Iguchi²

2012

SIAM J. Math. Anal.

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We consider a linear second order parabolic system with a third order dispersion term. This type of system arises when considering a nonlinear model equation describing the motion of a vortex filament with axial flow immersed in an incompressible and inviscid fluid. We prove the solvability of an initial-boundary value problem of the parabolic-dispersive system which allows application to the motion of a vortex filament. To do so, we propose a new regularization technique by adding a space-time derivative term.

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Section: Introductionmentioning

confidence: 99%

“…Onodera [8,9] proved the unique solvability for a geometrically generalized equation. Segata [11] proved the unique solvability and showed the asymptotic behavior in time of the solution to the Hirota equation, given by iq t = q xx + 1 2 |q| 2 q + iα q xxx + |q| 2 q x , (1.3) which can be obtained by applying the generalized Hasimoto transformation to the vortex filament equation. Since there are many results regarding the Cauchy problem for the Hirota equation and other Schrödinger type equations, it may feel more natural to see if the available theories from these results can be utilized to solve the initial-boundary value problem for (1.3), instead of considering (1.1) and (1.2) directly.…”

Section: Introductionmentioning

confidence: 99%

Motion of a vortex filament with axial flow in the half space

Aiki

Iguchi

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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On asymptotic behavior of solutions to Korteweg–de Vries type equations related to vortex filament with axial flow

Cited by 3 publications

References 22 publications

Conservation laws, exact traveling waves and modulation instability for an extended nonlinear Schrödinger equation

Conservation laws, exact traveling waves and modulation instability for an extended nonlinear Schrödinger equation

Solvability of an Initial-Boundary Value Problem for a Second Order Parabolic System with a Third Order Dispersion Term

Motion of a vortex filament with axial flow in the half space

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