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We remark that this polynomial, in scaled coordinates, is given by Gorshkov et al. From Fig. By considering when. In this article, we have derived a sequence of algebraically decaying rational solutions of the Boussinesq equation 1. The derivation of a representation of these special polynomials as determinants is currently under investigation and we do not pursue this further here.
We remark that other types of exact solutions of the Boussinesq equation 1. This is a fundamentally different hierarchy of solutions of the KPI equation 1. We also thank the reviewers for their helpful comments. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume 1. Article Contents. Rational solutions of the focusing NLS equation.
The Boussinesq equation. Rational solutions of the KP I equation. Rational solutions of the Boussinesq equation and applications to rogue waves Peter A Clarkson. Oxford Academic. Google Scholar. Ellen Dowie. Cite Citation. Permissions Icon Permissions.
Abstract We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. The average height of rogue waves is at least twice the height of the surrounding waves, are very unpredictable and so they can be quite unexpected and mysterious.
In recent years, the concept of rogue waves has been extended beyond oceanic waves: to pulses emerging from optical fibres Solli et al. There has been considerable interest in partial differential equations solvable by inverse scattering, the soliton equations , since the discovery by Gardner et al. In Section 2 , we discuss rational solutions of the focusing NLS equation 1.
In Section 3 , we discuss rational solutions of the Boussinesq equation 1. Further the generalized rational solutions have an interesting structure as they are comprised of a linear combination of four independent solutions of an associated bilinear equation. Rational solutions of the focusing NLS equation 1.
In this paper we consider uniqueness and multiplicity results for single-peak solutions for the nonlinear Schrödinger equation. For a suitable class of potentials V. [1] Ambrosetti A., Badiale M., Cingolani S., Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal. () | MR
The first two rational solutions of the focusing NLS equation 1. Dubard et al. A6 and eq. Furthermore, taking q[1],r[1], a0 and d0 into eq.
We need to parameterize T1 by the eigenfunctions associated with Xi. This purpose can be realized through a system of equations defined by its kernel, i. Solving this system of algebraic equations on a1, d1, bo, co , eq. Next, substituting a1, d1, b0, c0 into eq.
B8 , new solutions q[1] and r[1] are given as eq. Further, by using explicit matrix representation eq. Last, a tedious calculation verifies that T1 in eq. A5 in Appendix A. In the process of verification, it is crucial to use the fact that satisfies eq. So WKI.
Therefore T1 is the DT of eq. Kharif and E. Pelinovsky, "Physical mechanisms of the rogue wave phenomenon," Eur. B Fluids 22, Solli, C.
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