44 research outputs found

    On asymptotically equivalent shallow water wave equations

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    The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution

    The Riemann-Hilbert approach for the integrable fractional Fokas--Lenells equation

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    In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann-Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional NN-soliton solution in determinant form. Additionally, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as t|t|\to\infty, the fractional NN-soliton solution can be expressed as a linear combination of NN fractional single-soliton solutions

    A direct method for solving the generalized sine-Gordon equation II

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    The generalized sine-Gordon (sG) equation utx=(1+νx2)sinuu_{tx}=(1+\nu\partial_x^2)\sin\,u was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with ν=1\nu=-1. Here, we address the equation with ν=1\nu=1. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. Last, we provide a recipe for deriving an infinite number of conservation laws by using a novel B\"acklund transformation connecting solutions of the sG and generalized sG equations.Comment: To appear in J. Phys. A: Math. Theor. The first part of this paper has been published in J. Phys. A: Math. Theor. 43 (2010) 10520

    Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa- Holm equations

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    In this paper, we study peakon, cuspon, and pseudo-peakon solutions for two generalized Camassa-Holm equations. Based on the method of dynamical systems, the two generalized Camassa-Holm equations are shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, pseudo-peakons, and periodic cusp solutions. In particular, the pseudo-peakon solution is for the first time proposed in our paper. Moreover, when a traveling system has a singular straight line and a heteroclinic loop, under some parameter conditions, there must be peaked solitary wave solutions appearing
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