44 research outputs found
On asymptotically equivalent shallow water wave equations
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
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 -soliton solution in determinant form.
Additionally, we prove the fractional one-soliton solution rigorously. Notably,
we demonstrate that as , the fractional -soliton solution can
be expressed as a linear combination of fractional single-soliton
solutions
A direct method for solving the generalized sine-Gordon equation II
The generalized sine-Gordon (sG) equation
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 . Here, we address the equation with . 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
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