The higher grading structure of the WKI hierarchy and the two-component short pulse equation

A higher grading affine algebraic construction of integrable hierarchies, containing the Wadati-Konno-Ichikawa (WKI) hierarchy as a particular case, is proposed. We show that a two-component generalization of the Sch\" afer-Wayne short pulse equation arises quite naturally from the first negative flow of the WKI hierarchy. Some novel integrable nonautonomous models are also proposed. The conserved charges, both local and nonlocal, are obtained from the Riccati form of the spectral problem. The loop-soliton solutions of the WKI hierarchy are systematically constructed through gauge followed by reciprocal B\" acklund transformation, establishing the precise connection between the whole WKI and AKNS hierarchies. The connection between the short pulse equation with the sine-Gordon model is extended to a correspondence between the two-component short pulse equation and the Lund-Regge model

Complex solitons with real energies

Using Hirota’s direct method and Bäcklund transformations we construct explicit complex one and two-soliton solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT -symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT -symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota’s method break the PT -symmetric, whereas those obtained from Bäcklund transformations are PT -invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT -symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT -symmetrizable one-soliton solutions

Rogue wave solutions for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism

In this paper, generalized Darboux transformation for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism are constructed according to which rouge wave solutions of the equation are obtained. Influences of equation parameter on the evolution of rogue waves are discussed. With the aid of Mathematica, some special solutions are graphically illustrated which could help to better understand the evolution of rogue waves

On the conservation laws and invariant solutions of the mKdV equation

AbstractIn this paper, we consider modified Korteweg–de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation

Application of (G/G') -expansion method to the compound Kdv-burgers type equations

In this Letter, the (G'/G)-expansion method is proposed to seek exact solutions of nonlinear evolution equations. For illustrative examples, we choose the compound KdV-Burgers equation, the compound KdV equation, the KdV-Burgers equation, the mKdV equation. The power of the employed method is confirmed

Geometrical interpretations of Bäcklund transformations and certain types of partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Page 37 is missing from the original copy.Gauss' Theorema Egregium contains a partial differential equation relating the Gaussian curvature K to components of the metric tensor and its derivatives. Well known partial differential equations such as the Schrödinger equation and the sine-Gordon equation correspond to this PDE for special choices of K and special coördinate systems. The sine-Gordon equation, for example, can be derived via Gauss' equation for K = –1 using the Tchebychef net as a coördinate system. In this thesis we consider a special class of Bäcklund Transformations which correspond to coördinate transformations on surfaces having a specified Gaussian curvature. These transformations lead to Gauss' PDE in different forms and provide a method for solving certain classes of non-linear second order partial differential equations. In addition, we develop a more systematic way to obtain a coordinate system for a more general class of PDE, such that this PDE corresponds to the Gauss equation

Solution Proliferation of the Nonlinear Integrable Systems and Riccati Equation

According to the scheme of the Riemann-Hilbert transform, we first introduce a simple but local transformation by which the solution of nonlinear integrable equations are produced recursively ("proliferation"). The key of this step is to introduce a projection matrix consiting of solutions for a linear decoupling set correspgnding to the nonlinear equation. We next fomd that the relation between the projection matrix and solutions of the Riccati equation results in a fractional transformation of the problem. By this relation we show a sequence of proliferations constructed uniquely with quite algebraic manners, because two kinds of fractional transformations commute each other.リーマンヒルベルト変換の考え方に沿って，まず我々は簡単しかしローカルな変換を構成し，それによって非線形可積分系の解の逐次的構成法を与える。この方法の鍵は，解くべき非線形方程式に同値な線形連立系の解からなる射影行列を導くことである。次に我々は，射影行列とRiccati方程式の解の関係から，問題を分数変換に帰着させる。2個の分数変換は互いに可換である事実から，逐次的な増殖も全く代数的な計算で求まる事が示される

An equation admitting infinite true contact transformations

AbstractThe linear wave equation is shown to possess the unique property that if wn is a true contact transformation admitted by the wave equation, i.e., wn is not linear in the first derivatives of the dependent variable, then so is ∑nwn. We comment of the physical implications