stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

weak geodesic flow on a semidirect product and global solutions to the periodic hunter saxton system abstract we give explicit solutions for the two component hunter saxton system on the unit circle moreover we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re expression of the geodesic flow on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle these spatially and temporally periodic solutions turn out to be conservative

notes on solitary wave solutions of the generalized two component hunter saxton system abstract the goals of this paper are to employ the approach of dynamical system to obtain some new results of the following generalized two component hunter saxton system u t x x 2 u x u x x u u x x x x a u x 0 t u x 0 r a 0 which are not obtained in moon 2013

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

high spatial order finite element method to solve maxwell s equations in time domain this paper presents a finite element method with high spatial order for solving the maxwell equations in the time domain in the first part we provide the mathematical background of the method then we discuss the advantages of the new scheme compared to a classical finite difference time domain fdtd method several examples show the advantages of using the new method for different kinds of problems comparisons in terms of accuracy and cpu time between this method the fdtd and the finite volume time domain methods are given as well

2 parameter tau τ function for the first painleve equation topological recursion and direct monodromy problem via exact wkb analysis we show that a 2 parameter family of tau functions for the first painleve equation can be constructed by the discrete fourier transform of the topological recursion partition function for a family of elliptic curves we also perform an exact wkb theoretic computation of the stokes multipliers of associated isomonodromy system assuming certain conjectures

optimal solution and optimality condition of the hunter saxton equation this paper is devoted to the optimal distributed control problem governed by the hunter saxton equation with constraints on the control we first investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary conditions in contrast with our previous research the proof of solution mapping is local lipschitz continuous which is one big improvement second based on the well posedness result we find a unique optimal control and optimal solution for the controlled system with the quadratic cost functional moreover we establish the sufficient and necessary optimality condition of an optimal control by means of the optimal control theory not limited to the necessary condition which is another major novelty of this paper we also discuss the optimality conditions corresponding to two physical meaningful distributed observation cases

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

on the n 2 supersymmetric camassa holm and hunter saxton equations we consider n 2 supersymmetric extensions of the camassa holm and hunter saxton equations we show that they admit geometric interpretations as euler equations on the superconformal algebra of contact vector fields on the 1 2 dimensional supercircle we use the bi hamiltonian formulation to derive lax pairs moreover we present some simple examples of explicit solutions as a by product of our analysis we obtain a description of the bounded traveling wave solutions for the two component hunter saxton equation

periodic solitons of an equation for short capillary gravity waves we consider the periodic solitons of an equation for short capillary gravity waves although their profiles cannot be given a description in terms of a mathematical expression in closed form we show that they are transformed by a natural change of variables to the well known cnoidal periodic solitons of the kdv equation

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

wave breaking and global existence for the generalized periodic two component hunter saxton system article history in this paper we study the wave breaking phenomena and global existence for the generalized two component hunter saxton sys tem in the periodic setting we first establish local well posedness for the generalized two component hunter saxton system we ob tain a wave breaking criterion for solutions and results of wave breaking solutions with certain initial profiles we also determine the exact blow up rate of strong solutions finally we give as uf fi cient condition for global solutions

solitary wave solutions of the generalized two component hunter saxton system abstract the existence of solitary wave solutions of the generalized two component hunter saxton system is determined it is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

solitary wave solutions for singular non newtonian filtration equations this paper is mainly concerned with the solitary wave solutions for the singular non newtonian filtration equations a sufficient criterion for the existence of solitary wave solutions of the considered equation is provided via an innovative method of lyusternik schnirelmann category theory and variational methods some recent results in the literature have been complemented and improved

generalized hunter saxton equation and the geometry of the group of circle diffeomorphisms we study an equation lying mid way between the periodic hunter saxton and camassa holm equations and which describes evolution of rotators in liquid crystals with external magnetic field and self interaction we prove that it is an euler equation on the diffeomorphism group of the circle corresponding to a natural right invariant sobolev metric we show that the equation is bihamiltonian and admits both cusped and smooth traveling wave solutions which are natural candidates for solitons we also prove that it is locally well posed and establish results on the lifespan of its solutions throughout the paper we argue that despite similarities to the kdv ch and hs equations the new equation manifests several distinctive features that set it apart from the other three