existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

existence of a generalized solitary wave solution for water with positive bond number less than 13 abstract the objective of this paper is to give a rigorous answer to the question whether there exists a generalized solitary wave of elevation on water when the nondimensional surface tension coefficient the so called bond number is positive but less than 1 3 and the froude number is near 1 but greater than 1 the main difficulties to overcome are the small amplitude oscillations at infinity and the choice of appropriate banach spaces for the solution of the problem it is shown that an asymptotic expansion developed for this problem indeed yields an approximate solitary wave solution to the exact equations

a shallow water approximation to the full water wave problem we demonstrate that the system of the green naghdi equations as a two directional nonlinearly dispersive wave model is a close approximation to the two dimensional full water wave problem based on the energy estimates and the proof of the well posedness for the green naghdi equations and the water wave problem we compare solutions of the two systems showing that without restrictions on the wave amplitude any two solutions of the two systems remain close at least in some finite time within the shallow water regime provided that their initial data are close in the banach space hs hs 1 for some s as a consequence we show that if the depth of the water compared with the wavelength is sufficiently small the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper 2006 wiley periodicals inc

spectral synthesis and topologies on ideal spaces for banach algebras this paper continues the study of spectral synthesis and the topologies and r on the ideal space of a banach algebra concentrating on the class of banach algebras and in particular on l1 group algebras it is shown that if a group g is a finite extension of an abelian group then r is hausdorff on the ideal space of l1 g if and only if l1 g has spectral synthesis which in turn is equivalent to g being compact the result is applied to nilpotent groups fd groups and moore groups an example is given of a non compact non abelian group g for which l1 g has spectral synthesis it is also shown that if g is a non discrete group then r is not hausdorff on the ideal lattice of the fourier algebra a g

topological horseshoes in travelling waves of discretized nonlinear wave equations applying the concept of anti integrable limit to coupled map lattices originated from space time discretized nonlinear wave equations we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions in particular the coupled map lattices display spatio temporal chaos on the horseshoes

efficient pseudospectral flow simulations in moderately complex geometries a computationally efficient pseudospectral method is developed for incompressible flow simulations in two dimensional geometries involving periodicity in one direction and significant surface deformations a pseudoconformal mapping is used to map the flow domain into a rectangle thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated the time integration of the spectrally discretized two dimensional momentum equations is performed by a second order mixed explicit implicit time integration scheme the satisfaction of the continuity equation is obtained through the solution of a poisson equation for the pressure and the use of the influence matrix technique a highly efficient iterative solver has been developed for the solution of a generalized stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm which exhibits almost linear scalability requiring an ordernlog2nnumber of operations wherenis the number of unknowns numerical results are presented for two dimensional steady oscillatory and peristaltic flows within an undulating channel which agree well with previous results that have appeared in the literature

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for gravity capillary solitary waves where the first term in the series is the well known sech2 solution of the kdv equation the asymptotic series is used with nine terms included to investigate the effects of surface tension on the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for surface tension below a critical value the solitary wave with the maximum energy is obtained for large surface tension the series is also used to study the energy related to the solitary waves of depression energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom comparisons are also made with recent experiments

two dimensional travelling waves over a moving bottom abstract two dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied the bottom and the wave travel with a same speed the exact euler equations are formulated as a spatial dynamic system by using the stream function a manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions a homoclinic solution to the normal form of this reduced system persists when higher order terms are added which gives a generalized solitary wave the homoclinic solution connecting a periodic solution

the generation of radiating waves in a singularly perturbed korteweg de vries equation abstract we consider a fifth order kdv equation where the fifth order derivative term is multiplied by a small parameter it has been conjectured that this equation admits a non local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core we prove that this solution cannot be exactly steady and instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail the decay rate is calculated in the limit as the parameter tends to zero in order to verify the analytical results we have developed a high precision spectral method for numerical integration of this equation the analytical and numerical result show good agreement

nanopteron solutions of diatomic fermi pasta ulam tsingou lattices with small mass ratio abstract consider an infinite chain of masses each connected to its nearest neighbors by a nonlinear spring this is a fermi pasta ulam tsingou lattice we prove the existence of traveling waves in the setting where the masses alternate in size in particular we address the limit where the mass ratio tends to zero the problem is inherently singular and we find that the traveling waves are not true solitary waves but rather nanopterons which is to say waves which are asymptotic at spatial infinity to very small amplitude periodic waves moreover we can only find solutions when the mass ratio lies in a certain open set the difficulties in the problem all revolve around understanding jost solutions of a nonlocal schrodinger operator in its semi classical limit

orbits homoclinic to exponentially small periodic orbits for a class of reversible systems application to water waves in this paper a class of reversible analytic vector fields is investigated near an equilibrium for these vector fields the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

analytical and numerical studies of a singularly perturbed boussinesq equation we study the singularly perturbed sixth order boussinesq equation recently introduced by daripa and hua appl math comput 101 1999 159 motivated by their work we formally derive this equation from two dimensional potential flow equations governing the small amplitude long capillary gravity waves on the surface of shallow water for bond number very close to but less than 1 3 on the basis of far field analyses and heuristic arguments we show that the traveling wave solutions of this equation are weakly non local solitary waves characterized by small amplitude fast oscillations in the far field we review various analytical and numerical methods originally devised to obtain this type of weakly non local solitary wave solutions of the singularly perturbed fifth order kdv equation using these methods we obtain weakly non local solitary wave solutions of the singularly perturbed sixth order boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far field

three dimensional gravity capillary waves on water small surface tension case abstract this paper considers three dimensional gravity capillary waves on water of finite depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction the exact euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time like variable the existence of the solutions of the system is determined by two non dimensional constants the bond number b and the froude number f which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow assume that f 2 c 1 is the curve in the b plane on which the first two eigenvalues for three dimensional waves collide at the imaginary axis and the intersection point of c 1 with 1 is b 1 0 in this paper the case for 0 b b 1 and b near c 1 is considered a center manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations using the existence of a homoclinic orbit connecting to a two dimensional periodic called generalized solitary wave thereafter solution for the reduced system it is shown that such a generalized solitary wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants

asymptotic expansions for solitary gravity capillary waves in two and three dimensions high order asymptotic series are obtained for two and three dimensional gravity capillary solitary waves in two dimensions the first term in the asymptotic series is the well known sech 2 solution of the korteweg de vries equation in three dimensions the first term is the rational lump solution of the kadomtsev petviashvili equation i the two dimensional series is used with nine terms included to investigate how small surface tension affects the height and energy of large amplitude waves and waves close to the solitary version of stokes extreme wave in particular for small surface tension the solitary wave with the maximum energy is obtained for large surface tension the two dimensional series is also used to study the energy of depression solitary waves energy considerations suggest that for large enough surface tension there are solitary waves that can get close to the fluid bottom in three dimensions analytic solutions for the high order perturbation terms are computed numerically and the resulting asymptotic series to three terms is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension finally the above asymptotic method is applied to the benney luke bl equation and the resulting asymptotic series to three terms is verified to agree with the solitary wave solution of the bl equation

capillary gravity solitary waves with damped oscillations abstract capillary gravity solitary waves with damped oscillations are studied analytically the analysis follows the work of iooss and kirchgassner who proved that these waves exist for all values of the froude number smaller than one the water wave problem is reduced to a system of ordinary differential equations by using the center manifold theorem the normal form of this reduced system can be obtained and a good approximation to these waves for small amplitude is constructed the limit as the water depth becomes infinite is considered as a special case a comparison with existing numerical results is made for small amplitude waves

exact solitary water waves with vorticity the solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field the main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity each solution has a supercritical parameter value and decays exponentially at infinity the proof is based on a generalized implicit function theorem of the nash moser type the first approximation to the surface profile is given by the kdv equation with a supercritical value of the surface tension coefficient a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity