prescribing the center of mass of a multi soliton solution for a perturbed semilinear wave equation abstract we construct a finite time blow up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension given any integer k 2 and 0 r we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times although the result is similar to the unperturbed case in its statement our method is new indeed our perturbed equation is not invariant under the lorentz transform and this requires new ideas in fact the main difficulty in this paper is to prescribe the center of mass 0 r we would like to mention that our method is valid also in the unperturbed case and simplifies the original proof by cote and zaag 9 as far as the center of mass prescription is concerned

numerical computation of 2d sommerfeld integrals a novel asymptotic extraction technique abstract the accurate and efficient computation of the elements in the impedance matrix is a crucial step in the application of galerkin s method to the analysis of planar structures as was demonstrated in a previous paper it is possible to decompose the angular integral in the polar representation for the 2d sommerfeld integrals in terms of incomplete lipschitz hankel integrals ilhis when piecewise sinusoidal basis functions are employed since bessel series expansions can be used to compute these ilhis a numerical integration of the inner angular integral is not required this technique provides an efficient method for the computation of the inner angular integral however the outer semi infinite integral still converges very slowly when a real axis integration is applied therefore it is very difficult to compute the impedance elements accurately and efficiently in this paper it is shown that this problem can be overcome by using the ilhi representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi infinite integral the usefulness of this asymptotic extraction technique is demonstrated by applying it to the analysis of a printed strip dipole antenna in a layered medium

numerical computation of 2d sommerfeld integrals decomposition of the angular integral abstract spectral domain techniques are frequently used in conjunction with galerkin s method to obtain the current distribution on planar structures when this technique is employed a large percentage of the computation time is spent filling the impedance matrix therefore it is important to develop accurate and efficient numerical techniques for the computation of the impedance elements which can be written as two dimensional 2d sommerfeld integrals once the current distribution has been found then the near zone electric field distribution can be obtained by computing another set of 2d sommerfeld integrals the computational efficiency of the 2d sommerfeld integrals can be improved in two ways the first method which is discussed in this paper involves finding a new way to compute the inner angular integral in the polar representation of these integrals it turns out that the angular integral can be decomposed into a finite number of incomplete lipschitz hankel integrals which in turn can be calculated using series expansions therefore the angular integral can be computed by summing a series instead of applying a standard numerical integration algorithm this new technique is found to be more accurate and efficient when piecewise sinusoidal basis functions are used to analyze a printed strip dipole antenna in a layered medium the incomplete lipschitz hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi infinite integral

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

on the profile of energy concentration at blow up points for subconformal focusing nonlinear waves we consider singularities of the focusing subconformal nonlinear wave equation and some generalizations of it at noncharacteristic points on the singularity surface merle and zaag have identified the rate of blow up of the h1 norm of the solution inside cones that terminate at the singularity we derive bounds that restrict how this h1 energy can be distributed inside such cones our proof relies on new localized estimates obtained using carlemantype inequalities for such nonlinear waves these bound the lp 1 norm in the interior of timelike cones by their h1 norm near the boundary of the cones such estimates can also be applied to obtain certain integrated decay estimates for globally regular solutions to such equations in the interior of time cones

openness of the set of non characteristic points and regularity of the blow up curve for the 1 d semilinear wave equation we consider here the 1 d semilinear wave equation with a power nonlinearity and with no restriction on initial data we first prove a liouville theorem for that equation then we consider a blow up solution its blow up curve x mapsto t x and i 0 subset mathbb r the set of non characteristic points we show that i0 is open and that t x is c1 on i0 all these results fundamentally use our previous result in 19 showing the convergence in selfsimilar variables for x in i 0

a novel asymptotic extraction technique for the efficient evaluation of a class of double sommerfeld integrals when the galerkin method of moments mom is used in the numerical analysis of microwave microstrip networks elements of a network impedance matrix are required two dimensional 2d sommerfeld spectral integrals must be evaluated in such cases when semi infinite circuit feed lines are used certain specific types of sommerfeld integrals arise whose integrands oscillate rapidly and present serious numerical difficulties for their evaluation in this paper a novel accurate and much more efficient asymptotic extraction technique aet is developed that involves extracting an inner asymptotic angular integral which is then evaluated analytically before evaluating the outer integral numerically

multi speed solitary waves for the klein gordon schrodinger system with cubic interaction abstract considered in this report is the existence of multi speed solitary waves of a klein gordon schrodinger system with cubic interaction these solutions behave at large time as a couple of single solitary waves at different speeds this type of solutions has been investigated for nonlinear schrodinger system in 15 27 it is obtained by solving the system backward in time around a sequence of approximate multi speed solitary waves and showing convergence to a solution with the desired property the new ingredients of the proof are coercivity of the hessian of the action around each component of the multi speed solitary waves energy estimate and finite speed of propagation

numerical analysis of arbitrarily shaped probe excited single arm printed wire antennas a general integral equation technique is described for analysis of an arbitrarily shaped single arm printed wire antenna excited through a vertical probe a unified current integral equation is formulated on the basis of dyadic green s functions and the reciprocity theorem the current distribution is obtained by using a parametric moment method in which parameter segments are adopted for the printed wire instead of the commonly employed wire length segments the radiation field solution involving both the printed antenna and vertical probe is also presented the validity of the formulation is verified by comparing the numerically obtained input impedance and radiation patterns for a linear antenna and a meander antenna with measured data a circular open loop and an archimedian spiral are investigated to illustrate the applicability of the present technique

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

construction of solutions to the subcritical gkdv equations with a given asymptotical behavior we consider the subcritical generalized korteweg de vries equation ut uxx u4 x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v be in an adequate weighted sobolev space we construct a solution u t to the generalized korteweg de vries equation such that limt u t u t v j 1nrj t h1 0

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

a simple numerical solution for an optimal investment strategy for a dc pension plan in a jump diffusion model abstract in this paper we study an optimal investment strategy for a fund manager of a dc pension who wants to maximise the expected exponential utility of the terminal wealth in a market where the stock is a jump diffusion process using stochastic control theory we derive a hamilton jacobi bellman equation since the market is not complete the optimal strategy cannot be found in closed as is done in most literature on dc pension plans we characterise the optimal strategy as a solution of an integro ordinary differential equation which can easily be solved by a simple numerical method we investigate the impact of different jump parameters through numerical experiments using a familiar distribution of jumps

the blow up boundary for nonlinear wave equations consider the cauchy problem for a nonlinear wave equation du f u in n space dimensions n x with x continuously difierentiable introduction consider the nonlinear wave equation 0 1 uu u t au f u for x 6 r i 0 with the initial data 0 2 u x 0 x ul x 0 g x for x g rn it is well known that if f u is nonnegative and superlinear then in general a solution cannot exist for all times furthermore if t is the supremum of all times s such that a classical solution exists for all 0 t s then sup w jc t oo if t t

blowup solutions for a reaction diffusion system with exponential nonlinearities abstract we consider the following parabolic system whose nonlinearity has no gradient structure t u u e p v t v v e q u u 0 u 0 v 0 v 0 p q 0 in the whole space r n we show the existence of a stable blowup solution and obtain a complete description of its singularity formation the construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude in particular our analysis uses neither the maximum principle nor the classical methods based on energy type estimates which are not supported in this system the stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem

existence and classification of characteristic points at blow up for a semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension we first show the existence of a blow up solution with a characteristic point then we consider an arbitrary blow up solution u x t the graph x mapsto t x of its blow up points and scr s subset bbb r the set of all characteristic points and show that scr s has an empty interior finally given x 0 in scr s we show that in selfsimilar variables the solution decomposes into a decoupled sum of at least two solitons with alternate signs and that t x forms a corner of angle pi over 2 at x 0

multi travelling waves for the nonlinear klein gordon equation for the nonlinear klein gordon equation in r 1 d we prove the existence of multi solitary waves made of any number n of decoupled bound states this extends the work of cote and munoz forum math sigma 2 2014 which was restricted to ground states as were most previous similar results for other nonlinear dispersive and wave models

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

numerical computation of the incomplete lipschitz hankel integral je o a z abstract two factorial neumann series expansions are derived for the incomplete lipschitz hankel integral je 0 a z these expansions are used together with the neumann series expansion given by agrest in an algorithm which efficiently computes je 0 a z to a user defined number of significant digits other expansions for je 0 a z which are found in the literature are also discussed but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm

optimal investment policies for a firm with a random risk process exponential utility and minimizing the probability of ruin we consider a firm that is faced with an uncontrollable stochastic cash flow or random risk process there is one investment opportunity a risky stock and we study the optimal investment decision for such firms there is a fundamental incompleteness in the market in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy since the random risk process ensures that there is always a positive probability of ruin bankruptcy we therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin in particular we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level this policy also minimizes the probability of ruin we prove that when there is no risk free interest rate this policy is equivalent to the policy that maximizes utility from terminal wealth for a fixed terminal time when the firm has an exponential utility function this validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility when there is a positive risk free interest rate the conjecture is shown to be false we also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin

a blowup solution of a complex semi linear heat equation with an irrational power abstract in this paper we consider the following semi linear complex heat equation t u u u p u c in r n with an arbitrary power p p 1 we construct for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a moreover we also describe the asymptotics of the solution by the following final profiles u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a 0 as x a in addition to that since we also have u 1 0 t and u 2 0 t as t t the blowup in the imaginary part shows a new phenomenon unknown for the standard heat equation in the real case a non constant sign near the singularity with the existence of a vanishing surface for the imaginary part shrinking to the origin in our work we have succeeded to extend for any power p where the non linear term u p is not continuous if p is not integer in particular the solution which we have constructed has a positive real part we study our equation as a system of the real part and the imaginary part u 1 and u 2 our work relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term abstract we consider the following exponential reaction diffusion equation involving a nonlinear gradient term t u u u 2 e u x t r n 0 t 1 we construct for this equation a solution which blows up in finite time t 0 and satisfies some prescribed asymptotic behavior we also show that the constructed solution and its gradient blow up in finite time t simultaneously at the origin and find precisely a description of its final blowup profile it happens that the quadratic gradient term is critical in some sense resulting in the change of the final blowup profile in comparison with the case 0 the proof of the construction is inspired by the method of merle and zaag in 1997 it relies on the reduction of the problem to a finite dimensional one and uses the index theory to conclude one of the major difficulties arising in the proof is that outside the blowup region the spectrum of the linearized operator around the profile can never be made negative truly new ideas are needed to achieve the control of the outer part of the solution thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point we obtain the stability of the constructed solution with respect to perturbations in the initial data

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

pathwise stochastic optimal control this paper approaches optimal control problems for discrete time controlled markov processes by representing the value of the problem in a dual lagrangian form the value is expressed as an infimum over a family of lagrangian martingales of an expectation of a pathwise supremum of the objective adjusted by the lagrangian martingale term this representation opens up the possibility of numerical methods based on monte carlo simulation which may be advantageous in high dimensional problems or in problems with complicated constraints

asymptotic n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

construction of a blow up solution for the complex ginzburg landau equation in a critical case we construct a solution for the complex ginzburg landau equation in some critical case which blows up in finite time t only at one blow up point we also give a sharp description of its profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude the interpretation of the parameters of the finite dimension problem in terms of the blow up point and time allows to prove the stability of the constructed solution

the blow up rate for a non scaling invariant semilinear wave equations abstract we consider the semilinear wave equation 1 t 2 u u f u x t r n 0 t with f u u p 1 u log a 2 u 2 where p 1 and a r we show an upper bound for any blow up solution of 1 then in the one space dimensional case using this estimate and the logarithmic property we prove that the exact blow up rate of any singular solution of 1 is given by the ode solution associated with 1 namely u u p 1 u log a 2 u 2 unlike the pure power case g u u p 1 u the difficulties here are due to the fact that equation 1 is not scale invariant

construction of solutions to the l 2 critical kdv equation with a given asymptotic behaviour we consider the critical korteweg de vries equation ut uxx u x 0 t x r let rj t x qcj x xj cjt j 1 n be n soliton solutions to this equation denote u t the kdv linear group and let v h be with sufficient decay on the right that is 1 x0 v l for some 0 0 we construct a solution u t to the critical korteweg de vries equation such that lim t u t u t v n x j 1 rj t h1 0

construction of a stable periodic solution to a semilinear heat equation with a prescribed profile we construct a periodic solution to the semilinear heat equation with power nonlinearity in one space dimension which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude thanks to the geometrical interpretation of the finite dimensional parameters in terms of the blow up time and blow up point we derive the stability of the constructed solution with respect to initial data

existence and universality of the blow up profile for the semilinear wave equation in one space dimension in this paper we consider the semilinear wave equation with a power nonlinearity in one space dimension we exhibit a universal one parameter family of functions which stand for the blow up profile in self similar variables at a non characteristic point for general initial data the proof is done in self similar variables we first characterize all the solutions of the associated stationary problem as a one parameter family then we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm in the non characteristic case and to a finite decoupled sum of such a solution in the characteristic case finally in the case where this sum is reduced to one element which is the case for noncharacteristic points we use modulation theory coupled with a nonlinear argument to show the exponential convergence in the self similar time variable of the various parameters and conclude the proof this step provides us with a result of independent interest the trapping of the solution in self similar variables near the set of stationary solutions valid also for non characteristic points the proof of these results is based on a new analysis in the self similar variable 2007 elsevier inc all rights reserved

stable blow up dynamics for energy supercritical wave equations we study the semilinear wave equation displaystyle partial t 2 psi delta psi vert psi vert p 1 psi for p 3 with radial data in three spatial dimensions there exists an explicit solution which blows up at t t 0 given by displaystyle psi t t x c p t t frac 2 p 1 where c p is a suitable constant we prove that the blow up described by psi t is stable in the sense that there exists an open set in a topology strictly stronger than the energy of radial initial data that leads to a solution which converges to psi t as t to t in the backward lightcone of the blow up point t r t 0

pricing pension plans under jump diffusion models for the salary in this paper we consider the valuation of a defined benefit pension plan in the presence of jumps in the underlying salary and including the possibility of early retirement we will consider that the salary follows a jump diffusion model thus giving rise to a partial integro differential equation pide after posing the model we propose the appropriate numerical methods to solve the pide problem these methods mainly consists of lagrange galerkin discretizations combined with augmented lagrangian active set techniques and with the explicit treatment of the integral term finally we compare the numerical results with those ones obtained with monte carlo techniques

an asymptotic extraction technique for evaluating sommerfeld type integrals an asymptotic extraction process is developed which leads to an efficient method for the evaluation of the electric field from a hertzian dipole in a layered geometry this technique allows for a rapidly converging expression for the electric field which remains well behaved as the source and observation heights z z coincide a useful application of this method is in the calculation of mutual impedance between printed dipoles