By the semi inverse method proposed jihuan he, a variational principle is established for mongeampere equation. The concept behind the variational method of approximating solutions to the schrodinger equation is based on. Otherwise we only get linear convergence in the bregman distance. The semi inverse variational principle svp is utilized to secure bright soliton solutions to the governing model. The semi inverse method suggested by the author is one of the best and most convenient ways to deduce generalized variational principles with multivariables from 1 partial differential equations pde and boundary conditions bc, or 2 known variational principles with single variable or double variables, or 3 a suitable energy trialfunctional, without any crisis variational phenomenon. In order to use the semiinverse method to establish a lagrangianfor, we rst check some simple cases. Moreover, the results are also compared with hes homotopy. This paper studies optical solitons, in presence of higher order dispersion terms by the aid of hes semiinverse variational principle. Also, the obtained semi inverse variational principle has profound implications in physical understandings. Onedimensional harmonic oscilator how to do this using the variational principle. In this paper we will illustrate how to establish a variational principle for a nonlinear problem using the semiinverse method. This paper determines bright soliton solutions to the model by hes semi inverse variational principle svp. In soliton theory, we aim to search for the solitary wave solutions for nlpdes. Pdf in this paper an analytical expression for an optical soliton is obtained with the aid of hes semiinverse variational principle in the presence.
Variational semiblind sparse deconvolution with orthogonal. Many new exact traveling wave solutions are successfully obtained, which contain soliton solutions. Functionals are often expressed as definite integrals involving functions and their derivatives. The solution is a closed form 1soliton solution that is obtained by hes semiin. Variational principle for traveling waves in a modified. Semiinverse method hamiltonian approach variational approach duffing equation nonlinear schrodingers equation thomasfermi equation laneemden equation. Heroa auniversity of michigan, eecs department, ann arbor, usa. A note on the semiinverse method and a variational. Bright and exotic solitons in optical metamaterials by semi. Generalized variational principle for electromagnetic field. The semi inverse method suggested by the author is one of the best and most convenient ways to deduce generalized variational principles with multivariables from 1 partial differential equations.
To solve this semi blind deconvolution problem, prior distributions are specified for the psf and the 3d image. The method is applied to a novel deep architecture of variational autoencoders. Applications of hes semiinverse method, item and ggm to the. We employ an augmented lagrangian variational formulation for solving the outputleastsquares inverse problem. Variational methods for the solution of inverse problems. In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. However, it is well known that not every nonlinear partial differential equation admits a variational formula. This note reveals that the semiinverse method also works for a generalized kdvmkdv equation with nonlinear terms of any orders. He variational approach to impulsive differential equations using the semiinverse method 3. Optical solitons with quadraticcubic nonlinearity by semi. Based on this formulation, a solitary solution can. The variational crisis and its removal are also discussed, it also reveals that the semi inverse method is. In this paper, we apply hes semiinverse method to establish a variational theory for the boussi nesq system.
The purpose of this paper is to adopt a semi inverse method to search for the variational formulation from the governing equations. Introduction generally speaking, there exist two basic ways to describe a physical problem. This paper studies the nonlinear mongeampere equation in the form1. Semianalytical solutions of the nonlinear oscillator with a matrix lagrange multiplier hua cheng1 and yongyan yu2 abstract in this paper, a semianalytical method based on variational iteration formulae is proposed. Lagrange crisis and generalized variational principle for 3d. The mathematical method used to determine these solutions is usually called the semiinverse method. In, the semiinverse method is systematically studied and many examples are given to show how to establish a variational formulation for a nonlinear equation. The purpose of this paper is to adopt a semiinverse method to search for the variational formulation from the governing equations. The semiinverse variational principle svp is utilized to secure bright soliton solutions to the governing model. Among them, variational approaches, such as he,s semi inverse method 9 is a powerful tool to. Variational principle for a nonautonomous cubicquintic. Variational approach to impulsive differential equations using the semiinverse method. In this paper, variational principle for traveling waves in a modified kuramotosivashinsky equation is constructed by the semi inverse method.
From the given examples, we found that it is difficult to find a variational principle for nonlinear evolution equations with nonlinear terms of any orders. Generally, a nonlinear oscillator with a damping term admits no variational principle. A variational principle for the laneemden equation is established by hes semiinverse method. In this paper, hes semiinverse method is used to construct a family of variational principles for the long waterwave problem. Variational approach to impulsive differential equations. A comparative study of variational iteration method and he. Probabilistic representation and inverse design of. This paper studies optical solitons, in presence of higher order dispersion terms by the aid of. The variational crisis and its removal are also discussed, it also reveals that the semiinverse method is. Hybrid algorithms for solving variational inequalities, variational inclusions, mixed equilibria, and fixed point problems ceng, luchuan, petrusel, adrian, wong, muming, and yao, jenchih, abstract and applied analysis, 2014. The direct relationship between these two techniques is mathematically described. Journal of low frequency noise, semianalytical solutions.
Pdf dispersive optical solitons by the semiinverse variational. Dec 01, 2009 a generalized variational principle is deduced for the initial boundary value problem of the electromagnetic field with magnetic monopoles by hes semi inverse method, confirming the existence of the magnetic monopole. The mathematical method used to determine these solutions is usually called the semi inverse method. On the semiinverse method and variational principle. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations. The motivation for using tv is its wellestablished ability to recover sharp discontinuities. Via hes semiinverse method, a variational principle is established for coupled nonlinear. More notably, by introducing latent variable to encode metamaterial design, the proposed deep generative model offers interpretability and can utilize unlabeled data in a semisupervised learning strategy to assist the construction of the latent. Based on this formulation, a solitary solution can be easily obtained using ritz method.
Secondorder nonlinear oscillator equations are numerically investigated by this method. Variational semiblind sparse image reconstruction with. In this paper, variational principle for traveling waves in a modified kuramotosivashinsky equation is constructed by the semiinverse method. There is a known variational principle for the problem, which is. Semiinverse method and variational principle are explained in this chapter, and the hamiltonian and variational approaches are discussed in detail. The identification of the unknown f is similar to the identification of the. Conclusions in this paper the semiinverse method is applied to establish a variational formulation for the dirichlet boundary value problem with impulses.
V v 2 for v 0, v 2 2 for v 0, v3 6 for v2 2 v v 0, v 1 for v 2. In this paper, we apply hes semiinverse method to establish a variational theory for the boussinesq system. It is most frequently used to compute the ground state, but can be extended to compute the low lying excited states. From the given examples, we found that it is di cult to nd a variational principle for nonlinear evolution equations with nonlinearterms of any orders.
Siam journal on scientific computing siam society for. Letter to the editor a note on the semiinverse method and. Traveling wave hypothesis as well as the method of undetermined coefficients were adopted to extract these soliton solutions. Inverse problems of determining the unknown shape of. The variational principle contents 1 theory 2 the ground state of helium 3 the linear variational problem. Lagrange crisis and generalized variational principle for. Letter to the editor a note on the semiinverse method and a. By the semiinverse method proposed jihuan he, a variational principle is established for mongeampere equation.
Based on the established variational formulation, approximate solutions can be easily obtained by the ritzmethod. This allows calculating approximate wavefunctions such as molecular orbitals. They are kerr law, power law, parabolic law, dualpower law and finally the log law nonlinearity. In particular, we train deep variational autoencoders with latent variables at multiple levels of the hierarchy, where each stochastic variable is a 3d tensor a stack of featuremaps, and demonstrate that the method greatly improves performance. Variational principles for coupled nonlinear schrodinger equations. Since that point onwards, this method is commonly referred to as hes semiinverse variational principle. Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. Generalized variational principle for electromagnetic. Generalized variational principle for long waterwave. We demonstrate this method by improving inference networks of deep variational autoencoders. In, the semi inverse method is systematically studied and many examples are given to show how to establish a variational formulation for a nonlinear equation. We confirm that the method is very effective and gives the results in the concise form of variational functionals.
To solve this semiblind deconvolution problem, prior distributions are specified for the psf and the 3d image. The basis for this method is the variational principle the method consists of choosing a trial wavefunction depending on one or more parameters, and finding. Variational bayesian em the variational bayesian em algorithm has been used to approximate bayesian learning in a wide range of models such as. We present a variational bayesian method of joint image reconstruction and point spread function psf estimation when the psf of the imaging device is only partially known. Moreover, the results are also compared with hes homotopy perturbation method, liaos homotopy analysis. Several techniques including the adomian decomposition method, the variational iteration method. In order to establish an variational formulation, by the semiinverse method 3,4, we. Solitary solutions of the boitileonmannapempinelli.
Sep 20, 2019 a threedimensional 3d unsteady potential flow might admit a variational principle. Applications of hes semiinverse method, item and ggm to. Variational principle for some nonlinear partial differential equations with. Applications of hes variational principle and the first. Abstractthis paper studies optical solitons, in presence of higher order dispersion terms by the aid of hes semiinverse variational principle. Apr 29, 2009 variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. The obtained solutions arein good agreement withthe exact ones.
Conclusions in this paper the semi inverse method is applied to establish a variational formulation for the dirichlet boundary value problem with impulses. This note reveals that the semiinverse method also works for a generalized kdv mkdv equation with nonlinear terms of any orders. Bright and exotic solitons in optical metamaterials by. A note on the semiinverse method and a variational principle for the generalized kdvmkdv equation. Using some illustrative examples, it was shown that the semiinverse variational method is a powerful and straightforward solution method to find closedform for schrodingertype equations. Box 471, 1882 yanan xilu road, shanghai 200051, china abstract. Functions that maximize or minimize functionals may be found. The method can be extended to other impulsive problems with ease. This paper determines bright soliton solutions to the model by hes semiinverse variational principle svp. The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. There are five forms of nonlinearity that are studied in this paper. This is essentially a heuristic method that consists in formulating a priori a special ansatz on the geometric andor kinematical. Pdf the semiinverse method is used to establish a variational principle for the dirichlet boundary value problem with impulses. Analysis of ivps and bvps on semiinfinite domains via collocation methods maleki, mohammad, hashim, ishak, and abbasbandy, saeid, journal of applied mathematics, 2012.
Keywords variational theory, semiinverse method, solitary solution, exp function method, jacobielliptic function. Optical solitons with higher order dispersion by semiinverse variational principle by p. In 1, the semiinverse method is systematically studied and many examples are given to show how to establish a variational formulation for a nonlinear. Kerr law the kerr law of nonlinearity originates from the fact that a light. Pdf variational approach to impulsive differential. Lecture 8 wkb approximation, variational methods and the. Hes semiinverse method for soliton solutions of boussinesq. The semiinverse method to elucidate basic property of the semiinverse method 4, we consider a 2d incompressible and potential flow, its governing equations are.
Thus, this paper is thus going to address equation 1 on the most generalized setting so far, and the tool of integration is going to be the svp. On soliton solutions of the drinfeldsokolovwilson system. Sion by semiinverse variational principle pd green. The semiinverse method suggested by the author is one of the best and most convenient ways to deduce generalized variational principles with multivariables from 1 partial differential equations. A threedimensional 3d unsteady potential flow might admit a variational principle. The wkb approximation will be especially useful in deriving the tunnel current in a tunnel diode see brennan section 11. Variational principles for nonlinear fiber optics ning pan. He variational approach to impulsive differential equations using the semi inverse method 3. By hes semiinverse variational principle we have obtained dark and bright soliton wave solutions. In this method, an energy integral with an unknown f is used as a trial. Soliton solutions of the perturbed resonant nonlinear. Variational principles for some nonlinear wave equations. Lagrange multiplier method is the most used method, but the method becomes invalid for some special cases e. The pochhammerchreeequation consider the pochhammerchree equation governed by 23 u tt.
This variational characterization of eigenvalues leads to the rayleighritz method. This note shows that the semiinverse method in works also for the present problem, and it is concluded that the semiinverse method is a powerful mathematical tool to the construction of a variational formulation for a nonlinear equation. A generalized variational principle is deduced for the initial boundary value problem of the electromagnetic field with magnetic monopoles by hes semiinverse method, confirming the existence of the magnetic monopole. Semiinverse method of establishing generalized variational. In this paper, we have shown a possible connection between the semiinverse variational method and the evaluation of solutions of a given quantum system.
Improving variational inference with inverse autoregressive flow. These solutions might play important role in engineering and physics fields. The semiinverse method suggested by the author is one of the best and most convenient ways to deduce generalized variational principles with multivariables from 1 partial differential equations pde and boundary conditions bc, or 2 known variational principles with single variable or double variables, or 3 a suitable energy trialfunctional, without any crisis variational phenomenon. Also, the obtained semiinverse variational principle has profound implications in physical understandings. A generalized variational principle of 2d unsteady compressible flow around oscillating airfoils is established directly from the governing equations and boundaryinitial conditions via the semi. Xu succeed in establishing the needed variational principle forkm,n equations. Journal of low frequency noise, a short remark on the.
We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation tv regularization. A semiinverse variational method for generating the bound. A note on the semiinverse method and a variational principle for. Adomian decomposition method adm 7, hes semi inverse method 8, the tanh method, homotopy perturba tion methodhpm, sinhcosh method, the differential transform method and the variational iteration method vim 917. Apr 10, 2017 by hes semi inverse variational principle we have obtained dark and bright soliton wave solutions. The variational principle the variational principle quantum mechanics 2 lecture 5 igor luka cevi c ujjs, dept. Variational methods the variational technique represents a completely di. Jun 30, 2018 semi inverse method and variational principle are explained in this chapter, and the hamiltonian and variational approaches are discussed in detail. A note on the semiinverse method and a variational principle.
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