Showing papers on "Quintic function published in 2012"
TL;DR: In this paper, the quintic non-linear equation of motion is derived based on Hamilton's principle and solved by means of an analytical technique, namely the Homotopy analysis method.
Abstract: non-linear vibration analysis of beam used in steel structures is of particular importance in mechanical and industrial applications. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. Equation of transversal vibration of hinged–hinged flexible beam subjected to constant excitation at its free end is identified as a non-linear differential equation. The quintic non-linear equation of motion is derived based on Hamilton’s principle and solved by means of an analytical technique, namely the Homotopy analysis method. To verify the soundness of the results, a comparison between analytical and numerical solutions is developed. Finally, to express the impact of the quintic nonlinearity, the non-linear responses obtained by HAM are compared with the results from usual beam theory.
75 citations
TL;DR: The generalized regularized long wave (GRLW) equation is solved numerically by the Petrov-Galerkin method which uses a linear hat function and a quintic B-spline function as the test function and is found to be accurate and efficient.
Abstract: The generalized regularized long wave (GRLW) equation is solved numerically by the Petrov-Galerkin method which uses a linear hat function as the trial function and a quintic B-spline function as the test function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.
59 citations
TL;DR: In this article, the growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS) was studied, and by a combination of the normal form reduction and the upside-down I-method, they established α = 1 for a general power nonlinearity.
Abstract: We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish
$${\left\| {u(t)} \right\|_{{H^s}}} \le {(1 + \left| t \right|)^{a(s - 1) + }}$$
with α = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with α = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with α = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain’s idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.
54 citations
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TL;DR: In this paper, the authors considered the problem of large data scattering for the quintic nonlinear Schrodinger equation on R \times \T^2 and proposed a large scale profile that controls the asymptotic behavior of the solutions.
Abstract: We consider the problem of large data scattering for the quintic nonlinear Schrodinger equation on $\R \times \T^2$. This equation is critical both at the level of energy and mass. Most notably, we exhibit a new type of profile (a "large scale profile") that controls the asymptotic behavior of the solutions.
49 citations
TL;DR: In this paper, the quintic nonlinear Schrodinger equation (NLS) on the circle was considered and it was shown that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited.
Abstract: We consider the quintic nonlinear Schrodinger equation (NLS) on the circle i ∂ t u + ∂ x 2 u = ± ν | u | 4 u , ν ≪ 1 , x ∈ S 1 , t ∈ R . We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set (see Definition 1.1), which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomenon does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomenon cannot occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.
46 citations
TL;DR: In this article, the effects of quintic nonlinearity on ultrashort optical pulse propagation in non-Kerr media were studied by virtue of the Darboux transformation (DT) and symbolic computation.
46 citations
TL;DR: A procedure to isolate propagating wave patterns in nonlinear Schrödinger equations based on two integrals of motion is proposed, which identifies a "gray" solitary pulse, a "dark" localized mode with nonzero minimum in intensity on a continuous-wave background.
Abstract: Quintic derivative nonlinear Schr\"odinger equations arise in various physical contexts, notably in the study of hydrodynamic wave packets and media with negative refractive index. A procedure to isolate propagating wave patterns in such nonlinear Schr\"odinger equations is proposed which is based on two integrals of motion. As an illustration of the method, a ``gray'' solitary pulse, a ``dark'' localized mode with nonzero minimum in intensity on a continuous-wave background is identified.
43 citations
TL;DR: A new quintic B-spline collocation method for linear and nonlinear second order mixed boundary value problems and it is shown that the method is fourth order convergent.
41 citations
TL;DR: In this paper, a numerical method is proposed to approximate the solution of the nonlinear general Rosenau-RLW Equation, which is based on collocation of quintic B-splines over finite elements.
Abstract: In this paper a numerical method is proposed to approximate the solution of the nonlinear general Rosenau-RLW Equation. The method is based on collocation of quintic B-splines over finite elements so that we have continuity of the dependent variable and its first four derivatives throughout the solution range. We apply quintic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK54 scheme. This method needs less storage space that causes to less accumulation of numerical errors. The numerical approximate solutions to the nonlinear general Rosenau-RLW Equation have been computed without transforming the equation and without using the linearization. Illustrative example is included for different value of $p=2,3$ and $6,$ to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method.
36 citations
TL;DR: In this article, the authors obtained the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operator satisfying polynomial Heisenberg algebras.
Abstract: We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painleve transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.
32 citations
TL;DR: In this paper, the authors study the damped Navier-stokes equations in the whole 2D space and verify the global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly local spaces.
Abstract: We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field $u_0\in L^\infty(\mathbb R^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|\to\infty$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.
TL;DR: In this paper, a non-uniform rational B-spline interpolation algorithm with compensatory parameter to minimize feedrate fluctuation and contour error is presented. But the algorithm is not suitable for high speed precision machining of a part.
Abstract: The high-speed precision machining of a part requires minimal feedrate fluctuation and contour error, and parametric spline interpolations have proven to be superior over linear and circular interpolations. However, parametric spline interpolations may result in large feedrate fluctuation due to an inaccurate mapping between the spline parameter u and the displacement S. This paper presents a non-uniform rational B-spline interpolation algorithm with compensatory parameter to minimize feedrate fluctuation and contour error. Since the cubic or quintic polynomials expressing the u-S mapping are acquired, they can be calculated by two consecutive interpolation points with the continuity condition. Thus, the parameter u can be calculated quickly and accurately by substituting the desired displacement S into the cubic or quintic polynomials. Simulation shows that the feedrate fluctuation of the proposed algorithm is much smaller than that of Taylor interpolation algorithms and Feed correction algorithm.
TL;DR: In this paper, the authors consider the Gross-Pitaevskii equation on R 4 and the cubic-quintic non-linear Schrodinger equation (NLS) on R 3 with non-vanishing boundary conditions at spa- tial infinity.
Abstract: We consider the Gross-Pitaevskii equation on R 4 and the cubic-quintic non- linear Schrodinger equation (NLS) on R 3 with non-vanishing boundary conditions at spa- tial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.
TL;DR: A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a(t),b(t) are linear, and have five degrees of freedom, as with the Class I curves.
TL;DR: In this paper, a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section was studied, which is equivalent to the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X.
Abstract: We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C_8 and C_14 has five irreducible components. In the component corresponding to the existence of a tangent conic to the sextic degeneration curve of the quadric bundle, we prove that the general member is both pfaffian and has nontrivial Brauer class. Such cubic fourfolds also provide twisted derived equivalences between K3 surfaces of degree 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.
TL;DR: In this article, a solution for the quintic self-focusing nonlinearity and cubic-quintic 1D Gross Pitaeskii equation (GPE) in a harmonic confined potential is reported.
Abstract: By means of new general variational method we report a direct solution for the quintic self-focusing nonlinearity and cubic-quintic 1D Gross Pitaeskii equation (GPE) in a harmonic confined potential. We explore the influence of the 3D transversal motion generating a quintic nonlinear term on the ideal 1D pure cigar-like shape model for the attractive and repulsive atom-atom interaction in Bose Einstein condensates (BEC). Also, we offer a closed analytical expression for the evaluation of the error produced when solely the cubic nonlinear GPE is considered for the description of 1D BEC.
TL;DR: In this paper, a method based on the identification of coefficients of hyperbolic functions for the construction of the soliton solutions of the cubic and quintic nonlinear Ginzburg-Landau equations was proposed.
Abstract: We apply a method based on the identification of coefficients of hyperbolic functions for the construction of the soliton solutions of the cubic and quintic nonlinear Ginzburg–Landau equations. This effective method improves the solution when the nonlinearity increases.
TL;DR: In this article, a rigorous approach based on the Whitham modulation theory was employed to investigate dispersive shock waves arising in a high-order nonlinear Schr¨ odinger equation with competing cubic and quintic nonlinear responses.
Abstract: By employing a rigorous approach based on the Whitham modulation theory, we investigate dispersive shock waves arising in a high-order nonlinear Schr¨ odinger equation with competing cubic and quintic nonlinear responses. This model finds important applications in both nonlinear optics and Bose-Einstein condensates. Our theory predicts the formation of dispersive shocks with totally controllable properties, encompassing both steering and compression effects. Numerical simulations confirm these results perfectly. Quite remarkably, shock tuning can be achieved in the regime of a very small high order, i.e. quintic, nonlinearity.
26 Feb 2012
TL;DR: In this article, a finite element method involving collocation method with quintic B-splines as basis functions has been developed to solve 8th order boundary value problems, where the basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain.
Abstract: A finite element method involving collocation method with quintic B-splines as basis functions has been developed to solve eighth order boundary value problems. The fifth order, sixth order, seventh order and eighth order derivatives for the dependent variable are approximated by the central differences of fourth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.
TL;DR: The relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.
Abstract: Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.
TL;DR: In this paper, analytical spatial similaritons to a (2+1)-dimensional inhomogeneous cubic-quintic nonlinear Schrodinger equation with distributed diffraction and gain are derived when some compatibility conditions are satisfied.
TL;DR: The paper presents an interpolation scheme for G^1 Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler-Rodrigues frame of the curve coincides with the rotationsat the points.
TL;DR: In this article, it was shown that any meromorphic solution of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation is necessarily elliptic or degenerate elliptic.
Abstract: We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie’s lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.
TL;DR: In this article, the coupled cubic-quintic nonlinear Schrodinger equations describing the effects of quintic non-linearity on the ultrashort optical pulse propagation in non-Kerr media were investigated.
Abstract: Under investigation in this paper are the coupled cubic–quintic nonlinear Schrodinger equations describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained via the Ablowitz–Kaup–Newell–Segur scheme and the corresponding Darboux transformation is constructed. One-, two- and three-soliton solutions are presented and an infinite number of conservation laws are also derived. The features of solitons are graphically discussed: (i) head-on and overtaking elastic collisions of the two solitons; (ii) periodic attraction and repulsion of the bounded states of two solitons; (iii) energy-exchanging collisions of the three solitons.
TL;DR: The present study demonstrates that the linked-list algorithm provides the same accuracy as the all-pair algorithm for indoor PM concentration estimation and the computational efficiency is proved to be much better than the widely used all- Pair algorithm.
TL;DR: In this paper, the authors discuss local Hermite interpolation by C^2 quintic Powell-Sabin splines represented in a normalized B-spline basis and derive explicit formulae for the spline coefficients to interpolate given Hermite data.
TL;DR: In this article, the nonlinear Schrodinger equation is numerically solved using the collocation method based on quintic B-spline interpolation functions, which is demonstrated by standard test problems, such as a one-soliton solution, interaction of two solitons, and formation of a soliton.
Abstract: The nonlinear Schrodinger equation is numerically solved using the collocation method based on quintic B-spline interpolation functions. The efficiency and robustness of the proposed method are demonstrated by standard test problems, such as a one-soliton solution, interaction of two solitons, and formation of a soliton. This method is compared with both the analytical and numerical techniques in the computational section.
TL;DR: In this article, the authors investigated analytical solutions of the (2+1)-dimensional combining cubic-quintic nonlinear Schrodinger (CQNLS) equation by the classical Lie group symmetry method.
Abstract: We investigate analytical solutions of the (2+1)-dimensional combining cubic-quintic nonlinear Schrodinger (CQNLS) equation by the classical Lie group symmetry method. We not only obtain the Lie-point symmetries and some (1+1)-dimensional partial differential systems, but also derive bright solitons, dark solitons, kink or anti-kink solutions and the localized instanton solution.
Journal Article•
TL;DR: In this article, a finite element method involving collocation method with quintic B-splines as basis functions have been developed to solve sixth order boundary value problems, where the basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain.
Abstract: A finite element method involving collocation method with quintic B-splines as basis functions have been developed to solve sixth order boundary value problems. The sixth order and fifth order derivatives for the dependent variable are approximated by the central differences of fourth order derivatives. The basis functions are redefined into a new set of basis functions which in number match with the number of collocated points selected in the space variable domain. The proposed method is tested on several linear and non-linear boundary value problems. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Numerical results obtained by the present method are in good agreement with the exact solutions or numerical solutions available in the literature.