Invariant Variational Principles Research Articles

Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions

The generalized derivative higher order non-linear Schrödinger (DNLS) equation describes pluses propagation in optical fibers and can be regarded as a special case of the generalized higher order non-linear Schrödinger equation. We derive a Lagrangian and the invariant variational principle for DNLS equation. Using the amplitude ansatz method, we obtain the different cases of the exact bright, dark and bright–dark solitary wave soliton solutions of the generalized higher order DNLS equation. By implementing the modulation instability analysis and stability analysis solutions, the stability analysis of the obtained solutions and the movement role of the waves are analyzed. All solutions are analytic and stable.

Stability analysis of the soliton solutions for the generalized quintic derivative nonlinear Schrödinger equation

The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrödinger (DNLS) equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated.

Approximation solutions of derivative nonlinear Schrödinger equation with computational applications by variational method

The derivative nonlinear Schrodinger (DNLS) equation is a nonlinear dispersive model that appears in the description of wave propagation in a plasma. The existence of a Lagrangian and the invariant variational principle for two coupled equations are given. The two coupled equations is describing the nonlinear evolution of the Alfven wave with magnetosonic waves at much larger scale. A type of the coupled DNLS equations is studied by means of symbolic computation, which can describe the wave propagation in birefringent optical fibers. The functional integral corresponding to those equations is derived. We investigate the approximation solutions of the DNLS equation by choice of a trial function in the region of the rectangular box in two cases. By using this trial functions, the functional integral and the Lagrangian of the system without loss are found. The general case for the two-box potential can be obtained on the basis of a different ansatz, where we approximate the Jost function by series in the tanh function method instead of the piece-wise linear function.

A brief note on the analytical solution of Meshchersky’s equation within the inverse problem of Lagrangian mechanics

Meshchersky’s equation is a basic differential equation in the mechanics of variable-mass particles. This note particularly considers the case in which a one-dimensional and position-dependent mass particle is under the action of a potential force. The absolute velocity of mass ejection (or accretion) is supposed to be a linear function of the particle velocity. Within the formulation of the inverse problem of Lagrangian mechanics, an analytical solution of Meshchersky’s equation is here derived. The solution method follows from applying the concept of constant of motion of an extremum problem, which is a fundamental ground in the theory of invariant variational principles.

New exact solutions for the KdV equation with higher order nonlinearity by using the variational method

The Korteweg–de Vries (KdV) equation with higher order nonlinearity models the wave propagation in one-dimensional nonlinear lattice. A higher-order extension of the familiar KdV equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface. The variational approximation method is applied to obtain the solutions for the well-known KdV equation. Explicit solutions are presented and compared with the exact solutions. Very good agreement is achieved, demonstrating the high efficiency of variational approximation method. The existence of a Lagrangian and the invariant variational principle for the higher order KdV equation are discussed. The simplest version of the variational approximation, based on trial functions with two free parameters is demonstrated. The jost functions by quadratic, cubic and fourth order polynomials are approximated. Also, we choose the trial jost functions in the form of exponential and sinh solutions. All solutions are exact and stable, and have applications in physics.

Variational method for the derivative nonlinear Schrödinger equation with computational applications

The derivative nonlinear Schrödinger equation (DNLSE) arises as a physical model for ultra-short pulse propagation. In this paper, the existence of a Lagrangian and the invariant variational principle (i.e. in the sense of the inverse problem of calculus of variations through deriving the functional integral corresponding to a given coupled nonlinear partial differential equations) for two-coupled equations describing the nonlinear evolution of the Alfvén wave with magnetosonic waves at a much larger scale are given and the functional integral corresponding to those equations is derived. We found the solutions of DNLSE by choice of a trial function in a region of a rectangular box in two cases, and using this trial function, we find the functional integral and the Lagrangian of the system without loss. Solution of the general case for the two-box potential can be obtained on the basis of a different ansatz where we approximate the Jost function using polynomials of order n instead of the piecewise linear function. An example for the third order is given for illustrating the general case.

Dual conservation integrals and energy release rates

Simple derivation of the dual conservation integrals in small strain elasticity is presented, without the aid of Noether’s theorem on invariant variational principles. The derived integrals are related to the release rates of the potential and complementary potential energy associated with the defect motion. The analysis corrects the errors in earlier derivation of the relationship between the dual integrals and the release rates of the complementary potential energy. Selected examples in plane and anti-plane strain illustrate the calculation of dual integrals and their application. It is shown that the evaluation of dual integrals is of similar complexity to that of classical integrals, so that either can be used to determine the stress intensity factors or the forces between defects, without solving the corresponding boundary value problems. An advantage of combining the two calculations is discussed.

Variational method for the nonlinear dynamics of an elliptic magnetic stagnation line

The nonlinear evolution of the kink instability of a plasma with an elliptic magnetic stagnation line is studied by means of an amplitude expansion of the ideal magnetohydrodynamic equations. Wahlberg et al. [12] have shown that, near marginal stability, the nonlinear evolution of the stability can be described in terms of a two-dimensional potential U(X,Y), where X and Y represent the amplitudes of the perturbations with positive and negative helical polarization. The potential U(X,Y) is found to be nonlinearly stabilizing for all values of the polarization. In our paper a Lagrangian and an invariant variational principle for two coupled nonlinear ordinal differential equations describing the nonlinear evolution of the stagnation line instability with arbitrary polarization are given. Using a trial function in a rectangular box we find the functional integral. The general case for the two box potential can be obtained on the basis of a different ansatz where we approximate the Jost function by polynomials of order n instead of a piecewise linear function. An example for the second order is given to illustrate the general case. Some considerations concerning solar filaments and filament bands (circular or straight) are indicated as possible applications besides laboratory experiments with cusp geometry corresponding to quadripolar cusp geometries for some clouds and thunderstorms.

An invariant variational principle for canonical flows on Lie groups

In this paper we examine the existence of Lie groups, whose canonical geodesic flows are variational with respect to a left-invariant regular—but not necessarily quadratic (i.e., metric)—Lagrange function. We give effective necessary and sufficient conditions for the existence of an invariant variational principle generating the canonical flow. With these results, taken in conjunction with the classification of Lie algebras, we solve the inverse problem of invariant Lagrangian dynamics in dimensions up to four.

Maurer–Cartan forms and the structure of Lie pseudo-groups

This paper begins a series devoted to developing a general and practical theory of moving frames for infinite-dimensional Lie pseudo-groups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudo-group. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudo-group on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudo-group actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.

Conservation laws of a decagonal quasicrystal in elastodynamics

A special version of Noether's theorem for the sake of absolute invariance on invariant variational principles is used to obtain conservation laws of a three-dimensional solid periodically stacked in a two-dimensional quasiperiodic structure with decagonal symmetry. By applying Lie's infinitesimal criterion on the invariance to the Lagrangian density function under a continuous transformation group, it is found that there exist fourteen conservation laws (nine in physical space and five in material space). Conservation laws in physical space correspond to the known physical facts. Influence of the phason displacement field on the conservation laws in material space is presented. The infinitesimal symmetry-transformations from scale change and translation of coordinates as well as one coordinate rotation are verified to be true. Especially, although the angular momentum conservation laws do not contain the physical quantities in phason field, and the conservation of angular momentum of the phason displacement field itself does not hold, the physical quantities in phason field play an important role in the conservation law in material space derived from an infinitesimal coordinate rotation.

Covariant gauge-natural conservation laws

When a gauge-natural invariant variational principle is assigned, to determine canonical covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms—defining infinitesimal variations of sections of gauge-natural bundles—must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. Vice versa all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as canonical generators of covariant gauge-natural physical charges.

Conservation laws in non-homogeneous electro-magneto-elastic materials

Noether's theorem on invariant variational principles is applied for linear transversely isotropic non-homogeneous electro-magneto-elastic material. Invariant condition of the electromagnetic enthalpy is derived and expressed in original manner. Usually, inserting infinitesimal symmetry-transformations relating to material space of homogeneous materials into the invariant condition, one can obtain some expressions for the non-homogeneous material. These expressions indicate the relations between material inhomogeneity force and energy-momentum tensor. In order to obtain conservation laws of the non-homogeneous material, it follows from the invariant condition that the material coefficients have to satisfy a set of first-order linear partial differential equations with which there is a possible mathematical form of infinitesimal symmetry-transformation relating to material space. Satisfaction of these partial differential equations not only assures the existence of infinitesimal symmetry-transformations but also determines the number of them. Several independent and non-trivial conservation laws relating to material space with different non-homogeneous material coefficients are given. Some results of the path-independent integrals emanating from the conservation laws calculated around a crack tip are presented.

On conservation integrals in micropolar elasticity

Two conservation laws of nonlinear micropolar elasticity (Jk = 0 and Lk = 0) are derived within the framework of the Noether theorem on invariant variational principles, thereby extending the earlier authors' results from the couple stress elasticity. Two non-conserved M-type integrals of linear micropolar elasticity are then derived and their values discussed. A comparison with related work is also given.

On certain classes of exact solutions of Einstein equations for rotating fields in conventional and nonconventional form

Using the symmetry reduction approach we have herein examined, under continuous groups of transformations, the invariance of Einstein exterior equations for stationary axisymmetric and rotating case, in conventional and nonconventional forms, that is a coupled system of nonlinear partial differential equations of second order. More specifically, the said technique yields the invariant transformation that reduces the given system of partial differential equations to a system of nonlinear ordinary differential equations (nlodes) which, in the case of conventional form, is reduced to a single nlode of second order. The first integral of the resulting nlode has been obtained via invariant-variational principle and Noether’s theorem and involves an integration constant. Depending upon the choice of the arbitrary constant two different forms of the exact solutions are indicated. The generalized forms of Weyl and Schwarzschild solutions for the case of no spin have also been deduced as particular cases. Investigation of nonconventional form of Einstein exterior equations has not only led to the recovery of solutions obtained through conventional form but it also yields physically important asymptotically flat solutions. In a particular case, a single third order nlode has been derived which evidently opens up the possibility of finding many further interesting solutions of the exterior field equations.

Invariant variational principles and conservation laws for some nonlinear partial differential equations with constant coefficients – I

The invariance under a one-parameter infinitesimal transformation groups [1] has been proven for a number of nonlinear partial differential equations (NLPDEs) with constant coefficients, which appear in a wide variety of modelling physical phenomena/applications. The invariance identities of Rund [2] involving the Lagrangian and the generators of the infinitesimal Lie groups are utilized, for writing down the conservation laws via Noether's theorem. In order that the study becomes more exhaustive, we have applied the above technique to the cases arising from the generalized Klein–Gordon equation by transforming it to ordinary differential equation (ODE), to get on exact solution for it.

Invariant variational principles and conservation laws for some nonlinear partial differential equations with variable coefficients part II

The conservation laws via Noether's theorem have been derived for a coupled system of nonlinear partial differential equations (NLPDEs) which modelling physical phenomena represented by the generalized Korteweg–de Vries equation, the variable coefficients nonlinear Korteweg–de Vries equation and the variable coefficients nonlinear Schrödinger's equation. For proving the efficiency of the technique under consideration the stable and unstable nonlinear Schrödinger's equations have been examined for the conservation laws.

Conservation integrals in couple stress elasticity

Noether’s theorem on invariant variational principles is applied in the case of infinitesimal couple stress elasticity, thereby extending the analysis of Knowles and Sternberg (1972. On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187–211) beyond the range of classical elasticity. Two conserved integral quantities are deduced which generalize the J-integral and L-integral in the notation of Budiansky and Rice (1973: Budiansky, B. and Rice, J. R. (1973) Conservation laws and energy-release rates. J. Appl. Mech. 40, 201–203). An expression for an M-integral is also obtained, but it is demonstrated that there is no corresponding conservation law for this integral. Relationships of the derived path integrals to other similar quantities for couple stress elasticity which have appeared in the literature are discussed.

On invariant integrals in the Marguerre-von Kármán shallow shell

By employing the second-order Noether's theorem, several new invariant integrals have been derived for the non-linear shallow shell—the Marguerre-von Kármán shell. The dynamic effect is considered in the derivations. These invariant integrals are path-independent over the projection image of the middle surface of the shell in a Cartesian plane, in which the projection area of the middle surface of the shallow shell is maximum. The proposed invariant integrals can be used to evaluate the asymptotic field around a defect embedded in the shell. Unlike most other studies, the Lagrangian density of the invariant variational principle used here belongs to a mixed type variational principle.

GL(n,R) as a candidate for fundamental symmetry in field theory

We formulate generally covariant andGL(n,R)-invariant variational principles for the field of linear co-frames in an-dimensional manifold. The internalGL(n,R)-symmetry is the main difference between presented models and the metric-teleparallel theories of gravitation (including Einstein theory), invariant under the global Lorentz groupO(1,n−1) (local in Einstein theory). There is a structural similarity between our Lagrangians and those used in Born-Infeld electrodynamics. We discuss field equations and Lagrangian constraints. Field equations are nonempty because the existence of certain «vacuumlike» solutions is explicitly shown. These solutions are isomorphic to canonical forms of semi-simple Lie groups (just as in certain field-theoretic models studied by Toller, D’Adda, Nelson and Regge), or to appropriately deformed canonical forms of trivial central extensions of semi-simple Lie groups. The latter applies,e.g., to four-dimensional space-times. The normal-hyperbolic signature is nota priori assumed, however, ifn-4, it turns out to be a property of the most natural solutions; roughly speaking, it is implied by differential equations. On the other hand, if we assume that the signature is to be normal-hyperbolic, thenn=4 is the lowest nontrivial dimension. We present certain arguments in favour of the hypothesis thatGL(n,R) is a promising candidate for the symmetry group of generally covariant fundamental field theories. OurGL(n,R)-invariant Lagrangians for frames provide a first step towards developments in this direction.