Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with time-dependent frequency (one degree of freedom); geodesics of a spheroid (two degrees of freedom); Calogero-Moser-Sutherland system of interacting particles (three degrees of freedom). For each system, a local generalization of Liouville integrability is shown. Specifically, the variational point symmetries in a Lagrangian setting lead to corresponding locally conserved integrals which are found to commute in the Poisson bracket imported from the equivalent Hamiltonian setting. Action-angle variables are then introduced in the Lagrangian setting, which leads to explicit integration of the Euler-Lagrange equations of motion locally in time.

The Kepler problem in classical mechanics exhibits a rich structure of conserved quantities, highlighted by the Laplace-Runge-Lenz (LRL) vector. Through Noether's theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the part of the LRL vector providing a constant of motion beyond angular momentum and energy is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The present work derives the infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector and obtains the explicit form of the symmetry transformations that it generates. When combined with the rotation symmetries, the resulting symmetry group is shown to be the semi-direct product of SO(3) and R3. This stands in contrast to the SO(4) symmetry group generated by the LRL symmetries and the rotations. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quantities is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated.

Interactions of two solitary waves with an up and a down orientation in the modified Korteweg-de Vries equation are shown to produce rogue-like waves. For waves that asymptotically vanish, the maximum ratio between the height of the interaction profile and the height of the tallest incoming wave is 2.41 when the waves have approximately equal speeds, and this ratio decreases to 2 when the speed ratio is 1.5. For waves that approach a non-zero constant at infinity, the same ratio reaches a maximum of 2.65 when the speed ratio of the waves is 6.32.

A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.

For the timelike geodesic equations in Schwarzschild spacetime, three hidden conserved quantities were found recently, which are analogues of dynamical quantities related to the well-known Laplace-Runge-Lenz (LRL) vector in Newtonian gravity. In particular, the geodesic equations possess an LRL angle, an LRL Killing-vector time and an LRL proper-time, each of which is a conserved quantity for all timelike geodesics. The present work provides a natural symmetry interpretation for these three quantities by applying Noether's theorem in reverse to the geodesic Lagrangian. This yields three hidden symmetry transformations. They are shown to commute with the Killing isometries and act on the equatorial geodesics by separate shifts and scaling of the geodesic energy and angular momentum. Together with the Killing symmetries, these transformations comprise the complete Noether symmetry group of the timelike equatorial geodesic equations.

There is a well known correspondence between geometric curve flows in the Euclidean plane and solutions of the modified Korteweg-de Vries (mKdV) equation. For each type of mKdV travelling wave, the resulting geometric curve flows are derived here through a simple quadrature formula and studied in detail. These curve flows can be divided into two broad types: travelling loops, and rotating loops. Travelling loops are shown to arise from mKdV solitons, cnoidal (Jacobi cn) and dnoidal (Jacobi dn) waves, the latter being periodic. Rotating loops comprise asymptotically circular ones that are obtained from both mKdV solitary waves on a non-zero background and mKdV rational waves, as well as periodic ones that are produced by mKdV rational elliptic (cn and dn) waves. A specialization of periodic loops, both open and closed, is shown to yield rational cosine loops. An explicit description of each of these types of curve flows is used to characterize their main features, including the condition under which closed loops exist.

Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV scaling weights. A condition for linear differential operators to be a Lax pair over octonions is formulated and solved for the unknown coefficients in the polynomials.

In the energy functional of nonlinear Klein-Gordon models,sphalerons arise from a saddle point between true and false vacua. The spectral problem for the (in)stabilty of sphalerons in quartic models is shown be governed by a Heun differential equation, which has three finite regular singular points, after a certain change of variable. This allows an explicit formulation of the eigenfunctions and eigenvalues to be obtained in terms of local Heun functions. Good approximations are found for certain ranges of the parameter.

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg-de Vries equation with p-power nonlinearity coupled to a linear Schrodinger equation arising in many different physical applications. This method yields 22 solution families, with p=1,2,3,4. No solutions for p>1 were known previously in the literature. For p=1, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) p symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized H & eacute;non-Heiles form.

Sphalerons in nonlinear Klein-Gordon models are unstable lump-like solutions that arise from a saddle point between true and false vacua in the energy functional. Numerical simulations are presented which show the sphaleron evolving into an accelerating kink-antikink pair whose separation approaches the speed of light asymptotically at large times. Utilizing a nonlinear collective coordinate method, an approximate analytical solution is derived for this evolution. These results indicate that an exact solution is expected to exhibit a gradient blow-up for large times,caused by energy concentrating at the flanks of the kink-antikink pair.

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev–Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various wave profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi cn functions, all of which are symmetric. The cosine and cn symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension.

A weak formulation is devised for the K(m,n) equation which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.

A generalization of the KP equation involving higher-order dispersion is studied. This equation appears in several physical applications. As new results, the Lie point symmetries are obtained and used to derive conservation laws via Noether’s theorem by introduction of a potential which gives a Lagrangian formulation for the equation. The meaning and properties of the symmetries and the conserved quantities are described. Explicit sech-type line wave solutions are found and their features are discussed. They are shown to describe dark solitary waves on a background which depends on a dispersion ratio and on the speed and direction of the waves. The zero-background case is explored.

The ideal magnetohydrodynamic (MHD) equations and the ideal Chew–Goldberger–Low (CGL) plasma equations are studied using Lagrangian field theory methods. Action principles for the equations are developed and used to obtain conservation laws using Noether’s theorems. The Galilean group admitted by the equations leads to (i) energy, (ii) momentum, (iii) center of mass, and (iv) angular momentum conservation laws corresponding to the time translation, space translation, Galilean boosts, and rotational symmetries, respectively. The cross-helicity conservation law is a consequence of a fluid relabeling symmetry, and is local or non-local depending on whether the entropy gradient ( ∇ S ) is perpendicular to the magnetic field induction B or otherwise. The point Lie symmetries of the MHD and CGL equations consist of Galilean transformations and scalings. Noether’s second theorem is used to derive a wave action conservation equation for a linear non-WKBJ Alfvén waves model for stellar winds. Recent symmetry group investigation of the Lagrangian MHD equations in two space dimensions are discussed.

of Solutions to Partial Differential Equations

Westervelt’s equation is a nonlinear wave equation that is widely used to model the propagation of sound waves in a compressible medium, with one important application being ultra-sound in human tissue. Two fundamental aspects of the general dissipative version of Westervelt’s equation – symmetries and conservation laws – are studied in the present work by modern methods. Numerous results are obtained: new conserved integrals; potential systems yielding hidden symmetries and nonlocal conservation laws; mapping of Westervelt’s equation in the undamped case into a linear wave equation; hidden variational structures, including a Lagrangian and a Hamiltonian; a recursion operator and a Noether operator; contact symmetries; higher-order symmetries and higher-order conservation laws.

Two aspects of a widely used 1D model of blood flow in a single blood vessel are studied by symmetry analysis, where the variables in the model are the blood pressure and the cross-section area of the blood vessel. As one main result, all travelling wave solutions are found by explicit quadrature of the model. The features, behaviour, and boundary conditions for these solutions are discussed. Solutions of interest include shock waves and sharp wave-front pulses for the pressure and the blood flow. Another main result is that three new conservation laws are derived for inviscid flows. Compared to the well-known conservation laws in 1D compressible fluid flow, they describe generalized momentum and generalized axial and volumetric energies. For viscous flows, these conservation laws get replaced by conservation balance equations which contain a dissipative term proportional to the friction coefficient in the model.

Abstract Symmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the range of a symmetry action; a generalised Noether (pre-symplectic) operator constructed from any non-variational adjoint-symmetry. These results are illustrated here by considering five examples of physically interesting nonlinear PDE systems – nonlinear reaction-diffusion equations, Navier-Stokes equations for compressible viscous fluid flow, surface-gravity water wave equations, coupled solitary wave equations and a nonlinear acoustic equation.

In Schwarzschild spacetime, the timelike geodesic equations, which define particle orbits, have a well-known formulation as a dynamical system in coordinates adapted to the timelike hypersurface containing the geodesic. For equatorial geodesics, the resulting dynamical system is shown to possess a conserved angular quantity and two conserved temporal quantities, whose properties and physical meaning are analogues of the conserved Laplace-Runge-Lenz vector, and its variant known as Hamilton's vector, in Newtonian gravity. When a particle orbit is projected into the spatial equatorial plane, the angular quantity yields the coordinate angle at which the orbit has either a turning point (where the radial velocity is zero) or a centripetal point (where the radial acceleration is zero). This is the same property as the angle of the respective Laplace-Runge-Lenz and Hamilton vectors in the plane of motion in Newtonian gravity. The temporal quantities yield the coordinate time and the proper time at which those points are reached on the orbit. In general, for orbits that have a single turning point, the three quantities are globally constant; for orbits that possess more than one turning point, the temporal quantities are just locally constant as they jump at every successive turning point, while the angular quantity similarly jumps only if an orbit is precessing. This is analogous to the properties of a generalized Laplace-Runge-Lenz vector and generalized Hamilton's vector which are known to exist for precessing orbits in post-Newtonian gravity. The angular conserved quantity can be used to define a direct analogue of these vectors at spatial infinity.