Equation Uxy Research Articles

Conservation Laws for Hyperbolic Equations: Search Algorithm for Local Preimage with Respect to the Total Derivative

We propose an algorithm that allows one to eliminate flows from conservation laws for hyperbolic equations by expressing partial derivatives of these flows in terms of the corresponding densities. In particular, the application of this algorithm allows one to prove that the decreasing of order of at least one of Laplace y-invariants of the equation uxy = F(x, y, u, ux, uy) is a necessary condition for the function Fuy to belong to the image of the total derivative Dx by virtue of this equation. Thus, we obtain constructive necessary conditions for the existence of differential substitutions that transform a hyperbolic equation into a linear equation or into the Klein–Gordon equation.

On the geometric properties of the dispersionless Toda equation

The problem of constructing exact solutions, Noether symmetries, and conservation laws for the (2 + 1)-dimensional dispersionless Toda equation uxy = exp(−uzz) is considered.

Symmetries of Systems of the Hyperbolic Riccati Type

Let \({\mathfrak{G}} = \oplus _{{\mathfrak{i}} \in {\mathbb{Z}}} {\mathfrak{G}}_{\mathfrak{i}} \) be a Kac–Moody algebra, U(x,y) be a function defined in \({\mathfrak{G}}_{ - 1} \), and a be a constant element of \({\mathfrak{G}}_1 \). We prove that the equation Uxy = [[U,a],Ux] has two symmetry hierarchies connected by a gauge transformation. In particular, the well-known Konno equation appears in the case of the algebra \(A_1^{\left( 1 \right)} \). The corresponding symmetry hierarchies contain the nonlinear Schrodinger and the Heisenberg magnet equations.

Some Classification Results for Hyperbolic Equations F(x, y, u, ux, uy, uxx, uxy, uyy)=0

We provide a contact invariant characterization for equations of the formuxy+a(x, y, u)ux+b(x, y, u)uy+c(x, y, u)=0,uxy+a(x, y)ux+b(x, y)uy+c(x, y, u)=0,uxy+c(x, y, u)=0,uxy=0. We classify all equations of the form uxy+f(x, y, u, ux, uy)=0 for which the two Ovsiannikov's invariants are constants. These results include characterization of the Klein–Gordon equation uxy=u, the Liouville equation uxy=eu, and the class of Euler–Poisson–Darboux equations. It is shown that the wave equation uxy=0, Liouville equation, and the linear equation uxy=2u/(x+y)2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge–Ampére equation Darboux integrable at level one is equivalent to an equation of type uxy+f(x, y, u, ux, uy)=0. We prove that the hyperbolic Fermi–Ulam–Pasta (FPU) equation uyy=κ(ux)2uxx is contact equivalent to a linear equation of type uxy=c(x+y)u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type uxy=F(u, ux) that describe pseudo-spherical surfaces.

An analogue of the Moutard transformation for the Goursat equation $$\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } $$

We present a new Backlund-type transformation for the nonlinear equation\(\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } \) studied by E. Goursat. Goursat found a linearization transformation and some properties of this equation, which make it similar to the Moutard equation uxy=M(x, y)u. However, this Goursat transformation does not provide proper superposition formulas. We give the necessary extended superposition formulas.

On conservation laws and zero-curvature representations of the Liouville equation

Applying the first Noether theorem to the Liouville equation uxy=exp u, we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by 2*2 traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential parameters. Finally, we show that all known zero-curvature representations of the Liouville equation are equivalent (in a definite sense) to matrices of conservation laws.

On an old article of Tzitzeica and the inverse scattering method

Tzitzeica’s class of surfaces in R3 is considered in connection with the inverse scattering method for the associated equation uxy=eu−e−2u. The Bäcklund transformation for this equation is derived from Tzitzeica’s Darboux transformation for it.

On the zeros of solutions to nonlinear hyperbolic equations

SynopsisWe consider the hyperbolic equation uxy + c(x, y, u) =f(x, y) and the wave equationWe show that, under suitable conditions, there are bounded domains in which every solution to certain problems has a zero. Characteristic initial value problems and initial boundary value problems are considered.

Oscillation criteria for second order hyperbolic initial value problems

SynopsisIt is established that under certain restrictions the solution u of the equation uxy − gu = 0 satisfying u(x, 0) = p(x) and u(0,y) = q(y) in ([0, = ∞) × [0, ∞), changes sign in where (X, Y) is any point in the relevant region.

Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for Uxy = 0 in rectangles

SynopsisIt is well known that the Dirichlet problem for hyperbolic equations is a classical “not well posed” problem. Here we consider the Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for the hyperbolic equation uxy = 0 in all positions of the square and a class of rectangles. We also get a partial answer to the problem which deals with a ray that moves from any point on the boundary of a rectangle and is reflected on the boundary such that the angle between every ray and its reflection is π/2.

On existence, uniqueness, and numerical evaluation of solutions of ordinary and hyperbolic differential equations

After a preliminary survey of related results, a general uniqueness theorem for the ordinary differential equation dy/dx=f(x, y) is given in section 4. The general uniqueness theorem for the hyperbolic partial differential equation uxy=f(x, y, u), proved in section 5, is an exact analogue of the general uniqueness theorem for the ordinary differential equation dy/dx=f(x, y).

On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x, y, u, ux, uy)

This paper1 develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x0)=y0) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation uxy=f(x, y, u, ux, yy), u(x, y0)=σ(x), u(x0, y)=τ(y), where σ(x0)=τ(y0). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions σ(x) and τ(y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.