We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.

The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.

Summary: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation $\partial u/\partial x_0+ \Delta u= au^n+ \varepsilon f(u)$, where $a$ and $n$ are arbitrary real constants, $f$ is a smooth function, and $0< \varepsilon\ll 1$.

The authors give the approximate symmetries for the multidimensional Landau-Ginzburg equation delta 2u/ delta x2i+ delta u/ delta x4=a1+a2u+ in un where n in R and 0( in <

The authors investigate the Lie symmetry vector fields of the wave equation Square Operator u+f(x1, . . ., xn, u)=0 where f is some nonlinear smooth function and n>or=2. The Painleve test is considered for the construction of explicitly space- and time-dependent integrable one-space-dimensional nonlinear wave equations.

For arbitrary functions f1, f2 and f3 the anharmonic oscillator x+f1(t)x+f2(t)x+f3(t)x3=0 cannot be solved in closed form (i.e. the general solution cannot be expressed as elliptic functions). The authors apply the Painleve test to obtain the constraint on the functions f1, f2 and f3 for which the equation passes the test. The constraint on f1, f2 and f3 (i.e the differential equation which f1, f2, and f3 obey) is discussed and solutions are given.

The authors comment on the Lie point symmetries for the Khokhlov-Zabolotskaya equation as calculated by Roy Chowdhury and Nasker (1986), and demonstrate that their result for the coefficients of the vector field is correct but incomplete.

On the Coulomb potential in one dimension1996In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 29, no 8, p. 1767-1771Article in journal (Refereed)

Abstract [en]

A mathematically rigorous definition of the one-dimensional Schrödinger operator -d2/dx2 - γ/x is given. It is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half-axes. The investigated operator is compared with the Schrödinger operator containing the Coulomb potential -γ/|x

The differential operator -(d2/dx2) - (γ/x), γ ∈ ℝ, in one dimension is studied using distribution theory. It is proven that there exists a unique self-adjoint operator corresponding to the differential expression understood in the principle-value sense. Point interactions determined by the singular operator -(d2/dx2) - (γ/x) + αδ(x) are studied.