Exactly Solvable And Integrable Systems

Integrable equations in ( 1+1 ) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in ( 2+1 ) dimensions have also the analogous hierarchies to those in ( 1+1 ) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in the Painlevé test. Nevertheless, it may be concluded that the BS hierarchy is integrable.

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as t→±∞ (x/t∼O(1)) of the solution to the Cauchy problem for the modified non-linear Schrödinger equation, i ∂ t u+1/2 ∂ 2 x u+|u | 2 u+is ∂ x (|u | 2 u)=0 , s∈ R >0 , which is a model for non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent non-linear evolution equations; in particular, the derivative non-linear Schrödinger equation, i ∂ t q+ ∂ 2 x q−i ∂ x (|q | 2 q)=0 .

In this note, we compute Hirota bilinear forms arising from both homogeneous and principal realization of vertex representations of 2-toroidal Lie algebras of type A l , D l , E l .

A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax equations, exhibiting the integrability properties of such manifolds. A number of different field equations for such hypercomplex manifolds are derived, one of which is in Cauchy-Kovaleskaya form which enables a formal general solution to be given. Various other properties of the field equations and their solutions are studied, such as their symmetry properties and the associated hierarchy of conservation laws.

Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the A_2 root system.

By constructing the general six-parameter bright two-soliton solution of the integrable coupled nonlinear Schrodinger equation (Manakov model) using the Hirota method, we find that the solitons exhibit certain novel inelastic collision properties, which have not been observed in any other (1+1) dimensional soliton system so far. In particular, we identify the exciting possibility of switching solitons between modes by changing the phase. However, the standard elastic collision property of solitons is regained with specific choices of parameters.

By linearizing the inhomogeneous Burgers equation through the Hopf-Cole transformation, we formulate the solution of the initial value problem of the corresponding linear heat type equation using the Feynman-Kac path integral formalism. For illustration, we present the exact solution for the forcing term of the form: F(x,t)= ω 2 x+f(t). We also present the initial value problem solution for the case with a constant forcing term to compare with the known result.

Initial boundary value problem on a half-line for the Modified KdV equation is considered with the boundary conditions equal to zero at the origin and initial condition chosen arbitrary decreasing rapidly enough and this problem is plunged into the scheme of the inverse scattering method. Here the inverse scattering problem is reduced to the Riemann problem on a system of rays on the complex plane.

Discusses several integrability tests for nonlinear evolution equations.