Journal of Nonlinear Mathematical Physics

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa–Holm equation are distributional traveling wave solutions to the Camassa–Holm equation provided that the set u^-1(c), where c is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa–Holm equation. However, the two component Camassa–Holm equation has distinct from Camassa–Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of u^-1(c) is not zero. We provide examples of such solutions.

We show that we can apply the Hirota direct method to some non-integrable equations. For this purpose, we consider the extended Kadomtsev–Petviashvili–Boussinesq (eKPBo) equation with M variable which is

In this paper, we introduce a q-analogue of the Tricomi expansion for the incomplete q-gamma function. A general method is described for converting a power series into an expansion in incomplete q-gamma function. Also, we use the q-Tricomi expansion for giving a formal proof of the relation between the incomplete gamma function and the exponential integral. Finally, we formally deduce the q-Tricomi expansion via the q-Taylor expansion.

The flow of a superfluid film adsorbed on a porous medium can be modeled by a meromorphic differential on a Riemann surface of high genus. In this paper, we define the mixed Hodge metric of meromorphic differentials on a Riemann surface and justify using this metric to approximate the kinetic energy of a superfluid film flowing on a porous surface.

In this paper, we study the geodesic completeness of nondegenerate submanifolds in semi-Euclidean spaces by extending the study of Beem and Ehrlich [1] to semi-Euclidean spaces. From the physical point of view, this extend may have a significance that a semi-Euclidean space contains more variety of Lorentzian submanifolds rather than those of Lorentzian hypersurfaces in a Minkowski space as in [1]. From mathematical point of view, since there is no distinction in the analysis of geodesic completeness of Lorentzian submanifolds and nondegenerate submanifolds in a semi-Euclidean space, we treat the mathematically more general case of nondegenerate submanifolds in a semi-Euclidean space. The new ideas leading to this generalization are the sufficient conditions for algorithms in the proofs of the results in [1]. Indeed these sufficient conditions for the algorithms also work well for the nondegenerate submanifolds in a semi-Euclidean space.

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice introduced by J. Zhao etc. [1]. This implies that Pfaffianization and Bäcklund transformation is commutative for the Leznov lattice. Moreover, since the two-dimensional Toda lattice constitutes the Leznov lattice, it is obvious that the commutativity is also valid for it.

By using gauge transformations, we manage to obtain new solutions of (2 + 1)-dimensional Kadomtsev–Petviashvili (KP), Kaup–Kuperschmidt (KK) and Sawada–Kotera (SK) equations from nonzero seeds. For each of the preceding equations, a Galilean type transformation between these solutions u₂ and the previously known solutions u′₂ generated from zero seed is given. We present several explicit formulas of the single-soliton solutions for u₂ and u′₂, and further point out the two main differences of them under the same value of parameters, i.e., height and location of peak line, which are demonstrated visibly in three figures.

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition — that we show robust under suitable perturbations — allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.