A new analytical modelling for nonlocal generalized Riesz fractional sine-Gordon equation

AbstractIn this paper, a novel approach comprising the modified decomposition method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation utt-RDxαu+sinu=0 where RDxα is the Riesz space fractional derivative, 1≤α≤2. For α=2, it becomes classical sine-Gordon equation utt−uxx+sin u=0 and corresponding to α=1, it becomes nonlocal sine-Gordon equation utt−Hu+sin u=0 which arises in Josephson junction theory, where H is the Hilbert transform. The fractional sine-Gordon equation is considered as an interpolation between the classical sine-Gordon equation (corresponding to α=2) and nonlocal sine-Gordon equation (corresponding to α=1). Here the analytic solution of fractional sine-Gordon equation is derived by using the modified decomposition method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method

Topologically embedded helicoidal pseudospherical cylinders

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1–1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic functions. This solves a problem posed by Popov (2014 Lobachevsky Geometry and Modern Nonlinear Problems (Berlin: Springer)). As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. A (singular) pseudospherical helicoid is proved to be either a dense subset of a region bounded by two coaxial cylinders, a topologically immersed cylinder with helical self-intersections, or a topologically embedded cylinder with helical singularities, called for short a pseudospherical twisted column. Pseudospherical twisted columns are characterized by four phenomenological invariants: the helicity η ∈ Z2, the parity ε ∈ Z2, the wave number n ∈ N, and the aspect ratio d > 0, up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided

Periodic travelling waves in convex Klein-Gordon chains

We study Klein-Gordon chains with attractive nearest neighbour forces and convex on-site potential, and show that there exists a two-parameter family of periodic travelling waves (wave trains) with unimodal and even profile functions. Our existence proof is based on a saddle-point problem with constraints and exploits the invariance properties of an improvement operator. Finally, we discuss the numerical computation of wave trains.Comment: 12 pages, 3 figure

TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS

The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u)^2 − (3u^2 − 1)D_x^α (D_x^α u) + D_x^α(D_x^α(D_x^α(D_x^α u))) = 0 and the space-time fractional symmetric regularized long wave (SRLW) equation D_t^α(D_t^α u) + D_x^α(D_x^α u) + uD_t^α(D_x^α u) + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u))) = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica

Exact solutions for a local fractional DDE associated with a nonlinear transmission line

Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions (kink/antikink solitons, singular, periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform