Dimensional sinh-Gordon Equation Research Articles

General Soliton and (Semi-)Rational Solutions of a (2+1)-Dimensional Sinh-Gordon Equation

In this paper, we investigate solutions of a (2+1)-dimensional sinh-Gordon equation. General solitons and (semi-)rational solutions are derived by the combination of Hirota’s bilinear method and Kadomtsev-Petviashvili hierarchy reduction approach. General solutions are expressed as Ntimes N Gram-type determinants. When the determinant size N is even, we generate solitons, line breathers, and (semi-)rational solutions located on constant backgrounds. In particular, through the asymptotic analysis we prove that the collision of solitons are completely elastic. When N is odd, we derive exact solutions on periodic backgrounds. The dynamical behaviors of those derived solutions are analyzed with plots. For rational solutions, we display the interaction of lumps. For semi-rational solutions, we find the interaction solutions between lumps and solitons.

A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions

In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and traveling wave solutions of the (2+1)-dimensional sine-Gordon equation are constructed. The traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a similar manner. Meanwhile, conservation laws are derived.

Rational solutions of a (2+1)-dimensional sinh–Gordon equation

We study rational solutions of a (2+1)-dimensional sinh–Gordon equation. The approach is based on Hirota’s bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. The derived rational solutions are expressed in terms of determinants. The dynamics of the obtained rational solutions are investigated analytically and graphically.

The numerical solution of the two–dimensional sinh-Gordon equation via three meshless methods

In this paper three numerical techniques are proposed for solving the nonlinear sinh-Gordon equation. Firstly, we obtain a time discrete scheme then we use the radial basis functions (RBFs) collocation based on Kansa׳s approach, RBF-pseudospectral (PS) technique and moving least squares (MLS) methods to approximate the spatial derivatives. The aim of this paper is to show that the meshless methods based on the RBFs using collocation approach and MLS are suitable for the treatment of the nonlinear partial differential equations and also we compare the mentioned methods in terms of condition number of coefficient matrix and absolute value of error. Also, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

RIEMANN THETA FUNCTIONS SOLUTIONS WITH RATIONAL CHARACTERISTICS FOR (2+1)-DIMENSIONAL SINH-GORDON EQUATION

In this letter, we use the Riemann theta functions with rational characteristics and the Hirota bilinear method to construct quasi-periodic wave solutions for (2+1)-dimensional sinh-Gordon equation. This method not only conveniently obtains quasi-periodic solutions of nonlinear equations, but also directly gets the explicit expressions of frequencies, wave numbers, phase and amplitudes for the waves.

Double periodic wave solutions and breather-soliton solutions of the (n+1)-dimensional sinh-Gordon equation

In this paper, using a new method named binary F-expansion method and a simple transformation technique, we discussed the (n+1)-dimensional sinh-Gordon equation. Many double periodic wave solutions and breather-soliton solutions are obtained.

Integrable discretizations of the (2+1)-dimensional sinh-Gordon equation

In this paper, we propose two semi-discrete equations and one fully discrete equation and study them by Hirota's bilinear method. These equations have continuum limits into a system which admits the (2+1)-dimensional generalization of the sinh-Gordon equation. As a result, two integrable semi-discrete versions and one fully discrete version for the sinh-Gordon equation are found. Bäcklund transformations, nonlinear superposition formulae, determinant solution and Lax pairs for these discrete versions are presented.

Kac–Moody–Virasoro symmetry algebra and symmetry reductions of the bilinear sinh-Gordon equation in (2 + 1)-dimensions

By means of the formal series symmetry approach proposed in [1], infinite many symmetries and the corresponding Kac–Moody–Virasoro Lie symmetry algebra of a new bilinear (2 + 1)-dimensional sinh-Gordon equation are given. Then, the obtained symmetries are used to get the symmetry reductions of the model. From one of the special reduction we know that the bilinear form of the first member of the negative Kadomtsev–Petviashvili hierarchy is not only a (2 + 1)-dimensional sinh-Gordon extension but also a novel (2 + 1)-dimensional classical Boussinesq extension.

Double elliptic equation method and new exact solutions of the (n+1)-dimensional sinh-Gordon equation

The double elliptic equation method is presented for constructing exact traveling wave solutions of nonlinear partial differential equations. With the aid of Maple, more new exact solutions are obtained for the (n+1)-dimensional sinh-Gordon equation. These solutions contain logarithmic functions, Jacobi elliptic functions, hyperbolic functions, trigonometric functions, and their combinations.

A ()-dimensional sinh-Gordon equation and its Pfaffian generalization

We consider a (2+1)-dimensional sinh-Gordon equation and show its integrability through bilinear method. A Pfaffian version of the (2+1)-dimensional sinh-Gordon equation is also presented.

Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation

We study 3-dimensional minimal Lagrangian submanifolds of the 3-dimensional complex projective space ℂP3 (4) which admit a unit length Killing vector field whose integral curves are geodesics. We show that such Lagrangian submanifolds can be obtained from either horizontal holomorphic curves in ℝP3 (4) (or equivalently superminimal immersions of surfaces in S4 (1)) or from solutions of the two dimensional sinh-Gordon equation. In the latter case, we explicitly obtain the immersions in terms of elliptic functions in the case that the solutions of the sinh-Gordon equation depend only on one variable.