A direct method for solving the generalized sine-Gordon equation II

The generalized sine-Gordon (sG) equation $u_{tx}=(1+\nu\partial_x^2)\sin\,u$ was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to as I, we developed a systematic method for solving the generalized sG equation with $\nu=-1$. Here, we address the equation with $\nu=1$. By solving the equation analytically, we find that the structure of solutions differs substantially from that of the former equation. In particular, we show that the equation exhibits kink and breather solutions and does not admit multi-valued solutions like loop solitons as obtained in I. We also demonstrate that the equation reduces to the short pulse and sG equations in appropriate scaling limits. The limiting forms of the multisoliton solutions are also presented. Last, we provide a recipe for deriving an infinite number of conservation laws by using a novel B\"acklund transformation connecting solutions of the sG and generalized sG equations.Comment: To appear in J. Phys. A: Math. Theor. The first part of this paper has been published in J. Phys. A: Math. Theor. 43 (2010) 10520

Assessment of performance and impact of irrigation and water resources systems in Taiwan and Sri Lanka

Irrigation systems / Water scarcity / Water allocation / Water supply / Water quality / Water distribution / Decision making / Water management / Regression analysis / Performance evaluation / Sustainability / Irrigated farming / Agricultural development / Farm size / Irrigation management / Farmers' associations / Rain / Paddy cultivation / Crop yield / Costs / Productivity

Use of untreated wastewater in peri-urban agriculture in Pakistan: risks and opportunities

Water reuse / Waste waters / Water quality / Groundwater / Irrigation practices / Soil properties / Environmental effects / Conjunctive use / Pakistan / Haroonabad

A direct method of solution for the Fokas-Lenells derivative nonlinear Schr\"odinger equation: I. Bright soliton solutions

We develop a direct method of solution for finding the bright $N$-soliton solution of the Fokas-Lenells derivative nonlinear Schr\"odinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the collisions of bright solitons.Comment: To appear in J. Phys. A: Math. Theor. 45(2012) Ma

Parametric Representation for the Multisoliton Solution of the Camassa-Holm Equation

The parametric representation is given to the multisoliton solution of the Camassa-Holm equation. It has a simple structure expressed in terms of determinants. The proof of the solution is carried out by an elementary theory of determinanats. The large time asymptotic of the solution is derived with the fomula for the phase shift. The latter reveals a new feature when compared with the one for the typical soliton solutions. The peakon limit of the phase shift ia also considered, showing that it reproduces the known result.Comment: 14 page

Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in 2+1 dimensions

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: $u_t+u_{xxy}-3uu_y-3u_xv=0$ and $u_t+u_{xxxxy}-(5u_{xx}v+10u_{xy}u-15u^2v)_x=0$ where $v_x=u_y$, thereby yielding their one-periodic and two-periodic wave solutions describing one dimensional propagation of waves

First star survivors as metal-rich halo stars that experienced supernova explosions in binary systems

The search for the first stars formed from metal-free gas in the universe is one of the key issues in astronomy because it relates to many fields, such as the formation of stars and galaxies, the evolution of the universe, and the origin of elements. It is not still clear if metal-free first stars can be found in the present universe. These first stars are thought to exist among extremely metal-poor stars in the halo of our Galaxy. Here we propose a new scenario for the formation of low-mass first stars that have survived until today and observational counterparts in our Galaxy. The first stars in binary systems, consisting of massive- and low-mass stars, are examined using stellar evolution models, simulations of supernova ejecta colliding with low-mass companions, and comparisons with observed data. These first star survivors will be observed as metal-rich halo stars in our Galaxy. We may have identified a candidate star in the observational database where elemental abundances and kinematic data are available. Our models also account for the existence in the literature of several solar-metallicity stars that have space velocities equivalent to the halo population. The proposed scenario demands a new channel of star formation in the early universe and is a supplementary scenario for the origin of the known metal-poor stars

First Star Survivors as Metal-Rich Halo Stars that Experienced Supernova Explosions in Binary Systems

The search for the first stars formed from metal-free gas in the universe is one of the key issues in astronomy because it relates to many fields, such as the formation of stars and galaxies, the evolution of the universe, and the origin of elements. It is not still clear if metal-free first stars can be found in the present universe. These first stars are thought to exist among extremely metal-poor stars in the halo of our Galaxy. Here we propose a new scenario for the formation of low-mass first stars that have survived until today and observational counterparts in our Galaxy. The first stars in binary systems, consisting of massive- and low-mass stars, are examined using stellar evolution models, simulations of supernova ejecta colliding with low-mass companions, and comparisons with observed data. These first star survivors will be observed as metal-rich halo stars in our Galaxy. We may have identified a candidate star in the observational database where elemental abundances and kinematic data are available. Our models also account for the existence of several solar-metallicity stars in the literature having space velocities equivalent to the halo population. The proposed scenario demands a new channel of star formation in the early universe and is a supplementary scenario for the origin of the known metal-poor stars.Comment: 41 pages, 14 figures, accepted for publication in PASJ (to be published in open access