New double Wronskian solutions of the AKNS equation

Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction.     

Bubbling solutions for an anisotropic Emden–Fowler equation

We consider the following anisotropic Emden–Fowler equation where is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quantity can approach to as . These results show a striking difference with the isotropic case [ Constant].     

Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa equation

Based on Hirota bilinear method, four kinds of localized waves, solitons, breathers, lumps and rogue waves of the extended (3+1)-dimensional Jimbo–Miwa equation are constructed. Breathers are obtained through choosing appropriate parameters on soliton solutions, while lumps and rogue waves are derived via the long wave limit on the soliton solutions. The energy, phase shift, shape, and propagation direction of these localized waves can be influenced and controlled by parameters. Considering mixed cases of the above four types of solutions, we also give many kinds of interaction solutions in the same plane with different parameters or different planes with the same parameters. Dynamical characteristics of these localized waves and interaction solutions are further analyzed and vividly demonstrated through figures.     

Exact solutions of nonlocal Fokas–Lenells equation

In this paper we propose a nonlocal Fokas–Lenells (FL) equation which can be derived from the Kaup–Newell (KN) linear scattering problem. By constructing the Darboux transformation of nonlocal FL equation, we obtain its different kinds of exact solutions including bright/dark solitons, kink solutions, periodic solutions and several types of mixed soliton solutions. It is shown that the solutions of nonlocal FL equation possess different properties from the normal FL equation.     

Qualitative properties of solutions of an integral equation associated with the Benjamin–Ono–Zakharov–Kuznetsov operator

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions.     

Existence of weak solutions of the g-Kelvin–Voight equation

The 2D $g$-Navier–Stokes equations have the form $\frac{?u}{?t}?\nu \Delta u+u.?u+?p=f\text{in }\Omega$ with the continuity equation $?.\left(gu\right)=0\text{in }\Omega$ in a bounded domain $\Omega ?{\mathbb{R}}^2$ where $g=g\left(x_{1,}{x}_2\right)$ is a smooth real valued function defined on $\Omega$. We use the method described by Roh [J. Roh, $g$-Navier Stokes equations, Ph.D. Thesis, University of Minnesota, 2001] for the derivation of $g$-Kelvin–Voight equations represented by $\frac{?u}{?t}?\nu {\Delta }_{g}u+\frac{\nu }{g}(?g??)u?\alpha {\Delta }_g{u}_t+\frac{\alpha }{g}(?g??)u_t+u??u+?p=f\left(x\right)\text{ in }\Omega$$?.\left(gu\right)=0\text{in }\Omega$ We discuss the existence and uniqueness of weak solutions of $g$-Kelvin–Voight equations by the use of the well known Feado–Galerkin method.     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Primitive solutions of the Korteweg–de Vries equation

Theoretical and Mathematical Physics - We survey recent results connected with constructing a new family of solutions of the Korteweg-de Vries equation, which we call primitive solutions. These...     

Exact solutions of the generalized Sinh–Gordon equation

In this paper, we successfully derive a new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

Numerical conservative solutions of the Hunter–Saxton equation

BIT Numerical Mathematics - In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear...     

Asymptotic solutions of an extended Korteweg–de Vries equation describing solitary waves with weak or strong downstream decay in turbulent open-channel flow

Solitary waves in turbulent open-channel flow over a strip of enlarged bottom roughness are considered. Slightly supercritical, fully developed flow far upstream and downstream is assumed. An analysis by Schneider [1] yielded an extended Korteweg–de Vries equation that is solved in the present paper by matched asymptotic expansions up to second order. Solutions for the surface elevation and the location of the wave crest are given. It is shown that a minimum enlargement of the bottom roughness is required for the stationary wave to exist. The surface elevation exhibits either a weak downstream decay, i.e. a shallow ‘tail’, or, if the enlargement of the bottom roughness equals an eigenvalue, the strong decay of the classical soliton solution. Numerical results available for the latter case [2] are in very good agreement with the present analytical solution. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasideterminant solutions of the extended noncommutative Kadomtsev–Petviashvili hierarchy

We construct a nonauto Darboux transformation for the extended noncommutative Kadomtsev–Petviashvili (ncKP) hierarchy and consequently derive its quasi-Wronskian solution. We also obtain the quasi-Wronskian solution of the ncKP equation with self-consistent sources (ncKPESCS) as a by-product. Finally, we use the direct verification method to prove the quasi-Wronskian solution of the ncKPESCS.     

Blow-up of smooth solutions of the Korteweg–de Vries equation

The paper is devoted both to some initial–boundary value problems and to the Cauchy problem for the KdV equation.