Binary Bell polynomials approach to the integrability of nonisospectral and variable-coefficient nonlinear equations

Recently, Lembert, Gilson et al proposed a lucid and systematic approach to obtain bilinear B\"{a}cklund transformations and Lax pairs for constant-coefficient soliton equations based on the use of binary Bell polynomials. In this paper, we would like to further develop this method with new applications. We extend this method to systematically investigate complete integrability of nonisospectral and variable-coefficient equations. In addiction, a method is described for deriving infinite conservation laws of nonlinear evolution equations based on the use of binary Bell polynomials. All conserved density and flux are given by explicit recursion formulas. By taking variable-coefficient KdV and KP equations as illustrative examples, their bilinear formulism, bilinear B\"{a}cklund transformations, Lax pairs, Darboux covariant Lax pairs and conservation laws are obtained in a quick and natural manner. In conclusion, though the coefficient functions have influences on a variable-coefficient nonlinear equation, under certain constrains the equation turn out to be also completely integrable, which leads us to a canonical interpretation of their $N$-soliton solutions in theory.Comment: 39 page

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.Comment: Key Words: Incompressible Euler equations, the Clarkson-Kruskal method, similarity reductions, nonlinear exact solution

The GLM representation of the global relation for the two-component nonlinear Schr\"odinger equation on the interval

In a previous work, we show that the solution of the initial-boundary value problem for the two-component nonlinear Schr\"odinger equation on the finite interval can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we use a Gelfand-Levitan-Marchenko representation to derive an expression for the generalized Dirichlet-to-Neumann map to characterize the unknown boundary values in physical domain, which is different from the approach, in fact it analyzed the global relation in spectral domain, used in the previous work. And, we can show that these two representations are equivalent.Comment: arXiv admin note: text overlap with arXiv:1109.4937, arXiv:1008.5379 by other author

Long-time asymptotic behavior for the complex short pulse equation

In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the solution of $2\times 2$-matrix Riemann-Hilbert problem, from which an implicit one-soliton solution is obtained on the discrete spectrum. While on the continuous spectrum we further establish the explicit long-time asymptotic behavior of the non-soliton solution by using Deift-Zhou nonlinear steepest descent method.Comment: 32 page

A Riemann-Hilbert approach to the Harry-Dym equation on the line

In this paper, we consider the Harry-Dym equation on the line with decaying initial value. The Fokas unified method is used to construct the solution of the Harry-Dym equation via a $2 \times 2$ matrix Riemann Hilbert problem in the complex plane. Further, one-cups soltion solution is expressed in terms of solutions of the Riemann Hilbert problem.Comment: 17 page

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.Comment: 46 pages. accepted for publication J Nonl Math Phys, 2014. arXiv admin note: substantial text overlap with arXiv:1406.6153, arXiv:1207.0574, arXiv:1205.6062; and with arXiv:nlin/0105021 by other author

The Ostrovsky-Vakhnenko equation on the half-line: a Riemann-Hilbert approach

We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a 3\times 3 vector Riemann-Hilbert problem formulated in the complex plane of a spectral parameter z.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1204.5252, arXiv:1311.0495 by other author

On Negative Order KdV Equations

In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough bilinear B\"{a}cklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent to the Camassa-Holm equation through some hodograph transformations, but also closely related to the Ermakov-Pinney systems, and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The 1-kink wave solution is expressed in the form of $tanh$ while the 1-bell soliton is in the form of $sech$, and both forms are very standard. The collisions of 2-kink-wave and 2-bell-soliton solutions, are analyzed in details, and this singular interaction is a big difference from the regular KdV equation. Multi-dimensional binary Bell polynomials are employed to find bilinear formulation and B\"{a}cklund transformations, which produce $N$-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasi-periodic wave solutions of the NKdV equations. Furthermore, the relations between quasi-periodic wave solutions and soliton solutions are clearly described. Finally, we show the quasi-periodic wave solution convergent to the soliton solution under some limit conditions.Comment: 61 pages, 4 figure

Long-time asymptotic for the derivative nonlinear Schr\"odinger equation with decaying initial value

We present a new Riemann-Hilbert problem formalism for the initial value problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line. We show that the solution of this initial value problem can be obtained from the solution of some associated Riemann-Hilbert problem. This new Riemann-Hilbert problem for the DNLS equation will lead us to use nonlinear steepest-descent/stationary phase method or Deift-Zhou method to derive the long-time asymptotic for the DNLS equation on the line.Comment: 41 page

Reality problems for the Algebro-Geometric Solutions of Fokas-Lenell hierarchy

In a previous study, we obtained the algebro-geometric solutions and $n$-dark solitons of Forkas-Lenells (FL) hierarchy using algebro-geometric method. In this paper, we construct physically relevant classes of solutions for FL hierarchy by studying the reality conditions for $q=\pm \bar{r}$ based on the idea of Vinikov's homological basis.Comment: 27 page