Casoratian Solutions Research Articles

Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation

Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations

A bilinearisation-reduction approach is described for finding solutions for nonlocal integrable systems and is illustrated with nonlocal discrete nonlinear Schrödinger equations. In this approach we first bilinearise the coupled system before reduction and derive its double Casoratian solutions; then we impose reduction on double Casoratians so that they coincide with the nonlocal reduction on potentials. Double Caosratian solutions of the classical and nonlocal (reverse space, reverse time and reverse space-time) discrete nonlinear Schrödinger equations are presented.

N-soliton-like and double Casoratian solutions of a nonisospectral Ablowitz–Ladik equation

A bilinear form of a nonisospectral differential-difference equation related to the Ablowitz–Ladik (AL) spectral problem is derived by a transformation of dependent variables. Exact solutions to the resulting bilinear equation are found. The [Formula: see text]-soliton-like solutions and the double Casoratian solutions are derived by means of Hirota’s direct method and the double Casoratian technique, respectively. Moreover, the connection between those two classes of solutions is explored.

Rational-Like Solutions of a Differential-Difference Equation Related to Ablowitz-Ladik Spectral Problem

By using the Casoratian technique, we construct the double Casoratian solutions whose entries satisfy matrix equation of a differential-difference equation related to the Ablowitz-Ladik spectral problem. Soliton solutions and rational-like solutions are obtained from taking special cases in general solutions.

Solutions and Continuum Limits of Two Semi-Discrete Lattice Potential Korteweg-de Vries Equations* *Supported by National Natural Science Foundation of China under Grant No. 11371241 and National Natural Science Foundation of Shanghai under Grant No. 13ZR1416700

We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.

Solutions to the non-autonomous ABS lattice equations: Casoratians and bilinearization

In this paper non-autonomous H1, H2, H3 δ and Q1 δ equations in the Adler-Bobenko-Suris (ABS) list are bilinearized. Their solutions are derived in Casoratian form. We also list out some Casoratian shift formulae which are used to verify Casoratian solutions.

Generalized double Casoratian solutions to the four-potential isospectral Ablowitz–Ladik equation

Generalized solutions in double Casoratian form of the four-potential isospectral Ablowitz–Ladik equation possessing bilinear form are derived through a matrix method for constructing double Casoratian entries. A novel class of explicit solutions, such as soliton, rational-like, Matveev, Complexiton and interaction solutions, are obtained by letting the general matrix be some special cases. Interestingly, a periodic solution is deduced from the Complexiton solution.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.

Casoratian Solutions to Toda Lattice Via Its Bäcklund Transformation

Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bäcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bäcklund transformation, we can give transformations among many kinds of solutions.

Casoratian Solutions to a Non-isospectral Differential-Difference Kadomtsev–Petviashvilli Equation

In this paper, a non-isospectral differential-difference Kadomtsev–Petviashvilli equation (n-DΔKPE) is presented. Then, the Casoratian solutions of the n-DΔKPE are obtained by generalizing Casoratian conditions of the non-isospectral DΔKPE, single-soliton solution is also derived by using Hiorta's method.