In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and some special types of high dimensional models which possess Schwarz variants may have a second order Lax pair. The explicit Lax pairs for (1+1)-dimensional Kor...

It is well-known that many dynamical problems in physics and other natural fields are usually characterized by nonlinear evolution partial differential equations known as governing equations. In soliton theory, searching for an analytical exact solution to a nonlinear physical system has long been an important and interesting topic both for physicists and mathematicians since much physical info...

In this paper, we are concerned with the integrable decomposition for (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. By utilizing recursive relations and symmetric reductions, propose that (n2 − n1 + 1)-flow of AKNS hierarchy can be decomposed into corresponding n1-flow n2-flow (1+1)-dimensional hierarchy, both in coupled reduced cases. As an appropriate generalization, decompos...

Nonlocal reductions of a nonisospectral (2+1)-dimensional breaking soliton Ablowitz-Kaup-Newell-Segur equation are discussed on the base double Wronskian reduction technique. Various types solutions, including solutions and Jordan-block for resulting nonlocal equations derived. Dynamics these obtained analyzed illustrated.

We discuss the relation between matrix models and the Seiberg–Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean one-matrix model in the planar (large N) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham τ-function. The corresponding Whi...

Hammack & Segur [1] conducted a series of surface water-wave experiments in which the evolution of long waves of depression was measured and studied. This present work compares time series from these experiments with predictions from numerical simulations of the KdV, Serre, and several unidirectional and bidirectional Whitham-type equations. These comparisons show that accurate predictions come...