Soliton Solutions of a Coupled Derivative Modified KdV Equations

Abstract

The multisoliton solution to a system of nonlinear partial differential equations which is ex-, for i = 1, 2, · · · , N where the coefficients C jk are arbitrary constants, is expressed by the pfaffians.

New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

“…Considering a special case with a proper reduction, we solve the Gel'fand-Levitan-Marchenko equations to obtain the one-soliton solutions of the Sp(m)-invariant systems under the decaying boundary conditions. It should be emphasized that the soliton solutions of (1.4) obtained in this manner are indeed more general than the previously known solutions [12,13]. Although the accuracy of these one-soliton solutions can easily be verified by direct substitutions, unlike the case of the NLS equation, it is not easy to obtain such solutions directly without resorting to the inverse scattering method or other sophisticated methods in soliton theory.…”

“…For the reduced system to assume a concise form without the imaginary unit and fractions, we consider the following vector reduction (l 1 = 1, l 2 = M): 6) which also implies the simple relation qr = 0. System (2.2) is then reduced to a system of coupled derivative mKdV equations [12]:…”

“…However, in contrast to the U(m)-invariant and O(m)-invariant systems, little research has been conducted on integrable systems having invariance with respect to the symplectic group Sp(m); this refers to the group of 2m × 2m real/complex symplectic matrices, Sp(m, R) or Sp(m, C), in accordance with the attribute of the dependent variables. To the best of the author's knowledge, the only example of a 2m-component vector nonlinear PDE with Sp(m) invariance is the system of coupled derivative mKdV equations studied using the bilinear method by Iwao and Hirota [12], as well as its higher symmetries. It should be noted that system (1.4) is a natural multicomponent generalization of the two-component system [(1.4) with m = 1] derived within the framework of the Sato theory by Loris and Willox [13,14].…”

On travelling wave solutions of some nonlinear evolution equations

“…In mathematics, for a nonlinear evolution equation, usually the travelling wave solutions are considered. A number of methods were presented such as Hirota's bilinear methods [1], the inverse scattering transform [2], painlevé expansions [3], the real exponential method [4], the homogeneous balance method [5], Darboux transformation [6], Lie group method [7], the trial function method [8], the nonlinear transformation method [9], sine-cosine method [10], tanh-function and extended tanhfunction methods [11][12][13].…”

Soliton equations exhibiting Pfaffian solutions

“…The multisoliton solution to equation (3) is expressed in the same form as that of the coupled modified KdV equation [5]:…”