Infinitely many Lax pairs and symmetry constraints of the KP equation

Abstract

Starting from a known Lax pair, one can get some infinitely many coupled Lax pairs, infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways. In this paper, taking the well known Kadomtsev–Petviashvili (KP) equation as a special example, we show that infinitely many nonhomogeneous linear Lax pairs can be obtained by using infinitely many symmetries, differentiating the spectral functions with respect to the inner parameters. Using a known Lax pair and the Darboux transformations (DT), infinitely many nonhomogeneous nonlinear Lax pairs can also be obtained. By means of the infinitely many Lax pairs, DT and the conformal invariance of the Schwartz form of the KP equation, infinitely many new nonlocal symmetries can be obtained naturally. Infinitely many integrable models in (1+1)-dimensions, (2+1)-dimensions, (3+1)-dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the KP equation related to the infinitely many Lax pairs.

Non-Traveling Wave Solutions for the (2+1)-Dimensional Breaking Soliton System

“…3 1 2 3 , , , , , k k k r r r Substituting (30) into (23) leads to a soliton structure for the system (1)- (2). Figures 1(a)-(d) are the evolution plots of the solution (23) …”

“…Modern soliton theory is widely applied in many natural sciences [1][2][3][4] such as chemistry, biology, mathematics, communication, and particularly in almost all branches of physics like fluid dynamics, plasma physics, field theory, optics, and condensed matter physics, etc. [5][6][7][8].…”

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

“…both of which are integrable soliton equations, possessing, respectively, infiniteand finite-dimensional Lie symmetry pseudo-groups, [1,5,6,7,18,19,20]. Let us recall how the classical Lie symmetry method, [22], works in the context of the KP equation.…”

Bell-Like and Peak-Like Loop Solitons in (2+1)-Dimensional Dispersive Long Water-Wave System

“…Some special similarity solutions are also given in [11] by using symmetry algebra and classical theoretical analysis. The more general symmetry algebra, W ∞ , is given in [12]. In [13], Lou gave nine types of two-dimensional similarity reductions and thirteen types of ordinary differential equation reductions.…”