Interaction of Two Pulsatory Waves of the Korteweg-de Vries Equation in a Zigzag Hyperbolic Structure

A new exact solution for nonlinear interaction of two pulsatory waves of 
the Korteweg-de Vries (KdV) equation is computed by decomposition in an 
invariant zigzag hyperbolic tangent (ZHT) structure. A computational algorithm 
is developed by experimental programming with lists of equations and 
expressions. The structural solution is proved by theoretical programming with 
symbolic general terms. Convergence, tolerance, and summation of the ZHT 
structural approximation are discussed. When a reference level vanishes, the 
two-wave solution is reduced to the two-soliton solution of the KdV equation.

The structure of this paper is as follows.In section 2, experimental computation with lists of equations and expression is used to develop a computational algorithm in Maple™ and explore the convergence of the ZHT structural approximation.Theoretical computation with symbolic general terms is utilized in Section 3 to develop differentiation and multiplication formulas for the ZHT structure, show its invariance, and consider summation, convergence and tolerance of the structural approximation.The two-wave solution is visualized and compared with the two-soliton solution in Section 4, which is followed by a summary of main results in Section 5.

Experiment on
for nonlinear interaction of two pulsatory waves.Construct the structural solution in the ZHT structure of algebraic order 3 M = , which is used to illustrate a computational algorithm, where h is a reference level, which becomes the one-soliton solution ( ) ( ) ] for 0 h = .Consider an instance of (2) with , and 1 4 ν = . The ZHT structure then is ,

A ta A ta A ta Tb
A ta A ta Tb , where structural coefficients Secondly, computation of a temporal derivative of differential order 1 N = also returns the ZHT structure of algebraic order 5

T ta T ta T T ta T ta Tb t T ta T ta T ta Tb T ta T ta T ta Tb T ta T ta Tb
with a general term , m n m n T ta Tb and the following structural coefficients , Finally, a spatial derivative of order 3 N = again produces the ZHT structure of order     ,

Experimental Solution of an Algebraic KdV Problem
Consider an experimental solution for algebraic order 4, M = which is a smallest order required to avoid degeneration of the subsequent computational algorithm.Substitution of temporal derivative (7), product (11) of the two-wave solution and the first spatial derivative, third spatial derivative (9), and collection of structural coefficients reduce differential KdV Equation ( 1) to an experimental algebraic KdV equation ) )  are, respectively, In agreement with one-wave solution (3), structural coefficients of the two-wave solution are initialized by For the experimental solution, Solving Equations ( 14) and ( 15) with respect to Solving first and fifth equations of ( 19) with respect to ( ) ( ) Substitution of (20) in second, third, and fourth equations of system (19) reduces them to identities.
− , the binomial system also has five equations Substitution of the computed structural coefficients in the left-hand-side part of ( 13) returns an experimental remainder of the ZHT structural approximation ( ) Rate of convergence of the ZHT structural approximation is examined in Table 1, where a tolerance max , and a CPU time τ are given versus algebraic order M for various reference levels h .Table 1 was com- puted on a workstation in Maple 17.02 by using a six-core AMD-6300 processor with frequency 3.50 GHz and RAM 12.0 GB.The CPU time depends mainly on order of approximation M .Tolerance significantly depends upon M and h through interaction parameter q .For propagation celerities The experimental solutions of Section 2 were computed by experimental programming with lists of equations and expressions in the virtual environment of a global variable Eqe with 4 procedures of 185 code lines in total.

Formulation of the Theoretical Problem in the ZHT Structure
Compute theoretically a structural solution of (1) for nonlinear interaction of two pulsatory waves.Construct the structural solution in the ZHT structure of algebraic order M and differential order N ( ) is a structural coefficient, M and N are symbolic limits of sum- mation.Other variables and parameters are the same as in ( 2), but instead of experimental instances of section 2 they receive symbolic values to compute theoretically a general term of the structural solution.
In agreement with (26) and ( 2), a two-wave solution is constructed with algebraic order M and differential order 0 N = using a generalized Einstein notation for summation, which is extended for exponents, as follows:  16) is invoked, the two-wave solution is reduced to one-wave solution (3).

Theoretical Differentiation of the Invariant ZHT Structure
Primarily, computation a first spatial derivative of a binomial term n m ta Tb yields ( ) , .
Here, indices n and m , which are equal to powers of ta and Tb in the binomial derivative, define names of the matrix functions.Indices i and j which are equal to powers of ta and Tb in the binomial term i j ta Tb , determine definitions of the matrix functions.Similarly, computation of a first temporal derivative of the binomial term n m ta Tb gives ( ) , Thus, spatial and temporal derivatives (28) and (30) have the same structure but vary in differential binomial coefficients (29) and (31), respectively.
Increase in order of differentiation produces a two-dimensional (2d) differentiation cascade, which is shown in Figure 1.The one-dimensional (1d) differentiation cascade of invariant hyperbolic-secant structures is asymmetrical as the cascade spreads only towards higher powers [18].To the contrary of the 1d differentiation cascade, the 2d differentiation cascade spreads in symmetrical square waves, which resemble circular waves on the water surface generated by a point source.Similar to the 1d differentiation of the invariant trigonometric, hyperbolic, and elliptic structures [18], the 2d differentiation of an even order preserves structure of binomial derivatives and the 2d differentiation of an odd order converts structure of binomial derivatives to complementary ones.
Finally, computation a third spatial derivative of the binomial term ta Tb returns a binomial derivative of the following structure:  (   ) ( ) Similarly, a first temporal derivative of the two wave solution again yields the invariant ZHT structure of algebraic order M N + and differential order 1 N = ( ) where structural coefficients , and truncation conditions are given by (36).Truncation conditions for the first temporal derivative become , Finally, computation of a third spatial derivative of the two-wave solution produces the invariant ZHT structure of algebraic order M N + and differential order 3 N = ( ) where structural coefficients , , , , and truncation conditions are set by (36).Truncation conditions for the third spatial derivative are 0 for 2; 0, 0 for 3.
Equations ( 34), (38), and (41) show that the ZHT structure is invariant with respect to differentiation of orders 1 N = and 3 N = , which only modify algebraic orders and structural coefficients.The zigzag structure of twowave solution (27) and its derivatives (34) and (38) together with product of ( 27) and (34) are shown in Figure 2 in a virtual space of computational indices n and m of structural coefficients  ta Tb .Differentiation increases the width of the invariant ZHT structure and the effect of multiplication is similar to that of differentiation.

Theoretical Multiplication of the Invariant ZHT Structures
Continuation of spatial derivatives (34) and (38) in the invariant ZHT structure to any differential order N in the generalized Einstein notation gives where 0,1, , The spatial derivative of order 1 N − then becomes Differentiation of (45) with respect to x and reduction of all terms to a general term by substitutions , where structural coefficients of (44) are connected with structural coefficients of (46) by a recurrent relation .
Thus, the invariance of the ZHT structure with respect to spatial differentiation of any order is proved by induction.
Set up two spatial derivatives of the two-wave solution in the invariant ZHT structures of algebraic order M and differential orders 1 N and 2 N with structural coefficients for the first structure, for the second structure.Product of (48) with a binomial substitution , m n k l = = and (49) with a binomial substitution , where 0,1, , . Thus, the ZHT structure is also invariant with respect to multiplication since a general term of the product is the ZHT structure of algebraic order 1 2 2M N N + + and differentiation order 1 2 , where the structural coefficients are .
Summation of a general term of the product of two-wave solution (27) and first spatial derivative (34) by ( 51)-( 52) with 1 2 0, 1, 0,1, , 3  Structure of the two-wave solution is reduced to that of a spatial derivative by substitution Truncation conditions for product of the two-wave solution and the first spatial derivative become , Solving first and fifth equations of (62) with respect to ( ) ( ) where last two terms are obtained by induction.In agreement with the experimental solution of section 2.
Solving first equation of (64) with respect to ( ) ( ) where a structural coefficient
Conversion of (72) through two singular hyperbolic functions  When 0 h = , the two-wave solution reduces to the conventional form [6] of the two-soliton solution in two singular hyperbolic (75) converges slower than the ZHT structural remainder (69) because of the infinite limit of summation and truncation conditions (40), (56), and (43).So, the method of decomposition in the invariant ZHT structure is more robust than the method of expansion in the Taylor series.
Interaction of two pulsatory waves is visualized by spatiotemporal plots in Figure 3 for positive and negative reference levels h .Negative values of h considerably increase amplitudes and decrease dispersions of pulsatory waves compared with those of solitons with 0 h = , because pulsatory waves propagate on a more shallow water in this case.The effect of positive values of h is opposite and results in decrease of amplitudes and increase of dispersions.Similar to interaction of two solitons, interaction of two pulsatory waves is also conservative and preserves one-wave solutions before and after a nonlinear interaction at the moment of merging.Animations of the two-wave solution show that the merging process may be considered as a flow of a faster fluid of the first pulsatory wave into the second pulsatory wave with a slower fluid.

Conclusions
The analytical methods of solving PDEs by undetermined coefficients and series expansions are generalized by the computational method of solving nonlinear PDEs by decomposition in the invariant ZHT structure.The computational algorithm is developed by experimental computing using lists of equations and expressions implemented in four procedures of 185 code lines in total.Afterwards, the computational method is proved by theoretical computing with symbolic general terms implemented in 16 procedures with 923 code lines in total.The invariance of the ZHT structure with respect to differentiation and multiplication is shown by using 2d differentiation cascade of binomial structures and mathematical induction.Contrary to the asymmetric differentiation cascade in one dimension [18], the 2d differentiation cascade spreads in symmetric square waves.Compared with the 2d series expansion, the ZHT structure considerably saves computational resources and simplifies results since it implies a low-order polynomial in one dimension and a series expansion in another dimension.The invariance of the ZHT structure enables other computational applications in nonlinear PDEs with solutions approaching a constantor vanishing at infinity.
The ZHT structural approximation and remainder are computed theoretically to any algebraic order.Summation of the two-wave solution in the invariant ZHT structures is implemented and presented both through regular and singular hyperbolic functions.When a reference level vanishes, the two-wave solution is reduced to the twosoliton solution.Negative reference levels considerably increase amplitudes and decrease dispersions of pulsatory waves compared with those for solitons with a vanishing reference level.The effects of positive reference levels are opposite, i.e. amplitudes of pulsatory waves are decreased and dispersions are increased.

Interaction of Two Pulsatory Waves in the ZHT Structure 2 . 1 .
Formulation of an Experimental Problem in the ZHT StructureExplore experimentally a structural solution of the canonical form of the KdV equation are propagation variables, U and V are celerities, a and b are initial locations of first and second pulsatory waves, show a uniform convergence of the two-wave solution in the ZHT structure.

2 ,
+  , derivatives of which make a contribution to the general term of a zigzag derivative through binomial derivatives (28), (30), and (32).Substitute then the binomial derivatives of terms of a structural coefficient of the general term proportional to m Tb .In agreement with (5), a first spatial derivative of the two-wave solution is the invariant ZHT structure of algebraic order M N + and differential order 1 N = ( by(27).Conditions (36) result in truncation conditions for the first spatial derivative:
coincides with a virtual space of computational powers n and m of n m (40), (56), and (43) are invoked.The theoretical solutions of Section 3 in the invariant ZHT structure were computed using theoretical programming methods with symbolic general terms by the generalized Einstein notation in the virtual environment of a global variable Eqt with 16 procedures of 923 code lines in total.The theoretical formulas for two-wave solution (68), first spatial derivative (34)-(36), first temporal derivative (38)-(40), third spatial derivative (41)-(43), product of the two-wave solution and the first spatial derivative (53)-(56), and structural remainder (69)-(70) were justified by the correspondent experimental solutions for algebraic order 40 M = .
Y ν , three parameters , , h µ ν and two variables X and Y gives

Table 1 .
Convergence of the experimental solutions in the ZHT structure.