THE DIRECT METHOD IN SOLITON THEORY
CONTENTS BILINEARIZATION OF SOLITON EQUATIONS SOLITARY WAVES AND SOLITONS NONLINEARITY AND DISPERSION SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS LINEARIZATION OF NONLINEAR DIFFERENTIAL EQUATIONS ESSENTIALS OF THE DIRECT METHOD THE D-OPERATOR, A NEW DIFFERENTIAL OPERATOR BILINEARIZATION OF NONLINEAR DIFFERENTIAL EQUATIONS SOLUTIONS OF BILINEAR EQUATIONS TRANSFORMATION FROM BILINEAR TO NONLINEAR FORM DETERMINANTS AND PFAF?ANS INTRODUCTION PFAF?ANS EXTERIOR ALGEBRA EXPRESSIONS FOR GENERAL DETERMINANTS AND WRONSKIANS LAPLACE EXPANSIONS OF DETERMINANTS AND PL¨UCKER RELATIONS JACOBI IDENTITIES FOR DETERMINANTS SPECIAL DETERMINANTS PFAF?AN IDENTITIES EXPANSION FORMULAE FOR THE PFAF?AN (A1, A2, 1, 2, , 2N) 2 10 ADDITION FORMULAE FOR PFAF?ANS 2 11 DERIVATIVE FORMULAE FOR PFAF?ANS STRUCTURE OF SOLITON EQUATIONS THE KP EQUATION THE BKP EQUATION: PFAF?AN SOLUTIONS THE COUPLED KP EQUATION THE TWO-DIMENSIONAL TODA LATTICE EQUATION THE TWO-DIMENSIONAL TODA MOLECULE EQUATION B¨ACKLUND TRANSFORMATIONS WHAT IS A B¨ACKLUND TRANSFORMATION? B¨ACKLUND TRANSFORMATIONS FOR KDV-TYPE BILINEAR EQUATIONS THE B¨ACKLUND TRANSFORMATION FOR THE KP EQUATION THE B¨ACKLUND TRANSFORMATION FOR THE BKP EQUATION THE SOLUTION OF THE MODI?ED BKP EQUATION THE B¨ACKLUND TRANSFORMATION FOR THE TWO-DIMENSIONAL TODA EQUATION SOLUTION OF THE TWO-DIMENSIONAL MODI?ED TODA EQUATION BILINEARIZATION OF SOLITON EQUATIONS ONE-SOLITON SOLUTION 1 1 SOLITARY WAVES AND SOLITONS THE WORD ‘WAVE’ NORMALLY MAKES US THINK OF A WAVE TRAIN AS SHOWN IN FIGURE 1 1 HOWEVER, WHEN SUR?NG OFF A GENTLY SLOPING BEACH, WE OFTEN MAKE USE OF A SOLITARY WAVE (SEE FIGURE 1 2) A SOLITON IS A TYPE OF SOLITARY WAVE WHICH MAINTAINS ITS IDENTITY AFTER IT COLLIDES WITH ANOTHER WAVE OF THE SAME KIND LET US ?RST STUDY WAVE EQUATIONS WHICH DESCRIBE SOLITARY WAVES A WAVE EQUATION HAVING SOLITON SOLUTIONS HAS BOTH NONLINEARITY AND DISPER- SION BEFORE STUDYING HOW TO SOLVE SUCH A WAVE EQUATION, WE WILL INVESTIGATE ? ?X F (X, T) = 0 (1 2) 1 2 NONLINEARITY AND DISPERSION F (X, T) = 0 (1 3) IF WE SUPPOSE THAT IT HAS A PLANE WAVE SOLUTION F (X, T) ?EXP[I(?T ?KX)], THEN THE DISPERSION RELATION IS GIVEN BY ? = V0K ??K3, WHICH IS NONLINEAR WITH RESPECT TO K HENCE, THE PHASE VELOCITY IS DIFFERENT FROM THAT IN SECTION 1 2 1 AND IS GIVEN BY ? K = V0 ??K2, BILINEARIZATION OF SOLITON EQUATIONS WHICH DEPENDS ON A WAVE NUMBER K ON THE OTHER HAND, ITS GROUP VELOCITY IS GIVEN BY ?? ?K = V0 ?3?K2 WE REMARK THAT, IF ? > 0, BOTH VELOCITIES ARE LESS THAN V0 SINCE THE VELOCITY OF EACH OF THE PLANE WAVES WHICH MAKE UP AN INITIAL WAVE VARY WITH K, THE WAVE SPREADS OUT AS IT TRAVELS THIS SHOWS THAT LINEAR DISPERSIVE WAVES DO NOT PRESERVE THEIR ORIGINAL SHAPE THE TWO EXAMPLES WE HAVE DISCUSSED SO FAR ARE BOTH LINEAR DIFFERENTIAL EQUA- TIONS NEXT WE CONSIDER THE IN?UENCE OF NONLINEARITY 1 2 3 NONLINEAR NONDISPERSIVE WAVES WE CONSIDER, AS THE SIMPLEST EXAMPLE, THE NONDISPERSIVE WAVE EQUATION, ? ?T + V( F ) ? ?X F (X, T) = 0, (1 4) WHERE V( F ) = V0 + ?F M THIS EQUATION IS A NONLINEAR WAVE EQUATION IN WHICH THE SPEED V( F ) DEPENDS ON THE AMPLITUDE F EQUATION (1 4) HAS THE FORMAL SOLUTION F (X, T) = F (X ?V( F )T), AND IF V = V( F ) IS AN INCREASING FUNCTION IN F , THIS FORMULA TELLS US THAT A WAVE TRAVELS FASTER AS ITS AMPLITUDE INCREASES THEREFORE, AS ONE CAN SEE FROM FIGURE 1 3, THE WAVE STEEPENS AND THEN BREAKS PHYSICALLY SPEAKING, HOWEVER, BEFORE THE WAVE BREAKS, ITS GRADIENT |?F/?X| ?1 WHEN THIS HAPPENS, (1 4) BECOMES MEANINGLESS AND MUST BE REPLACED BY THE DIFFERENTIAL EQUATION (1 5) PRESENTED IN SECTION 1 2 4 T = T = A X FIGURE 1 3 STEEPENING OF A SOLITARY WAVE A WAVE WHICH IS SYMMETRICAL AT T = STEEPENS AND BREAKS BECAUSE OF THE DEPENDENCE OF THE WAVE SPEED ON ITS AMPLITUDE 1 2 NONLINEARITY AND DISPERSION F (X, T) = (1 5) POSSESSES A PULSE-LIKE WAVE SOLUTION WHICH TRAVELS WITH UNCHANGING SHAPE; A SOLITARY WAVE SOLUTION BEFORE ?NDING THIS SOLUTION BY MATHEMATICAL MEANS, LET US LOOK INTO THE PHYSICAL REASONS FOR THE EXISTENCE OF SUCH A SOLUTION WE HAVE SEEN IN SECTIONS 1 2 2 AND 1 2 3 THAT NEITHER A LINEAR DISPERSIVE SOLITARY WAVE NOR A NONLINEAR NONDISPERSIVE SOLITARY WAVE CAN EXIST WHY THEN CAN A SOLITARY WAVE SOLUTION EXIST FOR A WAVE EQUATION WHICH HAS BOTH NONLINEARITY AND DISPERSION? ASSUMING THAT THE SOLITARY WAVE SHOWN IN FIGURE 1 4 EXISTS, LET US INVES- TIGATE WHETHER THIS PULSE-LIKE WAVE CAN TRAVEL WITH UNCHANGING SHAPE TO THIS END IT IS NECESSARY, AT LEAST, THAT THE VELOCITIES AT THE TOP AND BASE OF THE WAVE HAVE THE SAME VALUE V IN ORDER TO INVESTIGATE FURTHER, WE INTRODUCE A NEW SPACE COORDINATE ? = PX ?T, WHERE V = /P AND P IS A FREE PARAMETER THE PA- RAMETER P IS SUCH THAT, AS IT INCREASES, THE PULSE BECOMES SHARPER IT IS CONVE- NIENT TO INTRODUCE ? TO DESCRIBE A WAVE WHICH TRAVELS AT A CONSTANT SPEED V FOR T = 0, ? IS PROPORTIONAL TO X (? = PX), AND, FOR T ?= 0, ? = P(X ?VT), BEING PROPORTIONAL TO X ?VT, TRAVELS AT A SPEED V IF THE MAXIMUM OF THE WAVE AMPLITUDE F IS A, OCCURRING AT ? = 0, THEN IN A NEIGHBOURHOOD OF THIS POINT WE HAVE F ?A(1 ?CONSTANT × ?2), BECAUSE HERE THE HEIGHT F CAN BE APPROXIMATED BY A QUADRATIC EXPRESSION IN ? FROM THIS EQUATION, WE HAVE ?3 F/?X3 ?0 AND THEREFORE, IN THE NEIGHBOURHOOD F ? BASE BASE TOP FIGURE 1 4 SPLITTING A SOLITARY WAVE INTO ITS TOP AND BASE